Al-Ahlia
New Edition
2010
Authors
H. NASSAR
N. BADR
G. KARROUM
A. MOARBES
M. EL ASMAR
K. ATIEH
C. MERHEB
Editors
AL- AHLIA
Composition and page
arrangement
Al-Ahlia Pre-press
Imprimerie
Les imprimeries modernes
AL- AHLIA
This collection : Puissance series is designed to induce attractive
practice in Mathematics and progressive development of the spirit of
research through motivation.
FORWARD
Our main purpose is to make mathematics easily accessible to
students, without putting obstacles that may provoke a negative reaction
from them :
This BOOK has the following objectives :
carry out operations with large numbers
work with fractions and decimals
carry on the study of statistics
measure time, mass, length and capacity
usage of calculators
multiply and divide decimals
find particular quadrilaterals and their diagonals
use the protractor to measure angles
determine the distance between two parallel lines
calculate areas
get familiar with usual solid forms.
We hope that this work will contribute to up-grading the teaching of
Mathematics and we will be pleased to receive all suggestions or criticism
that may help in improving this series.
The authors
3
4
1
Natural numbers (revision) - Multiples (revision)
8 - 14
Problems
15 - 18
2
Divisors
19 - 20
3
Divisibility rules by 2, by 5 and by 10
21 - 25
4
Divisibility rule by 3
26 - 27
5
Divisibility rule by 9
28 - 29
6
Remainder of the division by 3 and by 9
30 - 31
7
Divisibility rule by 4
32 - 33
8
Dividing by a three digit number
34 - 35
9
Angles
36 - 38
10 Fractions less than one - Fractions greater than one
39 - 42
11 Decimal numbers (reminder)
43 - 45
12 The thousandths and the ten-thousandths
46 - 47
13 Measuring angles (1)
48 - 49
14 Measuring angles (2)
50 - 52
15 Equal fractions - Simplified fractions
53 - 55
Problems
56
16 Comparing fractions (1) : same denominator
57 - 59
17 Comparing fractions (2) : same numerator
60
18 Comparing fractions
Problems
61 - 62
63
19 Comparing decimals
64 - 67
20 Addition and subtraction of decimals
68 - 73
TABLE OF CONTENTS
Chapters
5
TABLE OF CONTENTS
Chapters
6
Problems
21 The distance between two parallel lines
74 - 75
76 - 81
22 Multiplying by 10 ; 100 or 1,000
82
23 Dividing by 10 ; 100 or 1,000
83 - 85
24 Multiplying decimals - Operator technique
86 - 93
25 Multiplying a number by a multiple of 10, 100 or 1,000
94 - 95
Problems
26 Measure of length (Reminder)
27 Metric system - Measuring capacity
Problems
96
97
98 - 100
101
28 Polygons (revision)
102 - 103
29 Diagonals of a polygon
104 - 105
Problems
106 - 107
30 Addition and subtraction of fractions of same denominator (1)
108 - 109
31 Addition and subtraction of fractions (2)
110 - 113
Problems
114
32 Mixed number
115 - 116
33 Fraction of a number
117 - 120
Problems
121
34 Circle - Diameter of a circle
122 - 123
35 Dividing of two numbers - Appropriate quotient
124 - 125
36 Dividing a number by a number - Decimal quotient
126 - 128
37 Dividing a number by a number - Decimal quotient
129 - 130
(The dividend is less than the divisor)
38 Dividing a decimal by a number
Problems
39 The divisor ending with one or more zeros
40 Dividing a decimal by a decimal
Problems
41 Division - Decimal quotient to the nearest lower or higher
Problems
42 Surfaces and their area
Problems
131
132
133
134 - 135
136 - 137
138 - 139
140
141 - 143
144 - 146
43 Length of a circle (perimeter) - Area of a disc
147 - 152
44 Time - Duration (1)
153 - 154
45 Time - Duration (2)
155
Problems
156 - 158
46 Statistics
159 - 165
47 Homothecy
166 - 169
Problems
170
48 Cube
171 - 172
49 Pyramid
173
50 Pyramid of squared base
174
51 Pyramid of triangular base (tetrahedron)
175
52 Right prism
176
Revision problems
TABLE OF CONTENTS
Chapters
177-181
7
1
Natural numbers (revision)
Multiples (revision)
Activity
Set up and calculate : 999,999,999 + 1
Billions
period
h
t
o
Millions period
Thousands period
h
h
t
o
t
o
Ones
period
h
t
o
+
I get 1,000,000,000 as result.
I read : one billion.
Complete : 1,000,000,000 is the number that comes just after
........................................................
1
Complete with natural numbers.
a) 999,999,997 ; ............................................. ; 999,999,999 ; ............................................. ; 1,000,000,001 .
b) 999,999,995 + 5 = ............................................................................ .
999,999,990 + 10 = ......................................................................... .
2
Calculate.
999,999,900 + 100 = ............................................................................................. .
999,999,000 + 1,000 = ........................................................................................ .
8
999,000,000 + 1,000,000 = ............................................................................ .
3
What does the digit 5 represent in each of the numbers in the chart below ?
billions period
millions period
thousands period
h
o
h
t
o
h
t
o
h
t
o
x
5
1
2
6
2
4
3
0
7
2
y
7
2
5
3
1
2
4
6
0
9
5
1
9
7
0
8
3
3
6
4
0
3
8
3
0
4
4
3
2
0
0
1
z
t
5
t
ones period
x : ...................................................................................................
z : ...................................................................................................
y : ...................................................................................................
t : ....................................................................................................
4
Write in words the number placed in the chart below.
billions period
millions period
thousands period
ones period
h
t
o
h
t
o
h
t
o
h
t
o
3
6
4
5
1
2
8
0
0
0
0
0
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
What is the digit of ten billions ?
.......................................................................................................................................................................................................................................
What is the number of ten billions ?
.......................................................................................................................................................................................................................................
What digit is the hundred billions ?
.......................................................................................................................................................................................................................................
What is the number of hundred billions ?
.......................................................................................................................................................................................................................................
What is the number of millions ?
.......................................................................................................................................................................................................................................
9
5
Complete.
In
In
In
6
508,234,567
there are
2,731,000,000
there are
231,000,000,000 there are
.......................................................
thousands.
.......................................................
ten thousands.
.......................................................
hundred thousands.
.......................................................
millions.
.......................................................
ten millions.
.......................................................
hundred millions.
.......................................................
ten billions.
.......................................................
hundred billions.
.......................................................
billions.
Complete.
The number in digits
1,350,000,000
The number in words
one billion, three hundred, and fifty million.
two billions, one million, and six hundred thousand.
10,005,900,000
32,000,840,000
7
The number 7,421,058,069 is decomposed in this way :
7,421,058,069 = 7,000,000,000 + 400,000,000 + 20,000,000 + 1,000,000 + 50,000
+ 8,000 + 60 + 9
= 7 × 1,000,000,000 + 4 × 100,000,000 + 2 × 10,000,000
+ 1,000,000 + 5 × 10,000 + 8 × 1,000 + 6 × 10 + 9
Using the preceding example, decompose the following number :
40,305,920,700 = ............................................................................................................................................................................
............................................................................................................................................................................
= ............................................................................................................................................................................
............................................................................................................................................................................
10
8
9
Compare the following numbers, using the symbols < , > or = .
729,280 ............... 709,289
564,000 ............... 500,000 + 60,000 + 4,000.
1,692,045 ............... 1,962,045
900,000 + 1,000,000 ............... 900,000 + 800,000.
2,910,364 ............... 2,910,864
300,000 + 7,000,000 ............... 200,000 + 600,000.
400,352,614 ............... 400,342,614
764,000 – 43,200 ............... 764,000 – 53,200.
Calculate without setting up.
600,000 + 57,000 + 400 + 7 = ...................................................................................................................................................
20,000,000 + 700,000 + 9,000 = .............................................................................................................................................
35,000,000,000 + 900,000,000 + 5,000,000 = .........................................................................................................
10
Calculate each of the following numbers.
4 × 1,000,000 + 9 × 100,000 + 3 × 10,000 + 2 × 100 + 4 = ......................................................................
7 × 1,000,000 + 36 × 1,000 + 9 × 100 + 3 × 10 + 1 = .....................................................................................
8 × 1,000,000,000 + 5 × 1,000,000 + 7 × 10,000 + 9 = .................................................................................
11
Calculate.
1,270,985
2,601,780,000
1,104,880,000
89,867
+ 1,725,453,869
– 715,850,000
...................................
..............................................
..........................................
+
12
Set up and calculate (on the copybook).
a) 3,065,082 – 974,395
d) 3,845 + 98,439 + 932,378
b) 7,550,321 – 6,498,176
e) 1,389,374 + 125 + 1,398,798
c) 3,207,999,000 – 116,789,000
f) 986,852,000 + 17,584,000
11
13
14
Set up and calculate (on the copybook).
4,608 × 23
3,807 × 906
27,138 ÷ 6
6,325 divided by 7
2,478 × 94
1,895 × 875
108,544 ÷ 53
17,965 divided by 25
6,759 × 56
5,189 × 460
61,750 ÷ 38
31,291 divided by 91.
a) Write on the number line the multiples of 6 that are less than 50.
0
6
6×0
6×1
12
Write 23 between two consecutive multiples of 6.
................................................................................................................................................................................................................................
b) Write on the number line the multiples of 8 that are less than 50.
8×0
Write 23 between two consecutive multiples of 8.
................................................................................................................................................................................................................................
c) Write on the common multiples of 6 and 8 that are less than 50.
................................................................................................................................................................................................................................
d) What is the least common non-zero multiple of 6 and 8 ? .......................................................
15
a) Write the multiples of 2 that are less than 19.
................................................................................................................................................................................................................................
b) Write the multiples of 3 that are less than 19.
................................................................................................................................................................................................................................
c) What is the least common non-zero multiple of 2 and 3 ? .......................................................
12
16
a) Write the multiples of 5 that are less than 51.
................................................................................................................................................................................................................................
b) Write the multiples of 15 that are less than 51.
................................................................................................................................................................................................................................
c) What is the least common non-zero multiple of 5 and 15 ? ....................................................
d) What do you notice ? ...............................................................................................................................................................
................................................................................................................................................................................................................................
17
Write the common multiples of 4 and 5 included between 7 and 85.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
Find the multiples of 9 included between 400 and 430.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
18
a) Find the smallest multiple of 6 that is greater than 119 : ...........................................................
b) Find the greatest multiple of 6 that is less than 149 : ....................................................................
c) Is the sum of these two multiples a multiple of 6 ? Justify.
................................................................................................................................................................................................................................
d) Is their difference a multiple of 6 ? Justify.
................................................................................................................................................................................................................................
e) Is their product a multiple of 6 ? Justify.
................................................................................................................................................................................................................................
13
19
Complete.
a) The multiples of 6 that are less than 100 are :
................................................................................................................................................................................................................................
b) The multiples of 15 that are less than 100 are :
................................................................................................................................................................................................................................
c) The least common non-zero multiple of 6 and 15 is : .......................................................................
The greatest common multiple of 6 and 15 that is less than 100 is : ...............................
Is the sum of these two multiples a multiple of 6 and 15 ?
Justify.
..........................................................................................................................................................................................................................
..........................................................................................................................................................................................................................
Is it the same for their product ? Justify.
..........................................................................................................................................................................................................................
..........................................................................................................................................................................................................................
20
Observe the following example : 56 × 1,000 < 56,348 < 57 × 1,000.
56,348 is written between two consecutive multiples of 1,000.
Write each of the following numbers between two consecutive multiples of 1,000.
a) 908,751
;
b) 37,005
;
c) 99,914
a) ......................................................................................................................................................
b) ......................................................................................................................................................
c) ......................................................................................................................................................
d) ......................................................................................................................................................
14
;
d) 1,584,732.
Problems (Solve on your copybook)
1
Write each of the following numbers
between two consecutive even numbers.
3,450,601
;
297,413
;
347,007
; 1,999,999.
4
A company bought 13 television
sets for 735,000 L.L. each and
17 computers for 4,725,000 L.L. each.
The company paid with bills of
100,000 L.L.
2
How many bills are needed for this
Write each of the following numbers
purchase ?
between two consecutive odd numbers.
3,860,606
;
3,200,000
1,000,000
;
999,998.
;
5
Paul received 24,500 L.L. from his
mother and 25,500 L.L. from his father.
He bought 10 copybooks for 525 L.L.
each, a pencil case for 10,750 L.L. and
3
In a month, an airplane makes 5 times the
two books for 13,500 L.L. each.
flight BEIRUT-PARIS-BEIRUT, measuring
a) What sum of money did Paul receive ?
6,382 km, and 6 times the flight BEIRUT -
b) How much did he spend ?
ROME - BEIRUT, measuring 4,406 km.
c) How much does he still have ?
What is, in km, the distance covered by
this airplane in a month ?
7
6
Nabil has 496 bills of 5,000 L.L.,
28 bills of 20,000 L.L., 10 bills of
50,000 L.L. and 7 bills of 10,000 L.L.
What amount of money does Nabil
have ?
The monthly salary of an employee is 1,850,000 L.L.
a) What is his annual income ?
b) What is his annual saving, knowing that his monthly expenditure is 310,000 L.L. for rent,
655,000 L.L. for food and 450 000 L.L. for various expenses ?
15
Problems (Solve on your copybook)
8
A merchant bought 12 sheep for 175,500 L.L. each. He sold them all for 2,340,000 L.L.
a) What was the selling price of each sheep ?
b) How much profit did he make ?
9
A cake, a chocolate and a juice cost 8,500 L.L. The cake and the chocolate cost
5,500 L.L. The chocolate and the juice cost 6,500 L.L.
a) Calculate the price of the juce.
b) Calculate the price of the cake and that of the chocolate.
10
A greengrocer bought 350 fruit cases, each weighing 13 kg, for 5,800 L.L. the kg.
Transportation costed him 350,000 L.L.
a) What was the total cost ?
b) The fruit sale helped the greengrocer make a total profit of 10,010,000 L.L.
Calculate the selling price of each case.
11
A man intends to buy a house for 81,230,000 L.L.
He should pay 1,800,000 L.L. for registration.
He has 53,000,000 L.L. and will borrow the rest.
a) How much should he borrow ?
b) This man should settle the borrowed sum in 30 equal payments.
What would be the value of each payment ?
12
A merchant buys 650 m of velvet.
He sells half of it for 28,750 L.L. the meter and the rest for 25,000 L.L. the meter. His
profit is 2,112,500 L.L.
a) Calculate the selling price of the 650 meters.
b) Calculate the purchase price of the meter.
16
Problems (Solve on your copybook)
13
The planets of the solar system.
The indicated numbers show the distances between the planets and the sun.
a) Write these numbers in words.
b) Rewrite them in this order : from the farthest planet to the nearest one to the
sun.
Mars
229,000,000 km
Mercury
57,000,000 km
Uranus
2,878,000,000 km
Saturn
1,425,000,000 km
Jupiter
780,000,000 km
Earth
150,000,000 km
Pluto
5,913,000,000 km
Venus
108,000,000 km
Neptune
4,505,000,000 km
17
18
2
Divisors
Activity
Observe :
12 = 1 × 12
12 ÷ 1 = 12 ;
1 is a divisor of 12
12 ÷ 12 = 1 ; 12 is a divisor of 12
Complete :
12 = 2 × 6
12 ÷ 2 = ............... ; 2 is a divisor of ...........
12 = 3 × .........
12 ÷ .......... = ................ ;.................................................................
12 ÷ ............... = 2 ; .................................................................
12 ÷ .......... = ................ ;.................................................................
The divisors of 12 are :
12
÷
3
=
4
1 ; 2 ; 3 ; 4 ; 6 and 12 .
4
is
an
exact
quotient
of
the
division
of
12
by 3.
1
List all the divisors of 12. ....................................................................................................................................................
List all the divisors of 18. ....................................................................................................................................................
Is 3 a common divisor of 12 and 18 ? .................................................................................................................
What are the common divisors of 12 and 18 ? ........................................................................................
What are the greatest common divisor of 12 and 18 ? .................................................................
2
Is 7 a divisor of 84 ? Justify. ...........................................................................................................................................
Is 12 a divisor of 84 ? Justify. ......................................................................................................................................
Is 4 a common divisor of 12 and 84 ? Justify. .........................................................................................
................................................................................................................................................................................................................................
Is 16 a divisor of 84 ? Why ? ..........................................................................................................................................
Is 4 a common divisor of 12, 16 and 84 ? .....................................................................................................
19
3
List the divisors of 20. ............................................................................................................................................................
List the divisors of 30. ............................................................................................................................................................
What are the common divisors of 20 and 30 ? ........................................................................................
What is the greatest common divisor of 20 and 30 ? ......................................................................
4
Is 8 a divisor of 24 ? Justify. ...........................................................................................................................................
List the divisors of 8. ................................................................................................................................................................
List the divisors of 24. ............................................................................................................................................................
Are the divisors of 8 the divisors of 24 ? .......................................................................................................
5
Selim wants to arrange equally 1,309 apples in small boxes. Each box can contain a
number of apples between 9 and 13.
How many boxes does he need ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
6
List the multiples of 12 that are less than 61. ...........................................................................................
List the divisors of 12. ............................................................................................................................................................
List the divisors of 36. ............................................................................................................................................................
Are the divisors of 12 the divisors of 36 ? ....................................................................................................
Can we say that any divisor of 36 is a divisor of 12 ? ...................................................................
Justify your answer. ..................................................................................................................................................................
20
3
Divisibility rules
by 2, by 5 and by 10
Activity
Among the following numbers :
19 ; 20 ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 ; 29 ; 30.
a) Write those which are divisible by 2.
................................................................................................................................................................................................................................
Remember
All numbers whose ones digit is 0 , 2 , 4 , 6 or 8 are
divisible by 2 .
All numbers divisible by 2 are even.
b) Write those which are divisible by 5.
................................................................................................................................................................................................................................
Remember
All numbers whose ones digit is 9 or 5 are divisible by 5.
c) Write those which are divisible by 10.
................................................................................................................................................................................................................................
Remember
1
All numbers whose ones digit is 0 are divisible by 10.
Answer by true or false.
78,630 is divisible by 2 and by 5.
........................................
2 and 5 are divisors of 465.
........................................
2 and 10 are divisors of 3 460.
........................................
13,875 is divisible by 5 and by 10.
........................................
4,830 is divisible by 2; by 5 and by 10.
........................................
4,350 is divisible by 5 and by 10.
........................................
21
2
Place the following numbers in the table below.
95
;
112
;
360
;
530
;
2,108
;
3,205
Numbers divisible Numbers divisible Numbers divisible
by 10
22
by 2
by 5
;
190
;
2,003.
Numbers
divisible by 2 and by 5
3
Write the first 11 even numbers divisible by 5.
.......................................................................................................................................................................................................................................
4
Write the first 5 even numbers divisible by 5 and not by 2.
.......................................................................................................................................................................................................................................
5
Write the first 11 numbers divisible :
by 2 : ......................................................................................................................................................................................................................
by 5 : ......................................................................................................................................................................................................................
by 10 : ...................................................................................................................................................................................................................
6
What is the missing digit so that the number 379
would be divisible by
2 and 5 ?
.......................................................................................................................................................................................................................................
7
Write each of the numbers : 327 ; 792 ; 403 and 1,878 between :
a) two consecutive multiples of 10.
b) two consecutive multiples of 5.
................................................................................................
................................................................................................
................................................................................................
................................................................................................
................................................................................................
................................................................................................
................................................................................................
................................................................................................
23
8
Use the digits 8, 0, 5 (each digit is used only once) to form three-digit numbers
that are :
a) divisible by 2. .........................................................................................................................................................................................
b) divisible by 5. ........................................................................................................................................................................................
c) divisible by 10. .....................................................................................................................................................................................
9
Find the numbers less than 30 that if are :
a) divided by 2, give 1 as remainder. ...............................................................................................................................
...............................................................................................................................................................................................................................
b) divided by 5, give 1 as remainder. ...............................................................................................................................
c) divided by 2 and 5, give 1 as remainder. ..............................................................................................................
10
Hadi has as many male cousins as female cousins.
Their total number is less than 19 and divisible by 5.
How many female cousins does he have ?
.......................................................................................................................................................................................................................................
11
a) What numbers are divisible by 2 and included between 71 and 91 ?
.......................................................................................................................................................................................................................................
b) What numbers divisible by 5 and included between 71 and 91 ?
.......................................................................................................................................................................................................................................
c) What numbers are divisible by 10 and included between 71 and 91 ?
.......................................................................................................................................................................................................................................
24
12
I am a number greater than 51 and less than 61.
If you add 2, I will be divisible by 5.
What number am I ?
(Give all the possible solutions)
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
13
I am a three-digit number divisible by 10. My hundreds digit is divisible by 5. My tens
digit is 3 numbers less than my hundreds digit.
What number am I ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
25
4
Divisibility rule by 3
Activity
Calculate : 1,926 ÷ 3 = ...........................
Is the quotient exact ? ...........................
Complete : 1,926 is ...................................................... by 3.
The sum of the digits of number 1,926 is ........................... .
Is this sum divisible by 3 ? ...........................
Is the number 1,928 divisible by 3 ? ...........................
Is the sum of digits of number 1,928 divisible by 3 ? ...........................
Remember
1
A number is divisible by 3 if the sum of its digits is divisible by 3.
Write the first 11 numbers that are divisible by 3.
.......................................................................................................................................................................................................................................
2
Is the number 72 divisible by 3 ? Justify.
.......................................................................................................................................................................................................................................
3
Is the number 657 divisible by 3 ? Justify.
.......................................................................................................................................................................................................................................
4
The sum of digits of the number 13
is 8.
What is its ones digit ? .........................................................................................................................................................
Is the number divisible :
by 3 ? ............................................
26
by 2 ? ............................................
by 5 ? ............................................
5
The number 85
is divisible by 3.
Find the missing digit (give the three possibilities).
.......................................................................................................................................................................................................................................
6
The number 86
is divisible by 3.
Find the missing digit (give all the possibilities).
.......................................................................................................................................................................................................................................
7
The number 87
is divisible by 3.
Find the missing digit (give the four possibilities).
.......................................................................................................................................................................................................................................
8
I am a number included between 54 and 67, divisible by 2 ; by 5 and by 3.
What number am I ? ..........................................................................................................................................................................
9
The number 4 3
is divisible by 3 and by 5. The ones digit is not 5.
Find the missing digits (give all the solutions).
.......................................................................................................................................................................................................................................
10
Using the digits 5 ; 0 ; 9 ; 4 without repetition, write :
a) The smallest four-digit number : ...............................................................................................................................
b) The smallest four-digit number divisible by 2 : ......................................................................................
c) The greatest four-digit number divisible by 3 : .......................................................................................
d) The greatest four-digit number divisible by 5 but not divisible by 2 :
................................................................................................................................................................................................................................
e) The greatest four-digit number neither divisible by 5 nor by 2 : ......................................
27
5
Divisibility rule by 9
Activity
Calculate : 3,735 ÷ 9 = ...........................
Is the quotient exact ? ...........................
Complete : 3,735 is ...................................................... by 9.
The sum of the digits of number 3,735 is ........................... .
Is the sum divisible by 9 ? ...........................
Is the number 3,735 divisible by 9 ?
Is the sum of digits of the number 3,735 , divisible by 9 ? ...........................
Remember
1
A number is divisible by 9 if the sum of its digits is divisible by 9.
Write the first 11 numbers that are divisible by 9.
.......................................................................................................................................................................................................................................
2
Is the number 1,282 divisible by 9 ? Justify.
.......................................................................................................................................................................................................................................
3
Is the number 52,730 divisible by 9 ? Justify.
.......................................................................................................................................................................................................................................
4
Is the number 830,934 divisible by 9 ? Justify.
.......................................................................................................................................................................................................................................
5
Find all the numbers divisible by 9 :
a) included between 150 and 170 : ....................................................................................................................................
b) included between 2,000 and 2,020 : ..........................................................................................................................
c) having 36 as the number of tens : ................................................................................................................................
28
6
The number 15,34
is divisible by 9.
Find the missing digit.
.......................................................................................................................................................................................................................................
7
The number 4,37 ,256 is divisible by 9.
Find the missing digit (give the two possibilities).
.......................................................................................................................................................................................................................................
8
a) Find the smallest three-digit number, with different digits, divisible by 9.
................................................................................................................................................................................................................................
b) Find the greatest three-digit number, with different digits, divisible by 9.
................................................................................................................................................................................................................................
9
a) I am the greatest three-digit number divisible by 9 and by 10.
What number am I ? ..................................................................................................................................................................
b) I am the greatest three-digit number divisible by 9 and by 5, but not divisible
by 2.
What number am I ? ..................................................................................................................................................................
10
The number 6, 7
is divisible by 5 and by 9.
Find the missing digits (give all the solutions).
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
29
6
Remainder of the division
by 3 and by 9
Activity
a) Complete : 256 = 3 × ........................... + 1
1 is the ........................... of the .......................................... of 256 by 3.
The sum of the digits of the number 256 is : ........................... .
What is the remainder of the division of this sum by 3 ? .................
b) Complete : 256 = 9 × ........................... + 4
4 is the ........................... of the .......................................... of 256 by 9.
Is the sum of the digits of the number 256 divisible by 9 ? ........................... .
What is the remainder of the division of this sum by 9 ? ................
Remember
1
The remainder of the division of a number by 3 (or by 9) is equal to the
remainder of the division of the sum of its digits by 3 (or by 9).
Example.
Find the remainder of the division of
Remainder of the division of 758,257 by 3.
649,157 by 3.
7144424443
+5+8+2+5+7
34
↓
3+4
↓
7
↓
1
The remainder is 1.
To find the remainder Remainder of the division of 756,598 by 3.
of the division of a
7 5 6
5 9 8
natural number by 3, delete all the
12
13
digits 3, 6 and 9 and the digits whose
↓
the sum is a multiple of 3.
1
The remainder is 1.
Remark
30
2
Example.
Find the remainder of the division of
Remainder of the division of 438,175 by 9.
871,439 by 9.
4144424443
+3+8+1+7+5
28
↓
2+8
↓
10
↓
1+0
↓
1
The remainder is 1.
To find the remainder Remainder of the division of 648,953 by 9.
of the division of a
6 4 8
9 5 3
natural number by 9, delete all the
18
8
digits 9 and the digits whose the sum
↓
is a multiple of 9.
8
The remainder is 8.
Remark
3
4
Calculate the remainder of the division of each of the following numbers by 3.
5,296,123
432,764
974,562
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
Calculate the remainder of the division of each of the following numbers by 9.
765,432
7,432,107
497,643
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
.......................................................................
31
7
Divisibility rule by 4
Activity
Calculate : 128 ÷ 4 = ...........................
Is the quotient exact ? ...........................
Complete : 128 is ...................................................... by 4.
Is 28 divisible by 4 ? ........................... .
Is the number 534 divisible by 4 ? ........................... .
Is 34 divisible by 4 ? ...........................
Remember
A number is divisible by 4 if the number formed by its two last
digits on the right is divisible by 4.
1
Write the first 12 numbers divisible by 4.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
2
Without carrying out the division, indicate among the following numbers those
that are divisible by 4.
3,541 ; 3,004 ; 7,812 ; 5,122 ; 5,180 ; 6,596 ; 2,500 ; 3,000.
.......................................................................................................................................................................................................................................
32
3
Complete the table.
120
Divisible by 4
X
Divisible by 2
X
1,070
3,700
1,022
3,744
X
Divisible by 2 and 4
Divisible by 2 and not
divisible by 4
4
The number 386,9
is divisible by 4 and by 5.
Find the missing digit (give all the possibilities).
.......................................................................................................................................................................................................................................
5
The number 415,9 4 is divisible by 4.
Find the missing digit (give all the possibilities).
.......................................................................................................................................................................................................................................
6
Ordinary year or leap year.
An ordinary year counts 365 days (February : 28 days).
A leap year counts 366 days (February : 29 days).
A year is a leap year if it is divisible by 4.
In particular, all years that are divisible by a 100 are not leap unless if they are also
divisible by 400.
Examples
: The years 1600 and 1964 are leap years.
Application : Among the years between 1990 and 2017, indicate those which are
leap years. Justify.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
33
8
Dividing by a three digit number
Activity
Observe and complete.
5......
471
1......
251
24,963
The remainder is ....................... .
26,685
The remainder is ........................ .
The quotient is exact.
Compare
The Euclidean writing is :
the
remainder
with 251. ................................................
24,963 = ...............................................
The quotient is not exact.
The Euclidean writing is :
26,685 = ................................................
1
Calculate and write the non zero remainder.
133
34
4,305
201
8,442
319
32,539
Remainder : .........................
Remainder : .........................
Remainder : .........................
The Euclidean writing :
The Euclidean writing :
The Euclidean writing :
....................................................................
....................................................................
....................................................................
2
A number divided by 135 gives 209 as an exact quotient.
What is this number ?
.......................................................................................................................................................................................................................................
3
The division of a number by 241 gives 312 as a quotient and 173 as a remainder.
What is this number ?
.......................................................................................................................................................................................................................................
4
Use the four numbers 2 ; 15 ; 7 and 1 to form a euclidean writing.
.......................................................................................................................................................................................................................................
Same for the numbers 71 ; 3 ; 17 and 4.
.......................................................................................................................................................................................................................................
5
Carry out, give the euclidean writing and indicate the exact quotient.
305
12,810
191
7,069
415
11,670
................................................................
................................................................
................................................................
................................................................
................................................................
................................................................
303
19,647
415
84,206
607
46,713
................................................................
................................................................
................................................................
................................................................
................................................................
................................................................
35
9
Angles
In the adjacent figure, we draw through
point O the two semi-straight lines [Ox)
and [Oy).
To designate the adjacent angle, we
use the following notation :
xOy or yOx .
Point O is its vertex.
The semi-straight lines [Ox) and [Oy)
are its sides.
x
The
vertex of
the
angle
The sides of
the angle
O
y
c
m
A
K
n
d
The semi-straight lines [Ac) and [Ad)
The semi-straight lines [Km) and [Kn)
are perpendicular : cAd is a right
are opposite :
angle.
angle.
mKn is a straight
r
z
v
B
zBv is an angle that is smaller than
the right angle; zBv is an acute angle.
36
t
S
rSt is an angle that is greater than the
right angle; rSt is an obtuse angle.
1
A
Observe the adjacent figure and indicate the
angles by drawing
z
.
x
2
y
Observe the angles then complete the table.
x
c
A
u
O
y
T
d
f
p
m
The angle
S
n
I
xAy
cOd
q
uTv
v
mIn
l
B
pSq
fBl
Its vertex
Its sides
Among the above angles, name : an acute angle, an obtuse angle, a right
angle and a straight angle.
..................................................................................................................................................................................................................................
..................................................................................................................................................................................................................................
3
Draw an acute angle of vertex O and
of sides [Ou) and [Ov).
Draw an obtuse angle of vertex I
and of sides [Ix) and [Iy).
37
4
O
Write all the angles of the adjacent figure.
t
..............................................................................................................................
..............................................................................................................................
..............................................................................................................................
x
z
y
5
Find two right angles, two acute angles, two obtuse angles and a straight angle
in each of the figures below.
North
A
North-East
B
North-West
H
C
West
G
D
East
C
O
I
A
B
G
F
F
South-West
O
K
D
South-East
E
South
6
............................................................................................................
............................................................................................................
............................................................................................................
............................................................................................................
............................................................................................................
............................................................................................................
............................................................................................................
............................................................................................................
Draw the semi-straight lines [Ax) , [Ay) , [At) , [Az)
and [Am) knowing that :
xAy is acute, xAt is obtuse,
xAz is right, xAm is straight.
38
10
Fractions less than one
Fractions greater than one
Activity
Observe and complete by < or >
→ This band is divided into five equal
parts.
1
→ represent of this band
5
1
< 1
5
2
→ represent of this band
5
2
.................. 1
5
6
→ represent of this band
5
6
.................. 1.
5
Remember
When the numerator of a fraction is less than the
denominator, the fraction is less than one.
a
If a < b then < 1 .
b
When the numerator of a fraction is more than the
denominator, the fraction is greater than one.
a
If a > b then > 1 .
b
1
What do you call each part of the unit when it is divided into two equal parts ?
Into 4 equal parts ?
..................................................................................................................
2
; .................................................................................................................
Into how many parts should a unit be divided to get :
sixteenths ? ..........................................................................................................................................................................................
hundredths ? .......................................................................................................................................................................................
39
3
To get a unit, how many halves are needed ? ........................................................................................................
quarters ? .........................................................................
4
5
6
twentieths ? ..........................................................................
Write the appropriate fraction :
nine fifteenths : .................................................................
fifteen fortieths : .............................................................
thirteen twentieths : .....................................................
eight thousandths : .....................................................
three thousands : ...........................................................
seven sevenths : ...........................................................
fifteen tenths : ....................................................................
three halves : ....................................................................
Write a fraction equal to one and having :
85 for denominator : ....................................................
101 for denominator : ..............................................
307 for numerator : ....................................................
614 for numerator
: ...............................................
Give three fractions whose the numerator is the half of the denominator.
.......................................................................................................................................................................................................................................
7
8
9
Write a fraction less than one and having :
7 for denominator : ......................................................
5 for denominator : .....................................................
9 for numerator
8 for numerator
: ......................................................
: .....................................................
Write a fraction more than one and having :
4 for denominator : ......................................................
10 for denominator : .................................................
3 for numerator
13 for numerator
: ......................................................
: .................................................
Give three fractions more than one and having three as a denominator.
.......................................................................................................................................................................................................................................
40
10
Write four fractions less than one and having 5 as a numerator.
.......................................................................................................................................................................................................................................
11
Place the following fractions in the chart below.
8
17
12
;
13
7
;
9
9
;
27
28
98
105
;
;
65
65
;
104
99
;
49
49
7
.
6
;
Fractions
Fractions
Fractions
less than one
equal to one
greater than one
What fraction of the week remains at the end of the fourth day ?
.......................................................................................................................................................................................................................................
13
A biker leaves Beirut to go to Tripoli.
The distance between Beirut and Tripoli is 80 km.
What fraction of the road has the biker covered ?
Tripoli
29 km
........................................................................................................................................................
14
Draw a segment [AB] of 8 cm. Divide it into 8 equal parts. Above this segment, draw
3
9
two other segments, the first representing of [AB] and the second of [AB].
8
8
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
41
15
The half of a cake is divided into three equal pieces.
What fraction of the cake does each piece represent ?
..................................................................................................................................................................
16
What fraction of the year do 265 days represent ?
.......................................................................................................................................................................................................................................
17
What fraction of an hour remains after 45 minutes ?
.......................................................................................................................................................................................................................................
18
19
20
42
What fraction of the day passed :
a) at 10 o’clock in the morning ? ..................
b) at 6 o’clock in the evening ? .....................
......................................................................................................
....................................................................................................
Complete with the largest possible number.
<1
3
<1
20
3+
< 1
6
7–
> 1
4
18 –
> 1
10
90 +
< 1
100
120 –
> 1
100
10 –
> 1
5
Complete with the smallest possible number.
>1
4
>1
5
>1
10
8+
> 1
15
60 +
> 1
100
5×
> 1
20
30 +
> 1
50
7×
> 1
25
11
Decimal numbers
(reminder)
Activity
Observe and complete.
①
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
10
1
= 0.1 (one tenth)
10
5
10
5
= ........ (............................................)
10
10
10
10
= 1 (............................................)
10
11
10
12
10
11
= 1.1 (eleven tenths)
10
........................................................................
➁
1
1
1
1
1
1
1
1
1
1
1
1
1
100 100 100 100 100 100 100 100 100 100 100 100 100
1
100
1
= 0.01 (one hundredths)
100
6
100
6
= ........ (............................................)
100
10
100
10
= ........ (............................................)
100
11
100
12
100
11
=0.11(............................................)
100
12
= ........ (............................................)
100
274
1
13
5
49
; ; ; ; are decimal fractions.
10
10
100
100
100
0.1 ; 1.3 ; 0.05 ; 0.49 ; 2.74 are decimal fractions.
43
1
Write the following decimal fractions in the form of decimal numbers.
2
135
= ..............................
10
125
= ..............................
10
3
= ...............
100
556,794
= ............................................
100
101
= ..............................
100
1,998
= .........................
100
13
= ...................
10
1
= ........................................................
100
Write the following decimal numbers in the form of decimal fractions.
3
0.15 = .................................
145.3 = .............................
18.01 = .............................
8.25 = .................................
0.08 = .................................
468.19 = ..........................
130.05 = ..........................
0.4 = ....................................
Write in words (Example : 13.05 : Thirteen point zero five) .
0.09 : .........................................................................................................................................................................................................
300.14 : .........................................................................................................................................................................................................
650.81 : .........................................................................................................................................................................................................
261.31 : .........................................................................................................................................................................................................
3 001.71 : .........................................................................................................................................................................................................
4
Complete.
h
Ones
The tenths
The hundredths
t
o
digit
digit
4
3
2
The number
4.32
1.05
4
44
9
7
0
5
6
0
8
7
1
0
0
5
Complete the following chart :
The number
The tenths
digit
The tens
digit
The hundredths
digit
The hundreds
digit
The ones
digit
562.47
18.03
6.2
0.13
130.14
6
7
8
9
What does the digit 6 represent in each of the following numbers ?
43.6 : .......................................................
652.41 : ................................................
35.06 : .........................................
60.5 : .......................................................
86.03 : ................................................
6 001.33 : .........................................
Write in numbers.
Twenty ones and five hundredths : ................... Ten tens and one ones
: ..................
Twelve tenths
: ................... Fifty tens and six hundredths
: ..................
Nine hundredths
: ................... Fifty-three point one
: ..................
Complete with the smallest number possible so that the decimal fraction is a
natural number.
13 +
10
59 –
10
16 ×
10
128 –
100
235 +
100
121 –
10
125 ×
10
50 +
100
Hadi wants to buy a bicycle that costs 350,000 L.L.
He has only the tenth of its price. How much does he still need ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
45
12
The thousandths and
the ten-thousandths
A band is divided into 1,000
equal parts
.
The length of each part represents one thousandth of the
length of the band.
1
We write : and we read one thousandth .
1,000
1
is a decimal fraction which is equal to a thousandth of a
1,000
unit.
1
= 0.001
1,000
we read also zero point zero zero one.
1 is the thousandths digit.
0.001 is a decimal number which is equal to one thousandth
of a unit.
1
10,000
Same for : is a decimal fraction which is equal to ten
thousandths of a unit.
1
We write : = 0.0001 is a decimal number which is
10 000
equal to ten thousandths of a unit.
1 is the ten thousandths digit.
0.0001 is a decimal number equal to ten thousandths of a unit.
1
46
Write in the form of a decimal number.
3
= .........................
1,000
1
= .........................
1,000
347
= .........................
1,000
21,543
= .................................
10,000
3
= .....................
10,000
15
= .....................
10,000
347
= .....................
10,000
38,745
= .................................
10,000
2
3
4
Complete with ≠ or =
17
................... 0.17
1,000
135
............ 0.135
10,000
1,998
............ 0.1998
1,000
51,000
..................... 51
1,000
28
......... 0.0028
10,000
45
................... 0.45
1,000
140
................... 0.14
1,000
36,000
..................... 36
10,000
Write in the form of a decimal fraction.
0.04 = ..........................
0.109 = .......................
9.75 = ..........................
34.8733 = ..................................
0.15 = ..........................
3.245 = .......................
6.06 = ..........................
1.0009 = .....................................
Complete the numeration chart.
Ones
The number
digit
Ten
thousandths
digit
4
3
5.7043
Hundredths
Thousandths
digit
0
Tenths digit
h
t
o
5
7
9.0571
3
5
2
0
1
Write in the form of a decimal number the thousandths of each of the following
masses.
7t
................................................
6
15 kg
t
...........................................
900 kg
30,000 g
kg
..............................................
g
...........................................
kg
Count by thousandths from 7.357 to 7.363.
.......................................................................................................................................................................................................................................
7
To have a unit, how many thousandths are needed ?
.......................................................................................................................................................................................................................................
47
13
Measuring angles (1)
The unit of measure of an
angle is the degree : ° .
The adjacent geometric
instrument is called the
protractor.
The protractor is used to
measure the angle.
Center of the protractor
How do we measure the
adjacent angle xOy ?
Step 1
Place the center of the
protractor on O, the
vertex of the angle, and
the graduated 0° of the
protractor on one of the
sides of the angle, for
example [Oy).
Step 2
Look at the graduation
passing through the
second side [Ox).
This graduation is the
measure of the angle :
xOy = 50° .
48
1
Use your protractor to find in degrees the measure of each of the following
x
angles.
c
u
A
y
xAy = ...............................................
2
O
d
v
T
cOd = ..............................................
uTv = ...............................................
Measure the angles below (if necessary extend their sides) and complete.
z
O
m
y
t
x
A
xOt = ...............................................
B
n
zAy = ..............................................
mBn = ............................................
I
s
k
r
sIr = ..................................................
3
x
q
h
D
F
p
kFh = ..............................................
qDp = ..............................................
Arrange the following angles from the smallest to the largest, (use your
m
protractor to find their measure).O
y
I
n
z
A
t
s
k
p
B
r
D
h
F
q
.......................................................................................................................................................................................................................................
49
14
Measuring angles (2)
The protractor is used to
construct an angle whose
measure is given.
How do we construct
angle xOy = 70° ?
Draw one of its sides, for
example [Ox).
Place the center of the
protractor on O, the vertex
of the angle, and the zero
of the protractor on [Ox).
Count 70° from zero and
mark a point B.
Join O to B; [OB) is the
second side [Oy) of the
angle xOy .
1
Using a protractor, construct the following angles whose measure is given.
xOy = 45°
uBv = 120°
50
tIr = 100°
pDq = 180°
mSn = 30°
kFh = 83°
2
Measure all the interior angles of the polygons below. Calculate the sum of each
one.
BAC = .......................... ; ABC = .........................
A
ACB = ................................................................................
BAC + ABC + ACB = .......................................
C
....................................................................................................
B
D
....................................................................................................
E
....................................................................................................
....................................................................................................
G
F
I
H
....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
J
K
....................................................................................................
M
....................................................................................................
....................................................................................................
....................................................................................................
N
O
P
....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
Q
R
....................................................................................................
51
3
Construct an angle xOy of 75°° in each of the following cases.
O
x
O
4
x
O
x
What angle is added to 30°° to have a right angle ?
.......................................................................................................................................................................................................................................
5
Find the measure of the adjacent angle xOy .
x
..................................................................................................................................
O
Construct the semi-straight line in the
interior of xOy such that xOz = 20°° .
Calculate the measure of angle yOz .
..................................................................................................................................
y
6
Measure the adjacent angle xOy .
..........................................................................................................................
x
Construct the semi-straight line [Ot)
such that yOt = 180°° .
O
Calculate the measure of angle xOt .
y
..........................................................................................................................
7
In your copybook, draw the following angles.
x
O
B
I
v
r
y
8
u
t
In your copybook, construct: xOy =30° ; zAt = 2×xOy ; uBv = 3×xOy ; kMI = 4×xOy .
52
15
Equal Fractions
Simplified Fractions
The three rectangles are identical and the three coloured bands
are identical.
Complete.
........
3
........
of the rectangle ........ of the rectangle ........ of the rectangle
12
is light blue.
is grey.
is dark blue.
........
3
........
of the rectangle = of the rectangle = of the rectangle:
........
........
12
4
3 1
==
12 9 3
4÷4
1
=
12 ÷ 4 3
we simplifyd the fraction 12 4
Remember
3
Simplify 9
3 ÷ ........ 1
=
9 ÷ ........ 3
If we divide the numerator and the denominator of a fraction
by one common divisor, we obtain an equal fraction.
We say that we simplifyd the fraction.
1
Complete.
4÷2
4
= = .............................
10 10 ÷ 2
2
30 ........ ÷ ........
= = 45 ........ ÷ ........
..................
110 .....................
= = .....................
100 ........ ÷ 10
21
21 ÷ 7
= = ........................
35 ........ ÷ ........
.....................
70
= = .....................
105 ........ ÷ 35
240 ........ ÷ 120
= = ...................
120 ........ ÷ ........
53
2
3
Complete by ≠ or = (if necessary, simplify).
1
5
............... 10
2
1
3
............... 15
5
25
28
............... 21
20
44
33
............... 22
24
16
4
............... 4
3
3
1
............... 9
6
18
24
............... 12
16
3
39
............... 26
2
2
From a chocolate bar, Diana took and her sister took the rest.
4
Compare their parts.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
In a drawer, Walid found 3 bags, each containing 5 lollipops. He took 2 bags.
..........
Walid’s share is, therefore, of the bags.
..........
The number of the lollipops of the 3 bags is ................. .
Walid took .......... lollipops.
..........
Then, Walid’s share is of the lollipops.
..........
2 10
=
3 15
2×5
=
3 × 5 15 10
If we multiply the numerator and the denominator of a
Remember
fraction by the same non-zero number, then we obtain an
equal fraction.
4
54
Complete.
6
2
2×3
= = 5
5×
15
100
25
×
= = 6
×
4
= = 25
100
30
3
3 × 10
= = 8
8×
80
17
×
= = 4
×
12
7
= = 8
1,000
5
5×
45
= = 7
7×
63
13
×
39
= = 14
×
13
= = 4
100
5
6
7
Simplify, as much as possible.
26
26 ÷
13
= = 28
27
÷
= = 24
24 ÷
8
15
= ............................................................................................
35
63
= ............................................................................................
56
210
= .........................................................................................
330
304
= .........................................................................................
500
Complete by ≠ or = .
5
25
.......... 10
2
10
50
.......... 45
9
7
56
.......... 64
8
4
102
.......... 204
8
8
24
.......... 3
12
17
1
.......... 34
3
105
7
.......... 90
6
25
105
.......... 96
12
Write the multiples of 7 included between 1 and 45.
.......................................................................................................................................................................................................................................
Write the multiples of 13 included between 1 and 80.
.......................................................................................................................................................................................................................................
7
13
Give 5 fractions equal to .
....................................
8
....................................
;
....................................
;
....................................
;
....................................
;
....................................
;
....................................
;
....................................
7
Find 5 fractions equal to .
6
....................................
9
;
;
....................................
8
Give a fraction equals to and having 20 as a numerator.
12
.......................................................................................................................................................................................................................................
10
36
Give a fraction equals to and having the sum of the numerator and the
90
denominator equal to 28.
.......................................................................................................................................................................................................................................
11
Simplify, as much as possible.
105
48
14
87
;
;
;
42
15
42
39
;
729
.
630
55
Problems (Solve on your copybook)
1
Simplify.
24
80
a) ; 9
55
90
;
36
51
;
68
63
220
77
108
b) ; ; ; 42
110
33
99
2
6
99
1,998
999
c) ; ; ; 32
121
27
666
123
18
2,001
1,470
d) ; ; ; .
27
3
24
321
A farmer has 7,000 l of oil. He sold 35 barrels of 150 l each.
a) What fraction of oil did he sell ?
b) What fraction did he keep ?
3
A family spent 850,000 L.L. on a holiday. This expenditure consisted of 110,000 L.L. for
transport and 3 days at a hotel for 145,000 L.L. each.
a) What is the total amount of the other expenses ?
b) What fraction of the total expenses do transports represent ? the stay at the hotel ?
the other expenses ?
4
2.4 kg of dough is needed to prepare a cake of 2 kg.
What fraction of the dough does the rest represent ?
5
In a 20 cl glass, 10 cl of orange juice, 6 cl of pineapple juice and 4 cl of apple juice
are mixed.
What fraction of the mixture does each juice represent ?
56
16
Comparing fractions (1)
same denominator
Activity
Observe and complete .
The blue band represents a
The blue band represents a
unit.
unit.
The red band represents ............
The red band represents ............
of the blue band.
of the blue band.
The yellow band represents
The yellow band represents
............
of the blue band.
............
............
............
Which is the smallest, the red
Which is the greatest, the red
band or the yellow band ?
band or the yellow band ?
................................................................................
................................................................................
............
............
............
............
............
then < .
Remember
1
............
of the blue band.
............
............
............
............
then > .
If two fractions have the same denominator, then the one
with a smaller numerator is smaller than the other fraction.
Compare the fractions.
4
6
............... 7
7
13
11
............... 9
9
21
17
............... 35
35
3
25
............... .
25
3
57
2
3
4
5
Simplify then compare.
1
2
............... 6
12
20
6
............... 35
7
27
8
............... 45
20
15
24
............... 27
54
.................................................
.................................................
.................................................
..................................................
Write the following fractions in increasing order.
20
1
10
3
18
; ; ; ; 7
14
7
21
35
36
12
1
10
; ; ; 15
5
25
30
............................................................................................................
............................................................................................................
4
4
5
14
; ; ; 12
6
3
6
36
3
16
36
; ; ; .
11
11
66
33
............................................................................................................
............................................................................................................
Complete with a suitable number.
2
...
12
<<
5
5
15
1
2
9
<<
8
...
24
3
...
25
<<
11
11
55
10
3
...
<<
8
8
16
35
10
...
<<
22
11
55
25
6
...
<<.
26
13
65
Simplify then place the following fractions on a number line.
10
35
16
28
;
;
27
21
;
18
21
;
2
14
;
16
.
14
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
2
14
58
10
35
6
Place the decimal fractions on a number line.
30
100
;
2
5
;
5
10
;
110
100
;
20
100
;
15
.
25
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
20
100
7
30
100
4
3
From a bag containing marbles, Diana took and Mona took .
10
5
Who took less marbles ? Justify.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
8
Nabil bought a television set, a radio and a tape recorder. The price of the radio is
30
2
of that of the television set; the price of the recorder is of that the television
100
5
set.
Which is the cheapest apparatus ? Justify.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
59
17
Comparing fractions (2)
same numerator
Activity
Observe and complete .
The red part represent
............
of the first band.
............
The blue part
represents ............
of the second band.
............
Which of the two parts is the smallest ? .......................................................
............
then ............
< .
............
If two fractions have the same numerator, then the smallest
Remember
1
2
............
is the one with the greater denominator.
Compare the fractions.
13
13
............... 5
78
7
7
............... 9
8
3
12
............... 4
16
3
12
............... 4
20
7
28
............... 20
40
14
28
............... 15
34
10
5
............... 16
3
400
100
............... 100
20
35
25
............... 28
15
26
13
............... .
18
9
Write, in increasing order, seven fractions less than one and having 9 as a
numerator.
.......................................................................................................................................................................................................................................
3
Arrange in increasing order.
3
4
;
9
21
;
3
5
;
6
7
;
12
34
40
80
;
70
210
;
55
220
...................................................................
...................................................................
...................................................................
7
6
3
4
4
7
;
7
8
;
42
18
...................................................................
60
12
10
;
18
12
;
9
15
...................................................................
;
16
20
;
20
.
15
...................................................................
18
Comparing fractions
Activity
5
3
Let us compare and .
4
6
First method
Reduce
to
Second method
the
same
15
3
numerator : = 4 ..................
Reduce
to
same
denominator :
3 ..................
=
4
12
5
15
and = .
6 ..................
15
15
Compare and 20
18
5 ..................
and = .
6
12
9
12
Compare .........................................................................
3 5
so < .
4 6
the
10
and 12
.........................................................................
3 5
so < .
4 6
To compare two fractions, start by reducing to the same
Remember
1
numerator or to the same denominator.
Complete.
The given fractions
Writing them with the same denominator
Comparing
7
3
and 4
8
.......................................................
and .......................................................
.................................
9
5
.......................................................
and .......................................................
.................................
7
5
and 8
10
.......................................................
and .......................................................
.................................
14
12
and 7
9
.......................................................
and .......................................................
.................................
7
and
3
61
2
Complete.
The given fractions
3
Writing them with the same numerator
Comparing
2
3
and 3
4
.......................................................
and .......................................................
.................................
4
and
7
3
5
.......................................................
and .......................................................
.................................
3
13
and 9
2
.......................................................
and .......................................................
.................................
Simplify, reduce to the same denominator, then compare.
4
45
18
and 25
10
35
17
and 25
30
10
36
and 80
40
...................................................................
...................................................................
...................................................................
...................................................................
...................................................................
...................................................................
Simplify, reduce to the same numerator, then compare.
5
14
18
and 35
24
33
55
and 27
35
12
11
and 36
55
...................................................................
...................................................................
...................................................................
...................................................................
...................................................................
...................................................................
Compare each fraction to 1, then arrange by using the symbol <
2
3
6
,
5
4
17
23
,
25
18
123
79
,
101
138
...................................................................
...................................................................
...................................................................
...................................................................
...................................................................
...................................................................
In each of the following cases, show which fraction is greater.
62
7
11
and : ..........................
12
18
2
4
and : ......................
97
191
14
7
and : .............................
15
6
...................................................................
...................................................................
...................................................................
Problems (Solve on your copybook)
1
20
Simplify the fraction . Can you find an equal fraction having 45 as
75
denominator ?
2
42
Find all the fractions equal to and whose numerator and denominator are less
84
than 10.
3
1
In each of the following cases, find a fraction equal to .
7
a) the numerator is equal to 20.
b) the denominator is equal to 126.
4
From all the fractions smaller than 1 and having as numerator and denominator
two of the following four numbers : 3, 5, 8 and 12.
5
From all the fractions greater than 1 and having as numerator and denominator
two of the following four numbers : 2, 5, 7 and 9.
6
3
2
of the students in a class practice football and practice volleyball.
8
5
Which group is larger ?
7
7
3
Two barrels have the same capacity. The first contains and the second .
4
8
Which barrel contains less liquid ?
8
5
In a race, a contestant accomplished of the course before withdrawing. Another
9
4
accomplished .
7
Which one resisted more ?
9
4
2
Two fields are identical; of the first and of the second are plowed.
3
5
Which one has a greater plowed surface ?
63
19
Comparing decimals
Activity
① Write the numbers 13.5 and 12.9 in a form of decimal
fractions :
13.5 = .....................................
12.9 = .............................................. .
Compare these decimal fractions :
.....................................................................................................................................................................
Then deduce the comparison of the numbers 13.5
and 12.9.
.....................................................................................................................................................................
13.5
whole part
decimal part
12.9
Remember
The greater of two decimals is the one with the greater whole part.
➁ Write the numbers 12.31 and 12.8 in a form of decimal
fractions with same denominator :
...............................
12.31 = ...............................
12.8 = .
...............................
...............................
Compare these decimal fractions :
.....................................................................................................................................................................
Then deduce the comparison of the numbers 12.31 and
12.8.
.....................................................................................................................................................................
Remember
64
If two decimals have the same whole part, then the greater is the one with
the greater decimal part; after reducing the decimal part to the same
number of digits, and adding zeros if necessary.
1
2
3
4
Delete the unnecessary zeros and write the number.
3.50 = ...................
01500 = ...................
12.800 = ...................
1.10 000 = ................................
03.03 = ...................
00.010 = ...................
60.5000 = ...................
1.00010 = ................................
Complete by < or >
14.15 ................... 11.153
0.146 ................... 0.1452
18.9 ................... 18.09
13.165 ................... 13.201
116.1 ................... 116.09
132.4 ................... 133.95
152
................... 15.2
100
35,900
35.9 ................... 1,000
45
0.0045 ................... 100
1 ................... 0.9
0.75 ................... 1
1.001 ................... 1
Compare.
Arrange the following decimals from smallest to largest.
0.14
;
0.142
;
0.13
;
0.215
;
0.291
;
0.295
;
0.09 .
.......................................................................................................................................................................................................................................
5
Arrange the following decimals in descending order.
13.324
;
13.325
;
13.4
;
13.2569
;
13.33
;
13.099 .
.......................................................................................................................................................................................................................................
6
Given the number 73,659 ; place a decimal point to obtain :
a) a number less than 10 : .............................................................................................................................................................
b) a number included between 100 and 1,000 : ....................................................................................................
65
7
Compare by using the symbols > or <
63
5
12.5 and : ....................................................................................................................................................................................
38
3
12.5 and : ....................................................................................................................................................................................
19
3
6.1 and : ........................................................................................................................................................................................
8
7
and 1.14 : ........................................................................................................................................................................................
23
7
and 2.24 : ....................................................................................................................................................................................
8
18
58
Place the following numbers on a number line : 3.4 ; ; ; 2.5 .
20
5
2
9
3
4
Can you find a number with one decimal included between 2.5 and 2.6 ?
.......................................................................................................................................................................................................................................
10
Can you find a number with two decimals included between 3.64 and 3.65 ?
.......................................................................................................................................................................................................................................
11
Attention
For example, to insert three numbers of two decimals between 6.4 and 6.5
we write : 6.4 = 6.40
and
6.5 = 6.50 ; where :
6.40 < 6.41 < 6.42 < 6.49 < 6.50.
Insert five numbers formed of two decimals between 0.1 and 0.2.
.......................................................................................................................................................................................................................................
12
Insert four numbers formed of two decimals between 2.1 and 2.15.
.......................................................................................................................................................................................................................................
66
13
Insert three numbers formed of three decimals between 5.12 and 5.126.
.......................................................................................................................................................................................................................................
14
Write nine numbers of four decimals included between 7.321 and 7.322.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
15
Write each of the following numbers between the closest two numbers formed
of two decimals.
16
..................
< 3.542 < ..................
....................
< 0.207 < ...................
.........................
..................
< 0.193 < ..................
..................
< 5.4921 < ..................
......................
< 1.569 < ..........................
< 2.09774 < ......................
Attention
Rounding a decimal to a whole number means finding the nearest whole number
to it. If its tenths digit is 5, then we round to the next whole number.
1
1.6
1.5
;
0.9
2
;
1.495
2.5
;
2.1
;
3
1.855
;
2.7
The dots represent the places of the above numbers.
Match each circled number with the appropriate dot.
Round the following numbers to the nearest whole number.
17
1.6 : .....................................................
1.495 : ..............................................
1.855 : ..............................................
0.9 : .....................................................
2.1 : .....................................................
2.7 : .....................................................
Round the following decimals to the nearest whole number :
437
15
439
4,351
501
: ....................... : ....................... : ....................... : ....................... : .......................
100
10
10
10,000
1,000
67
20
Addition and subtraction
of decimals
Reminder
① To add two or more decimals, line up
➁ To subtract two decimals, line up the
the decimal points, write, if necessary,
decimal points, write, if necessary,
zeros in the decimal part, then add.
zeros in the decimal part, then subtract.
141.79 + 26 + 13.735
68.71 – 46.652
141.790
+ 26.000
68.710
+ 13.735
– 46.652
181.525
22.058
then :
then :
141.79 + 26 + 13.735 = 181.525
1
68.71 – 46.652 = 22.058
Calculate.
7.641
642.58
4359.13
+ 637.75
+
..................................
......................................
2
986.80
+
34.27
..................................
+ 659.870
+
86.435
......................................
Set up and calculate.
342.041 + 16.318
68
875.90
500.783 + 39.187
752.47 + 34.95 + 69.23
3
Set up and calculate.
99 + 314.8
4
197.86 + 296
Calculate.
702.36
–
49.04
..................................
5
3,498 + 13.367 + 564.52
9,056.20
–
967.47
8,104.13
5,010,00
768,00
–3,467.38
......................................
.......................................
–
......................................
Set up and calculate.
461.812 – 78.9
6
3,002.8 – 875.96
10,507 – 9,677.4
Work out without setting up.
7.4 + 1 = ............... 13.8 + 1 = ...............
5.82 + 1 = ........................... 12.345 + 1 = ...........................
3.7 + 1 = ............... 25.3 + 1 = ...............
99.99 + 1 = ...........................
9.871 + 1 = ...........................
2.8 + 1 = ............... 63.3 + 1 = ............... 999.99 + 1 = ........................... 99.999 + 1 = ...........................
7
Work out without setting up.
8.4 – 2 = ........................................
19.61 – 3.21 = .........................................
27.67 – 14.67 = ........................................
5.8 – 3 = .........................................
25.87 – 3.23 = ........................................
300.17 – 19.07 = .........................................
9.2 – 9 = ........................................
99.56 – 9.45 = .........................................
69
8
In each case, find the whole number to complete the equation.
9
7.2 +
= 9.2
18.7 +
= 19.7
27.82 +
= 30.82
8.6 +
= 11.6
13.55 +
= 18.55
125.73 +
= 150.73.
In each case, find the decimal to complete the equation.
10
11
12
4.2 +
= 4.3
13.5 +
= 15.8
125.11 +
= 125.12
35.04 +
= 35.09
6.7 +
= 6.8
25.3 +
= 30.6
60.61 +
= 66.62
52.16 +
= 55.26
347.20 +
= 350.30.
Complete.
0.3 + 0.7 = .................................
0.45 + 0.55 = .................................
0.32 + 0.68 = .................................
0.75 + 0.25 = .................................
0.35 + 0.65 = .................................
0.38 + 0.62 = .................................
1 – 0.5 = .................................
1 – 0.25 = .................................
1 – 0.01 = .................................
1.45 + 3.55 = .................................
7.3 + 2.70 = .................................
13.45 + 7.55 = .............................
4.4 + 11.6 = .................................
5.75 + 3.25 = .................................
17.32 + 2.68 = .............................
3.5 – 1.5 = .................................
9.2 – 6.2 = .................................
12.93 – 2.93 = .............................
46.82 – 27.9
39.9 – 27.27
Complete.
Set up and calculate.
73.57 + 24.43
70
13
Set up and calculate.
345.121 + 24.13
14
15
800 – 399.999
819 – 415.007
Complete.
100 –
= 2.75
100 –
= 81.81
1 000 –
= 0.1
100 –
= 13.13
1 000 –
= 13.3
100 –
= 0.99.
Set up and calculate.
23.73 + 14.64
14.64 + 23.73
8.173 + 6.213
6.213 + 8.173
What do you notice ?
.......................................................................................................................................................................................................................................
Remember
During addition, the order of the decimal numbers does not
affect the sum.
71
16
Set up and calculate.
(2.51 + 3.16) + 5.27
2.51 + (3.16 + 5.27)
(16.8 + 13.24) + 15.2
16.8 + (13.24 + 15.2)
What do you notice ?
.......................................................................................................................................................................................................................................
Remember
72
To facilitate the addition of several decimal numbers we can
group them in groups of two.
17
18
Calculate.
5 + (0.5 + 0.5) = .......................................................
(0.4 + 0.6) + 1 = .......................................................
(0.3 + 0.7) + 6 = .......................................................
4 + (0.25 + 0.75) = ......................................................
(0.2 + 0.8) + 7 = .......................................................
(0.35 + 0.65) + 4 = ......................................................
Group appropriately and calculate.
0.65 + 0.35 + 0.9 = ......................................................
7.4 + 3.64 + 9.36 = .....................................................
3.7 + 2.48 + 1.52 = ......................................................
2.15 + 4.85 + 3.8 = .....................................................
12.64 + 1.71 + 3.29 = ......................................................
6.9 + 1.44 + 4.56 = .....................................................
8.92 + 4.7 + 1.08 = ...................................................... 0.73 + 0.88 + 0.27 = .....................................................
14.31 + 9.4 + 3.69 = ......................................................
19
2.24 + 3.5 + 7.76 = .....................................................
AB = 3.65 m.
B
C
BC is 9 dm longer than AB.
CD is 125 cm longer than BC.
a) Calculate BC and CD.
A
............................................................................................................................
............................................................................................................................
D
b) Calculate, in meters, the distance from
A to D passing through B and C.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
73
Problems (Solve on your copybook)
1
To make curtains, Julie needs 19.5 m of material.
The salesman gives her 2.35 m more.
What is the length of the material that Julie gets ?
2
Nada weighs herself alone; the scale indicates 43.5 kg.
Then, she weighs herself with her cat in her arms; the scale indicates 48.69 kg.
What is the mass of the cat ?
3
Diana is thinking of a number. She adds 16.82 , to this number, then subtracts 6.12
and gets 19.
What is the number Diana is thinking of ?
4
Hoda wants to sew a skirt and a dress.
For the skirt, she needs 0.85 m of material and for the dress 1.5 m more.
What length of the material should Hoda buy ?
5
To water the garden, Nabil thought of using a hose 7.5 m long. Since the length was
not enough to reach the end of the garden, Nabil added two pieces of hose : the first
2.35 m and the second 1.75 m.
What is the new length of the watering hose ?
6
Hadi measures 0.2 m less than Jana, who measures 0.15 m less than Rola. The last
one measures 1.49 m.
Find Jana’s and Hadi’s height.
7
At the market, I bought 1.84 kg of beef, 0.275 kg of cheese, 4.5 kg of fruit, 1.75 kg
cabbage and 2.35 kg of carrots.
How many kilograms did I buy ?
74
8
The figure below shows the sketch of a garden.
Calculate the garden’s perimeter.
17.53 m
5.2 m
3.9 m
entrance
8m
4.25 m exit
8.32 m
10.4 m
9
From 6.38 m of material, a tailor uses 1.35 m; 225 cm and 1,450 mm.
Does he still have enough material to sew a jacket of 1.85 m ?
10
A biker should cover 201.2 km in three days.
He covers 76.5 km the first day and 7,800 m less the second day.
How much does he still have to cover the third day ?
11
A sugar bag weighs 100 kg. A merchant sells 42.75 kg to a customer and 14.85 kg to
another.
Can the merchant sell 35 kg of this bag to a third customer ?
12
The perimeter of a garden is 305.75 m. To fence it, three thread rolls of 150 m each
are used.
a) How much thread remains ?
b) If we want to fence the garden with a second row, how much thread do we still
need ?
75
21
The distance between
two parallel lines
Activity
Observe
the two
adjacent parallel lines
(xy) and (zt).
x
y
Place a point A on
z
t
(xy).
Use the T-square to construct line (D) passing through A and perpendicular to
(zt). Verify that (D) is perpendicular to (xy).
(D) cuts (zt) at B; place point B.
Use your graduated ruler then complete : AB = .......... mm = .......... cm.
AB is the distance between the two parallel lines (xy) and (zt).
Attention
(AB) is perpendicular to (xy) and to (zt).
1
In each of the following cases, choose a point on one of the two parallel lines,
draw a perpendicular line passing through this point to the other line, then
measure the distance between these two parallel lines.
x
u
v
y
z
m
n
The distance between (uv) and (mn) = ......... mm.
a
t
..............................................................................................................
g
c
s
h
b
d
t
76
..............................................................................................................
..............................................................................................................
2
In each of the following cases, identify the distance of the parallel lines.
M
AC E
J
P
K
R
I
B
3
D
F
G
L
H
N
Q
....................................................................
....................................................................
....................................................................
....................................................................
....................................................................
....................................................................
Use
the T-square to measure the
distance from each of the points A, B, C,
and E to (d).
(d)
E
The distance from A to (d) is ............................. mm.
B
........................................................................................................................
A
........................................................................................................................
C
........................................................................................................................
Draw line (AB). What can you say about
lines (AB) and (d) ? .............................................................
Draw line (CE). What can you say about
lines (CE) and (d) ? .............................................................
What can you say about the three lines (AB), (d) and (CE) ? ...............................................
................................................................................................................................................................................................................................
4
Let (D) be a straight line and A a point not
belonging to (D).
AB is the distance from A to (D).
Place a point C above (D) such that the
distance from C to (D) equals to AB. What
can you say about lines (AC) and (D) ?
A
B
(D)
.......................................................................................................................................................................................................................................
77
To construct a line passing through a point A and parallel to a line (D)
Remember
:
measure the distance from A to (D),
take on the side of A a point C such that the distance from C to
(D) equals to that from A to (D),
draw line (AC).
(AC) is a line passing through A and parallel to (D).
5
Draw through A the parallel to (D) in each of the following cases.
A
(D)
A
(D)
A
(D)
6
B
Draw (D1) passing through A and parallel
A
to (D),
(D2) passing through B and parallel to (D),
(D3) passing through E and parallel to (D),
(D)
(D4) passing through F and parallel to (D).
What can you say about lines (D), (D1),
(D2), (D3) and (D4) ? ..............................................................
..........................................................................................................................
78
E
F
7
(D1), (D2), (D3) and (D4) are four parallel straight
lines. The distance between two successive
lines is 1.5 cm.
Write all the possible couples of parallel
lines, each time indicating their distance.
(D1)
(D2)
(D3)
(D4)
(D1), (D2) : their distance is equal to 1.5 cm;
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
8
Observe the adjacent figure.
(D1)
A
Construct line (D3) passing through A and
perpendicular to (D2).
(D2)
Using your set-square, verify that (D3) and
(D1) are perpendicular.
What can you say about (D1) and (D2) ? ..........................................................................................................
Observe the adjacent figure.
Construct line (D6) passing through B and
perpendicular to (D5).
B
(D4)
Using your set-square, verify that (D6) and
(D4) are not perpendicular.
(D5)
Are lines (D4) and (D5) parallel ? ................................................................................................................................
Attention
To know if two lines (xy) and (zt) are parallel :
take a point on one of them (for example (xy)),
construct line (D) passing through this point and perpendicular to
(zt),
if (D) is perpendicular to (xy), then (xy) and (zt) are parallel.
(D)
x
z
A
y
t
79
9
In each of the following cases, do the same as the preceding exercise to know
if (xy) and (zt) are parallel.
x
x
y
z
t
x
z
z
y
y
t
t
.......................................................................
10
.......................................................................
.......................................................................
(d)
Observe the line (d).
Construct a line parallel to (d), of a
distance 2 cm from (d).
Are there others ? If yes, construct them.
11
The distance between the two
parallel lines (xy) and (zt) is 5 cm,
that between the parallel lines (zt)
and (uv) is 3 cm.
Draw the figure (there are two
possibilities).
What can you say about the
lines (xy) and (uv) ? What is
their distance ?
..........................................................................................................
..........................................................................................................
..........................................................................................................
80
..........................................................................................................
x
y
x
y
12
Draw two parallel lines (D1) and (D2)
having a distance of 6 cm.
Draw the segments [AB], [CD], [EF]
and [GH] each having one extremity
on (D1) and another on (D2) and such
that one of them is perpendicular to
(D1) and to (D2).
Measure the length of each of the
drawn segments.
............................................................................................................
............................................................................................................
Which segment is the shortest ?
What does it represent ?
............................................................................................................
............................................................................................................
13
Observe the adjacent figure.
Measure the length of each of the segments [AB], [CD], [EF] and [GH].
....................................................................................................
(D1)
A
C
E
G
B
D
F
H
....................................................................................................
Are (D1) and (D2) parallel ?
(D2)
....................................................................................................
14
Observe the adjacent figure.
Measure the length of each of the segments [AB] and [CD].
....................................................................................................
x
A
Are lines (xy) and (zt) parallel ?
....................................................................................................
B
z
C
y
D
t
81
22
Multiplying by
10 ; 100 or 1,000
Activity
Observe.
3 × 10 = 30
4 × 100 = 400
2 × 1,000 = 2,000
0.3 × 10 = 3
0.4 × 100 = 40
0.2 × 1,000 = 200
0.03 × 10 = 0.3
0.04 × 100 = 4
0.02 × 1,000 = 20
To multiply a decimal number by 10 , move the decimal point one digit to
the right.
To multiply a decimal number by 100 , move the decimal point two digits
to the right.
To multiply a decimal number by 1,000 , move the decimal point three
digits to the right.
Use zeros if necessary.
Remember
1
2
Calculate.
21.4 × 10 = .......................
45.2 × 100 = .......................
2.004 × 1,000 = ..........................
0.9 × 10 = .......................
10.01 × 100 = .......................
1.05 × 1,000 = ..........................
0.1 × 10 = .......................
0.01 × 100 = .......................
0.01 × 1,000 = ..........................
3.5 × ....................... = 35
0.01 × ....................... = 0.1
1.05 × ....................... = 105
0.02 × ....................... = 2
0.140 × ....................... = 14
0.1 × ....................... = 1
Complete.
4.06 × ....................... = 40.6
82
6.4 × ....................... = 640
0.09 × ....................... = 90.
23
Dividing by
10 ; 100 or 1,000
Activity
Observe and complete.
5.3 × 10 = 53
5.3 × 100 = 530
5.3 × 1,000 = 5,300
53 ÷ 10 = ............
530 ÷ 100 = ............
5,300 ÷ 1,000 = ............
To divide a number by 10, by 100 or by 1,000, place the decimal point
leaving one, two or three digits in the decimal part.
(Use zeros if necessary).
Remember
3.45 × 10 = 34.5
3.45 × 100 = 345
3.45 × 1,000 = 3,450
34.5 ÷ 10 = ............
345 ÷ 100 = ............
3,450 ÷ 1,000 = ............
To divide a decimal number by 10, by 100 or by 1,000, move the decimal
point one, two or three digits to the left.
Remember
1
(Use zeros if necessary).
Work out.
35 ÷ 10 = ....................
0.1 ÷ 10 = ..................
....................
÷ 10 = 0.09
0.1 × 10 = ....................
6.9 ÷ 10 = ....................
10 ÷ 10 = ..................
....................
÷ 10 = 3
20.5 × 10 = ....................
0.4 ÷ 10 = ....................
200 ÷ 10 = ..................
....................
÷ 10 = 0.5
0.01 × 10 = ....................
83
2
Work out.
358 ÷ 100 = ........................
0.1 ÷ 100 = ...........................
10 ÷ 100 = ........................
0.25 × 100 = ........................
37 ÷ 1,000 = ...........................
1,000 ÷ 100 = ........................
10 ÷ 100 = ........................
2.5 × 100 = ...........................
4.5 ÷ 100 = ........................
÷ 100 = 1
0.025 × 100 = ........................
504.1 ÷ 100 = ........................
...........................
1 ÷ 100 = ........................
3
4
32.9 ÷ 100 = ...........................
...........................
÷ 100 = 0.1
Work out.
4 962 ÷ 1,000 = ......................
5 ÷ 100 = ......................
56,000 ÷ 1,000 = ......................
0.1 × 1,000 = ......................
503 ÷ 1,000 = ......................
1,000 ÷ 1,000 = ......................
32 040 ÷ 1,000 = ......................
0.01 × 1,000 = ......................
74 ÷ 1,000 = ......................
100 ÷ 100 = ......................
1 ÷ 1,000 = ......................
7.2 × 1,000 = ......................
1 ÷ 100 = ......................
10 ÷ 1,000 = ......................
150 ÷ 1,000 = ......................
Complete the table.
Decimal fraction
5
10
7
100
39
10
40
10
500
10
8
720
1
1,000 1,000 1,000
1.7
3.82
5.09
1.004
1
0.1
Decimal number
5
Complete the table.
Decimal number
0.01
0.005
Decimal fraction
6
A number multiplied by 1,000 gives 1. What is this number ?
.......................................................................................................................................................................................................................................
84
7
Complete.
34 ÷ ..................... = 3.4
0.56 × ..................... = 56
8
9
1 ÷ ..................... = 0.01
1.40 × ..................... = 14
12.01 ÷ ..................... = 1.201
60.504 × ................. = 60,504
Tick the products and the quotients that are integers.
0.21 × 100
504 ÷ 10
620 ÷ 10
3.50 × 10
2.359 × 1,000
400 ÷ 10
1,200 ÷ 100
0.2 ÷ 10
0.001 × 100
A car consumes 98.43 liters of essence per 1,000 km.
How much does it consume per km ?
.......................................................................................................................................................................................................................................
How much liters does it consume in 100 km ?
.......................................................................................................................................................................................................................................
10
100 meters of iron wire weigh 45 kg.
What is the mass of one meter of iron wire ?
.......................................................................................................................................................................................................................................
11
A merchant puts 1,355 lollipops in boxes of 100 lollipops each.
How many boxes does he use ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
12
A cook prepares jam. She uses 6.5 kg of apricots to which she adds 3,600 g of sugar.
After the cooking, she fills the jam in a jar weighing 900 g. When the jar is full it
weighs 8.3 kg.
Calculate the mass of jam lost during the cooking ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
85
24
Multiplying decimals
Operator technique
Activity
Observe.
1.3
Technique
×
2.4
two decimals
5 2
2 6
3.1 2
two decimals
To multiply two decimal numbers :
do the multiplication ignoring the decimal point.
place the point in the result so that the number of digits to the right would be
Remember
1
equal to the sum of the number of digits in the decimal parts.
Calculate.
1.63
×
8
...........................
2
Set up and calculate.
85.8 × 19
103.7 × 128
13.5 × 8
125.43 × 16
86
34.6
×
7
...........................
0.39
×
6
...........................
1.527
×
9
..............................
3
Calculate : 16 × 25 = ..................................
Deduce : 16 × 2.5 = ................................ 16 × 0.25 = .................................... 25 × 1.6 = ..................................
4
Calculate : 163 × 17 = ..............................
Deduce : 16.3 × 17 = ............................ 163 × 170 = .................................. 163 × 0.17 = ...........................
5
6
7
Using the given products, calculate the other product.
307 × 45 = 13,815
213 × 27 = 5,751
3,009 × 132 = 397,188
30.7 × 4.5 = .............................
21.3 × 2.7 = .................................
30.09 × 13.2 = .................................
3.07 × 0.45 = ........................
2.13 × 2.7 = .................................
300.9 × 1.32 = .................................
0.307 × 4.5 = ..........................
21.3 × 0.027 = ..........................
3.009 × 0.132 = ..............................
13.5 × 17 = ...............................
625 × 13.8 = ...............................
456 × 13 = ...........................................
135 × 1.7 = ...........................
625 × 138 = .................................
456 × 1.3 = ..........................................
1.35 × 17 = ...............................
6.25 × 138 = ...............................
456 × 0.13 = ......................................
135 × 0.17 = ...........................
62.5 × 138 = ...............................
4.56 × 13 = ..........................................
Calculate.
What is the double of 56 ? ........................................................................................................................................................
Deduce that of 5.6 and of 0.56. ............................................................................................................................................
.......................................................................................................................................................................................................................................
8
Calculate.
3.7
× 4.3
×
30.56
4.8
9.08
2.17
4.07
× 9.2
× 8.04
× 0.92
87
9
10
11
Calculate.
30 × 0.1 = ..........................
500 × 0.01 = .............................
9,263 × 0.001 = .....................................
7 × 0.1 = ..........................
38 × 0.01 = .............................
787 × 0.001 = .....................................
69 × 0.1 = ..........................
6 × 0.01 = .............................
3,000 × 0.001 = .....................................
0.5 × 0.1 = .........................
0.4 × 0.01 = .............................
0.19 × 0.001 = ......................................
1.3 × 0.1 = .........................
5.02 × 0.01 = .............................
30.25 × 0.001 = ......................................
0.04 × 0.1 = ......................
24.1 × 0.01 = .............................
8,547.3 × 0.001 = ......................................
Calculate.
Complete the table.
The number
× 0.1
÷ 10
× 0.01
÷ 100
× 0.001
÷ 1,000
5,783
605
30.6
0.25
Observe the results and deduce the following rules :
to multiply by 0.1 is the same as to divide by .....................................................................................................
to multiply by 0.01 is the same as to ............................................................................................................................
to multiply by 0.001 is the same as to .........................................................................................................................
88
12
Observe the example and complete :
43 × 0.2 = 43 × (2 × 0.1) = (43 × 2) × 0.1 = 86 × 0.1 = 8.6.
Remember
For example, to multiply a number by 0.2 , we multiply this number by 2 and we
divide the product obtained by 10.
17 × 0.2 = .........................................................................................................................................................................................................
35 × 0.6 = .........................................................................................................................................................................................................
2.4 × 0.3 = ........................................................................................................................................................................................................
0.32 × 0.4 = ....................................................................................................................................................................................................
3.27 × 0.5 = ....................................................................................................................................................................................................
13
Observe the example and complete :
4.9 × 0.02 = 4.9 × (2 × 0.01) = (4.9 × 2) × 0.01 = 9.8 × 0.001 = 0.098.
Remember
For example, to multiply a number by 0.02 , we multiply this number by 2 and
we divide the product obtained by 100.
27 × 0.03 = ......................................................................................................................................................................................................
2.31 × 0.04 = .................................................................................................................................................................................................
0.7 × 0.07 = ....................................................................................................................................................................................................
Deduce a rule to multiply a number by 0.005.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
Complete.
13 × 0.005 = ..................................................................................................................................................................................................
21.7 × 0.002 = .............................................................................................................................................................................................
89
14
Each kg of lamb’s leg costs 13,250 L.L.
What is the price of 1.75 kg ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
15
Dad bought 0.750 kg of cheese for 18,550 L.L. each kg and 0.5 kg of coffee for
13,750 L.L.
How much did he pay ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
90
16
To bring water to a garden from the spring, a farmer used 32 pipes of 3.5 m each and
17 pipes of 2.75 m each.
What is the distance from the garden to the spring ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
17
What is the price of 0.75 m of fabric if the meter costs 31,500 L.L. ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
18
What quantity of water do 163 bottles of 1.25 l each contain ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
19
Mineral water is delivered in boxes of 34 bottles of 1.5 l each.
What quantity of mineral water is transported in a truck loaded with 209 boxes ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
20
With an electric wire of 325 m, we can make 100 extension cords.
How many meters of electric wire are needed to prepare 20 extension cords ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
91
21
A photo 52 cm long and 36 cm wide is to be framed with a rod worth 6,750 L.L. the
meter.
What should the length of the rod be ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
How much is paid if the cost of labor is 3,500 L.L. ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
22
One meter of iron wire weighs 2.75 kg.
Calculate the mass of wires of lengths 0.5 m and 102.5 m.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
23
A shopkeeper bought 75.35 kg of pears for 1,750 L.L. each kg. He sold 47.65 kg for
2,650 L.L. each kg and the rest for 750 L.L. each kg.
What is the total selling price ?
Did he gain or lose ? How much ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
92
24
The length of a rectangular piece of land is 88.40 m and the width measures 23.55 m
less than its length.
1) Calculate the width and the perimeter of the piece of land.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
2) When fencing the land, we leave 3.5 m as entrance. This fence is made of 5 rows
of barbed wire.
What is the length of the barbed wire needed ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
3) What is the price of barbed wire if the meter costs 90.5 L.L. ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
25
Observe.
8.3 > 1
1.4 × 8.3 = 11.62
11.62 > 1.4. Then 1.4 × 8.3 > 1.4
0.6 < 1
1.4 × 0.6 = 0.84
0.84 < 1.4. Then 1.4 × 0.6 < 1.4.
Complete by less or greater.
Multiplying a number n by a number greater than 1, we obtain a number ......................................
than n.
Multiplying a number n by a number less than 1, we obtain a number ......................................
than n.
Without calculating, compare by using one of the symbols < , > or =
37.08 × 0.5 .................... 37.08
37.08 × 1.51 .................... 37.08
97.13 × 37.8 .................... 97.13
97.13 × 0.98 .................... 97.13
102.1 × 1 .................... 102.1
102.1 × 0.99 .................... 102.1
93
25
Multiplying a number
by a multiple of 10, 100 or 1,000
Activity
Complete.
2.314 × 20 = (2.314 × 2) × 10 = .......................................................... × 10
= ..........................................................
2.314 × 200 = (2.314 × 2) × .................. = ............................................ × ...................
= ............................................
2.314 × 2,000 = (2.314 × 2) × .................. = ................................... × ...................
= ...................................
For example, to multiply a number by 20, we multiply this number by 2 and we
multiply the product obtained by 10.
Remember
1
Calculate.
5.4 × 20 = .........................................................................................................................................................................................................
7.2 × 30 = .........................................................................................................................................................................................................
0.8 × 50 = .........................................................................................................................................................................................................
1.13 × 40 = ......................................................................................................................................................................................................
2
Write a rule to multiply a number by 200.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
Calculate : 7.1 × 200 = ..............................................................................................................................................................
13 × 300 = ................................................................................................................................................................
94
0.7 × 500 = ..............................................................................................................................................................
3
Write a rule to multiply a number by 2,000.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
Calculate : 7 × 2,000 = ..............................................................................................................................................................
3.1 × 5,000 = .........................................................................................................................................................
2.36 × 4,000 = ......................................................................................................................................................
4
Calculate.
200 × 3.4
= ..................................................................................................................................................................................
200 × 3.47
= ..................................................................................................................................................................................
200 × 3.476
= ..................................................................................................................................................................................
35,000 × 0.04
= ..................................................................................................................................................................................
35,000 × 0.4
= ..................................................................................................................................................................................
35,000 × 0.004 = ..................................................................................................................................................................................
35,000 × 0.0004 = ..................................................................................................................................................................................
5
What is the price of 0.750 kg of meat at the rate of 20,000 L.L. each kg ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
6
One object weighs 0.0105 kg.
How much does 1 million of these objects weigh in tons ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
95
Problems (Solve on your copybook)
1
Each Monday, 18,000 copies of a sports magazine are printed. Each copy has a 0.4 cm thickness.
What would be the height of a pile of copies issued during a year ? (There are 52 weeks per
year).
2
Olive oil is delivered in boxes of 6 cans each. A can contains 4.9 l.
What quantity of oil does a truck, containing 200 boxes, transport ?
3
From a barrel containing 200 l of water, 45 bottles of 0.5 l each, 62 bottles of 0.75 l each, and 83
bottles of 1.25 l each are drawn off.
What is the quantity of water that remains in the barrel ?
4
A farmer surrounds a meadow having a perimeter of 783.45 m with a fence worth 2,350 L.L. per
meter.
What is the cost of the fence ?
5
Dad bought 2.750 kg of beef for 16,500 L.L. the kg and 1.5 kg of lamb meat for 28,000 L.L. the kg.
How much did he pay ?
6
A farmer loads 63 boxes, each weighing 18.5 kg, and 39 others, each weighing 12.8 kg, in a lorry
that can transport 1,250 kg.
Is the vehicle overloaded ? If yes, what is the overload ?
7
A salesman sells perfume in flasks of 0.27 l for 18,000 L.L. each and in flasks of 0.49 l for
25,000 L.L. each. A merchant orders 123 big flasks and 97 small flasks.
What quantity of perfume does he receive ?
How much should he pay ?
8
A man’s step measures 0.65 m. From his house, this man makes 1,400 steps to go to the office and
2,000 steps to go to the movies.
How far is the man’s house from the office ? from the movies ?
96
26
Measure of length
(Reminder)
The units of measure of length are :
The meter, its multiples and its submultiples.
The multiples of meter
Name
kilometer
hectometer
decameter
The submultiples of meter
meter
decimeter
centimeter
millimeter
Symbol
Each unit is 10 times the following unit.
1 m = 1,000 mm.
1
1 mm = m = 0.001 m
1,000
1
2
3
4
1 km = 1,000 m.
1
1 m = km = 0.001 km
1,000
Reduce.
4 m = ........................... cm.
13 m 3 cm = .............................. cm.
515 mm = ...................... cm.
13 m = ....................... cm.
4 m 225 cm = .......................... cm.
13 mm = ......................... cm.
Change to the unit given.
200 cm = .................... m.
2,135 cm = ..................................... m.
1,357 mm = .................... m.
18 cm = ....................... m.
4,300 cm = ..................................... m.
17 mm = ............................ m.
Find the missing unit.
42 cm = 420 ....................
548 mm = 5.48 .............
7,008 mm = 7 ........... and 8 ..............
7.8 cm = 78 ......................
175 mm = 17.5 .............
1,349 mm = 1.349 .................................
Change to units given.
8.25 cm = ............................. mm.
4,500 m = ............... km.
0.5 m = ........................... mm.
3.5 dam = ............................. mm.
5.5 hm = ......................... km.
2,145 dm = ................... km.
134 m = .................................... km.
0.5 km = ............................. m.
2.7 dam = ...................... km.
97
27
Metric system
Measuring capacity
The liter, the submultiples, and the multiples of liter are the units of volume.
The multiples
of liter
The submultiples
of liter
Name
hectoliter
decaliter
liter
deciliter
centiliter
milliliter
Symbol
hl
dal
l
dl
cl
ml
100 l
10 l
1l
1
l = 0.1 l
10
1
l = 0.01 l
100
1
l = 0.001 l
1,000
The main unit is the liter (l).
Each unit is equal to 10 times the following unit.
The multiples of liter are the hectoliter (hl) and the decaliter (dal).
1 hl = 10 dal = 100 l or 1 l = 0.1 dal = 0.01 hl.
The submultiples of liter are the deciliter (dl), the centiliter (cl) and the milliliter (ml).
1 l = 10 dl = 100 cl = 1,000 ml ou
1 dl = 0.1 l ; 1 cl = 0.01 l ; 1 ml = 0.001 l.
These units form the metric system of capacity.
1
98
Complete.
5 l = ....................................... dl
200 l = ................................ hl
25 dal = ................................ dl
125 l = ................................ dal
725 l = ................................ dal
16 hl = ................................... cl
2 l = ....................................... cl
650 dl = ............................. l
4,300 ml = ......................... l
2
3
Complete.
3.95 dl = ........................... cl
0.275 l = ........................... cl
20 cl = ................................. dal
12.5 l = ............................... hl
78.5 dal = ........................ hl
1,200 dl = ........................ hl
1,845 ml = ...................... dl
0.05 l = ............................... cl
2 hl = .................................... ml
Complete.
2l
= ...................................... dl = ...................................... cl = ...................................... ml.
3l
= ...................................... dal = ...................................... hl.
5 hl = ...................................... dal = ...................................... l.
2 dal = ...................................... l = ...................................... dl = ...................................... cl.
4 ml = ...................................... cl = ...................................... dl = ...................................... l = ...................................... dal.
4
How many cl of oil do you need too fill 5 bottles of 1l each ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
5
A barrel containing 210 l of wine is emptied into bottles of 75 cl each.
How many bottles are filled ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
6
7
The unit is the liter. Write in the form of a decimal number.
3
l
5
;
1
l
4
;
30 cl
;
3
l
10
;
27
l
100
;
837
dl
1,000
;
.............
;
.............
;
.............
;
.............
;
.............
;
.............
;
8,579
l.
10,000
.............
.
A juice bottle of capacity 1.5 l is half filled. What is the volume of the juice in cl ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
99
8
A bottle contains 1.5 l of nimeral water.
Find the cpacity of 12 bottles in dl.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
9
For her birthday, Rola bought 24 juice bottles of 25 cl each.
She invited 29 friends.
How many bottles should Rola also buy if each guest has the right to have 2
glasses of 0.5 l each ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
10
To fill a small pool, 52 water buckets of 10 l each are needed.
What is the capacity of the pool ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
11
A metal barrel, weighing 17.4 kg when empty, contains 0.5 hl of gasoline. One liter of
gasoline weighs 730 g.
Calculate the weight of the full barrel.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
12
In a coffee shop, a juice is sold for 1,500 L.L. in a 4 dl or 1,800 L.L. in a 5 dl glass.
Calculate the selling price of one liter of juice in each case.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
100
Problems (Solve on your copybook)
1
We connect 3 chords tip to end.
4
The first measures 6.5 m. The second
From a 7 m long iron rod, we cut a 3.8 m
piece, and then we cut another piece whose
measures 22.5 dm more than the first. The
length is half of the first.
third measures 175 cm less than the first.
What is the total length of the three
a) Calculate the length of the second piece
chords, in meters ?
in cm.
b) Calculate the length of the iron rod that
remains.
2
A merchant buys 36 liters of oil for 7,500
L.L. per liter.
He fills the oil in bottles of 75 cl each.
5
An empty box weighs 1.7 kg.
a) How many full bottles can we obtain ?
We fill it with butter packs each weighing
b) He sells each bottle for 6,750 L.L.
450 g. The full box now weighs 23.3 kg.
Calculate the profit obtained from this
a) What the mass of all the butter packs
selling.
in grams ?
b) How many butter packs were put in
the box ?
3
a) Complete the text below with the
following numbers : 75 ; 5.4 ; 3.6 .
6
An empty barrel weighs 12.5 kg.
We fill it with 12 gallons of wine each
A truck weighs ... tons. When
containing 4.5 liters.
charged with ... bags of flour, the
a) How many liters of wine were poured
truck weighs ... tons.
in the barrel ?
b) Each liter of wine weighs 0.9 kg.
b) Calculate the mass of 1 bag of flour in
What is the mass of the full barrel in
kilograms.
kilograms ?
101
28
Polygons
(revision)
1
In your copybook, redraw the following polygons and write the name of each one.
A
B
E
I
F
D
K
P
C
L
H
M
G
Q
S
O
N
R
2
102
J
Use the necessary geometric instruments to construct :
a) a parallelogram ABCD of sides measuring 6 cm and 4 cm.
b) un rectangle EFGH of sides measuring 4 cm and 3 cm.
c) a rhombus MNPQ of side 4 cm.
d) a square IJKL of side 3 cm.
a)
b)
c)
d)
3
Complete the chart with True or False.
Name of the
quadrilateral
Opposite sides The sides have The 4 sides are
are parallel the same length
equal
The 4 angles
are right
Parallelogram
Rectangle
Rhombus
Square
4
a) Place in each box two points A and B such that AB = 3 cm.
b) In the first box, add points C and D such that ABCD is a parallelogram.
c) In the second box, place points M and N such that MNAB is a rhombus.
d) In the third box, add points K and L such that ABKL is a rectangle.
e) In the fourth box, place points I and J such that ABIJ is a square.
103
29
Diagonals of a polygon
Activity
Observe the adjacent polygon.
A
B
How many vertices does it have ?
...............................................................................................................................
Name them. ..........................................................................................
C
...............................................................................................................................
E
D
A and B are two consecutive vertices. Can you find the others ? What are they ?
(Give all the possibilities)
......................................................................................................................................................................
A segment joining two consecutive vertices called a side of the polygon.
a) Name all the sides of the polygon ABCDE.
..............................................................................................................................................................
b) A and C are non-consecutive vertices. If you join them, you will obtain the
diagonal [AC] of the polygon ABCDE.
Join the non-consecutive vertices of polygon ABCDE, then name its
diagonals.
..............................................................................................................................................................
..............................................................................................................................................................
104
1
Trace in red the sides and in green the diagonals of the following polygons.
E
G
F
A
H
I
B
D
C
J
M
N
L
K
R
O
Q
2
P
Draw the diagonals of the following polygons.
E
F
H
G
J
I
A
D
K
B
C
M
L
105
Problems (Solve on your copybook)
1
Construct a parallelogram EFGH of sides 3 cm and 4 cm.
Draw its diagonals and designate by I their common point.
Use the compass to verify that the diagonals of the parallelogram intersect at their midpoint.
2
Construct a rectangle RSTU of sides 4 cm and 5 cm.
Draw its diagonals and designate by K their common point.
Use the compass to verify that they intersect at their midpoint and that they have the same
length.
3
Construct a rhombus IJKL of sides 4 cm.
Draw its diagonals and designate by M their intersection point.
Verify that they are perpendicular and that they intersect at their midpoint.
4
Draw a square ABCD. Name its diagonals and verify that they intersect at their midpoint O, that
they are perpendicular, and that they have the same length.
5
Complete the following table with “True” or “False”.
The name of the
quadrilateral
Parallelogram
Rectangle
Rhombus
Square
106
The diagonals have
the same midpoint
The diagonals have
the same length
The diagonals are
perpendicular
6
Construct a parallelogram ABCD of sides 8 cm.
What can you say about the figure that you have drawn ?
Draw its diagonals. What can you say about these diagonals ?
7
Construct a quadrilateral that is not a rhombus and whose diagonals are
perpendicular.
8
Construct a parallelogram whose diagonals measure 7 cm. What can you say
about the figure ?
9
I am a quadrilateral.
Guess my name in each of the following cases :
a) My diagonals are perpendicular, having the same length and the same midpoint.
I am a : ............................................................................................................................................................................................................
b) My diagonals intersect at their midpoint;
I am a : ............................................................................................................................................................................................................
c) My diagonals are perpendicular, having the same midpoint;
I am a : ............................................................................................................................................................................................................
d) My diagonals have the same length and the same midpoint;
I am a : ............................................................................................................................................................................................................
107
30
Addition and subtraction of
fractions of same denominator (1)
Activity
Observer and complete.
A ruler is divided into seven equal parts.
What fraction does one part of the ruler represent ? ............
............
What fraction does the blue part of the ruler represent ? ............
............
What fraction does the red part of the ruler represent ? ............
............
What fraction does the coloured part of the ruler represent ?
............
............
2 3 2+3 5
+==
7 7
7
7
5 2 5–2 3
–==
7 7
7
7
5 3 5–3 2
–==
7 7
7
7
1
108
Work out.
5
3
+ = ...................................
13 13
3 2
+ = ..........................................
8 8
9
4
+ = ...................................
11 11
11
8
+ = ...................................
27 27
5 5
+ = ..........................................
9 9
9
16
+ = ...................................
17 17
2
3
4
5
Work out and simplify the answer.
3 5
+ = ............................................................................
8 8
11
7
+ = .....................................................................
26 26
4 2
+ = ............................................................................
9 9
20 14
+ = .....................................................................
17 17
5 9
+ = ............................................................................
7 7
13
9
+ = .....................................................................
11 11
Work out and simplify the answer.
7
5
– = .....................................................................
12 12
14 5
– = .........................................................................
9
9
7 5
– = ............................................................................
8 8
117
12
– = ..............................................................
105 105
Complete each equation by a suitable number.
..........
12
8
–=
15
15
15
..........
14
– =1
6
6
15
10
..........
+=
21
21
21
8
..........
+ =1
36
36
Complete each equation by a suitable number.
12
..........
7
+=
13
13
13
..........
5
+=1
9
9
13
..........
23
–=
10
10
10
5
7
..........
–=
13
13
13
9
..........
–=2
2
2
3
3
..........
+=
10 10
20
109
31
Addition and subtraction
of fractions (2)
Activity
11
to the same denominator:
① Reduce the fractions 35 and 7
3
= ............
5 ............
11
= ............
7
............
11 3
............
............
Their sum is : + = ............
+=
7
5 ............
............
............
11 3
............
............
Their difference is : – = ............
–=
7
5 ............
............
............
➁ Complete : 68 = 4
..........
5 6
Calculate : + = ............
+ ............
= ............
= .................
4 8 ............ ............ ............
5 6
– = ...........................................................................
4 8
1
+=
➂ Calculate : 2 + 13 = 3
3
3
..........
..........
1
2 – = .............................................................................
3
1
2
110
Work out.
3 2
+ = ............................................................................
7 3
4
13
+ = .....................................................................
21 14
5 4
+ = ............................................................................
8 9
16
5
– = .....................................................................
9
18
15 2
– = .........................................................................
7
5
65
8
+ = .....................................................................
35 21
11 2
+ = ........................................................................
13 5
3 1
– = ............................................................................
4 3
Work out.
5
3 – = ..............................................................................
8
17
1 + = ..........................................................................
8
8
+ 12 = ..........................................................................
7
2
6 – = ..........................................................................
12
3
4
5
6
Simplify and work out..
4
2
+ = ........................................................................
12 6
2
6
+ = .....................................................................
18 27
18 4
– = .........................................................................
12 6
2
39
+ = .....................................................................
14 21
4
33
+ = .....................................................................
32 24
64 27
– = .....................................................................
52 39
Work out.
5
+ 3 = ..............................................................................
4
21
5 + = ..........................................................................
13
5
2 – = ..............................................................................
7
2
16 – = ..........................................................................
3
Reduce to the same denominator then calculate.
7
2
– = .........................................................................
10 5
26
3
+ = .....................................................................
30 10
5 2
+ = ............................................................................
6 3
5 2
+ = ............................................................................
4 3
7 3
– = ............................................................................
8 4
17
7
– = .....................................................................
24 18
Simplify and calculate.
9
16
+ = .......................................................................................................................................................................................................
21 28
16
9
– = .......................................................................................................................................................................................................
22 33
15 12
+ = .......................................................................................................................................................................................................
35 21
28 18
– = .......................................................................................................................................................................................................
52 30
27 40
+ = .......................................................................................................................................................................................................
57 76
65
14
– = ....................................................................................................................................................................................................
100 45
111
7
Simplify and calculate.
16 12
+ = .......................................................................................................................................................................................................
12 16
12 12
+ = .......................................................................................................................................................................................................
21 16
18 16
– = .......................................................................................................................................................................................................
15 40
250
63
+ = ..............................................................................................................................................................................................
1 000 90
21 21
+ = .......................................................................................................................................................................................................
63 36
21
25
– = ....................................................................................................................................................................................................
45 100
8
Work out.
2 5
+ + 2 = ...................................................................................................................................................................................................
3 6
4
1
+ 3 + = ...................................................................................................................................................................................................
3
5
3 2 5
+ + = .................................................................................................................................................................................................
5 3 6
4 18
+ + 2 = ................................................................................................................................................................................................
6
2
11
3
+ + 1 = ............................................................................................................................................................................................
5
10
9
112
Work out.
3
5 – = .....................................................................................
4
3
5 + = .....................................................................................
4
11
18 + = ..............................................................................
5
2
8 – = .....................................................................................
5
105
13 – = ...........................................................................
26
1
1 – = .....................................................................................
3
10
Work out.
–=
5 + 10 2
3
9
1
.........................................................................................................................................................................................
9
1
2
– + = .....................................................................................................................................................................................
4
12 24
=
3 – 4 + 12
7
1
3
9
1
7
– – = .....................................................................................................................................................................................
5 10
10
–1=
7 – 14 13
11
.........................................................................................................................................................................................
2
.......................................................................................................................................................................................
21
3
+ 9 – = ...........................................................................................................................................................................................
8
4
Work out.
1
5 2
+ – = ............................................................................................................................................................................................
3
3 9
2 + 8 – 4 =
............................................................................................................................................................................................
1 – 3 + 6 =
..............................................................................................................................................................................................
1
6
2
3
1
7
1
1 + – = ...........................................................................................................................................................................................
14 2
1 5
3 – + = ..............................................................................................................................................................................................
3 6
5
8 – 2 – = ................................................................................................................................................................................................
3
1
7 – + 1 = ................................................................................................................................................................................................
4
5 + 1 – 5 =
9
2
..............................................................................................................................................................................................
113
Problems (Solve on your copybook)
1
2
2
Jad ate then of a cake.
9
12
1) What fraction of the cake did Jad eat ?
2) What fraction of the cake remained ?
2
1
1
2
A father spends of his salary on schooling, on food and on rent.
5
7
10
1) What fraction of the salary do these expense represent ?
2) What fraction of his salary remaines ?
3
2
2
A farmer palnted of his garden with cabbages and with lettuce.
9
5
1) What fraction of the garden did the farmer plant ?
2) Compare the planted part to the non planted part of his garden.
4
Simplify and compare.
17
36
; 21
42
18
;
15
3 + 5 13
;
4
3 + 18 14
18
; 105
70
17
;
26
+
26 39 90
;
81
9 + 27 5
6
2
4
4
9
8
5
12
13
Work out then arrange in increasing order.
4 1
1 1
1 5
+
+
+.
;
;
5 2
2 3
2 3
6
1
Zeina gave of her money to her brother and too her sister.
5
10
What fraction does the given money represent ?
7
8
3
In the exam, of the students in a class got more than 10 and got 10.
12
18
What fraction do the students who got 10 or more represent ?
114
32
Mixed number
Activity
What is the greatest multiple of 5 less than 17 ? .......................
17
5
15
5
15 + ..........
5
............
............
............
............
Complete : = = + = 3 + 17
5
The greatest whole number less than is .................... .
2
3 is called mixed number
5
2
17
We write : = 3 ;
5
5
1
2
3
Write each of these fractions in the form of a mixed number.
9
= ................................................................................................
2
250
= .........................................................................................
35
5
= ................................................................................................
2
180
= .........................................................................................
25
11
= ............................................................................................
5
107
= .........................................................................................
27
Write each of the following mixed numbers in the form of a fraction.
7
3 = = .....................................................................................
5
17
3 = .......................................................................................
2
2
8 = .......................................................................................
11
4
4 = ..........................................................................................
3
1
3 = ..........................................................................................
8
1
2 = ..........................................................................................
3
Place the following numbers on a number line.
3
1
3
1
1
1
2 ; 1 ; 1 ; 1 ; 2 ; 2.
4
4
2
4
2
4
1
2
3
115
4
Write each of the following fractions in the form of a mixed number.
13
= ............................................................................................
4
13
= ............................................................................................
2
23
= ............................................................................................
5
15
= ............................................................................................
18
16
= ............................................................................................
3
11
= ............................................................................................
5
Match each fraction to the suitable place on a number line.
15
18
0
5
116
13
4
1
11
5
2
16
3
3
4
13
2
23
5
5
6
Compare by using the signs < , > or =
3
2
1 .......... 1 5
5
1
2
2 .......... 2 10
5
6
3
4 .......... 4 4
8
2
3
2 .......... 1 5
5
7
6
3 .......... 3 10
10
1
1
2 .......... 2 5
4
7
33
Fraction of a number
A ribbon of 63 cm is divided into 9 equal parts.
1
9
63
9
The fraction of the ribbon measures cm = 7 cm.
4
9
63 × 4
9
63
9
The of the ribbon measures 7 cm × 4 = cm × 4 = cm
= 28 cm.
63 × 4
4
of 63 is : = 28.
9
9
1
2
3
4
Calculate.
2
of 120
3
4
of 120
5
13
of 120
10
19
of 120.
12
...............................................
...............................................
...............................................
...............................................
25
of 10,000
100
63
of 10,000
100
75
of 15,000
100
40
of 235.
100
..................................................
..................................................
..................................................
..................................................
Calculate.
Calculate.
the half of 210
one third of 210
one fifth of 210
one seventh of 210
......................................................
......................................................
......................................................
......................................................
1
A farmer sold of a herd of 90 sheep.
3
How many sheep does he still have ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
117
5
1
A week for a student consists of 30 hrs. of attendance in class. is dedicated to math.
6
3
to French and the same to Arabic.
10
Calculate in hours the duration of each course.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
6
10
Flour gives of its mass for bread.
9
What mass of bread do we get with 360 kg of flour ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
7
1
In my class, there are 32 students. eat at the cafeteria.
4
How many students do not eat at the cafeteria ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
8
The dimensions of a rectangular garden are 64 m and 36 m.
Calculate its perimeter.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
7
10
Calculate the length of the fence and its price.
A fence is placed on of the perimeter for a price of 2,750 L.L. the meter.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
118
9
2
The age of the second boy is the age of the oldest. The youngest is 6 years old.
3
The age of the oldest is double that of the youngest.
How old is each boy ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
10
The thickness of one sheet is 35 hundredths of the mm.
Calculate, in mm, the thickness of 9 congurent sheets.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
11
A student bought a copybook for 625 L.L. and a ball-point pen. The price of the ball7
point pen is of that of the copybook.
5
Calculate the price of the ball-point pen.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
How much did the student pay ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
12
5
A motorist drove 600 km. in 3 days. On the first day, he covered of the distance;
12
4
on the second day, he covered .
7
Calculate the distance covered each day.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
119
13
4
1
A piece of meat of 1,470 g contains bones and fat.
6
5
Calculate the mass of bone and the mass of fat in this piece of meat.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
If this piece is boned and the fat removed, what would the mass of meat be ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
14
2
Chadi bought a refrigerator for 1,860,000 L.L. He paid of its price at the order and
3
1
at the delivery.
4
Calculate the sum of money already paid.
How much does she still have to pay ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
120
Problems (Solve on your copybook)
1
Diana and Sara should cover the same distance to go to school.
2
4
Diana covers of the distance in 10 min, Sara covers in 12 min.
5
9
Which one goes faster to school ?
2
3
2
A merchant had 40,000 l of oil. He sold the first week and of the rest the second
8
5
week.
How much oil did he sell the first week ?
How much oil did he sell the second week ?
How many barrels of 75 l each can he still sell ?
3
5
Two tanks, having the same capacity, contain gasoline. The first contains and the
6
7
second .
9
1) Which tank contains more gasoline ?
2) Calculate the quantity of gasoline in each tank, knowing that the capacity of
each is 2,700 l .
Then verify the result of the preceding question.
4
3
Three runners are competing in a race : Hadi covers of the course, whereas Fadi
5
2
covers , and Rami covers the half.
3
1) Indicate the order of the runners.
2) The race is on a 1,800 m. Calculate the distance covered by each runner and
verify the answer you have given in number 1.
5
Ziad had 20,000 L.L. He spent the quarter of the sum to buy icecream and
2
to buy magazines.
5
1) What fraction of the sum does his purchase represent ?
2) What fraction of the sum does he still have ?
3) With the rest of the sum, can he buy a book for 8,000 L.L. ?
6
1
4
Three brothers divided a sum of money among them : the first got and the second .
9
3
How much did the youngest get ? Who has the most` ?
121
34
Circle - Diameter of a circle
Activity
a) Draw a circle of center O and of
radius 3 cm.
b) Mark a point B on this circle; the straight
line (OB) cuts again the circle in A; mark
point A.
c) Complete :
O
OB = ............. cm ;
OA = ............. cm ;
AB = ............. cm.
d) What did the center O represent to
the segment [AB] ?
.............................................................................................................................................................................
[AB] is a diameter of the circle; AB is the length of this diameter :
AB = 2 × OB = 2 × OA.
e) Mark a point M on this circle.
Can we say that the segment [AM] is a diameter of this circle ?
.............................................................................................................................................................................
f) Draw the diameter that passes through M.
Remember
1
The diameter of a circle is the double of its radius.
a) Calculate the radius of a circle of diameter 20 m.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
b) Calculate the diameter of a circle of radius 5.7 m.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
122
2
Observe the adjacent figure.
Name :
the center of this circle ....................................................................................................
the radius of this circle .....................................................................................................
F
E
C
O
the diameter of this circle ..............................................................................................
A
Draw the diameter that passes through point C.
3
B
Draw a square ABCD of side 3 cm.
Its diagonals intersect in O.
Draw the circle of center O and of radius OA.
Did this circle pass through the points B, C and D ?
Justify.
....................................................................................................................................................................................................
....................................................................................................................................................................................................
4
Draw a parallelogram ABCD of side AB = 3 cm ; BAD = 60°° and
AD = 4 cm.
Its diagonals [AC] and [BD] intersect in O.
Draw the circle of center O and of radius OA.
Did this circle pass through the points B, C and D ? Justify.
..............................................................................................................................................................................................
..............................................................................................................................................................................................
5
Draw a rectangle ABCD of side AB = 3 cm and AD = 4 cm.
Its diagonals [AC] and [BD] intersect in O.
Draw the circle of center O and of radius OA.
Did this circle pass through the points B, C and D ? Justify.
..............................................................................................................................................................................................
..............................................................................................................................................................................................
123
35
Dividing of two numbers
Appropriate quotient
Activity
The quotient to the nearest higher number
3l
3l
3l
The quotient to the nearest lower
number
Oil
23 l
3l
3l
3l
3l
23 m of cloth
How many cans of 3 l are needed to empty
the barrel of 23 l ?
Then, give the euclidean division of 23 by 3.
How many sheets of 3 m can be
made with this cloth ?
Then, give the euclidean division
of 23 by 3.
............................................................................................................................
.............................................................................................
............................................................................................................................
.............................................................................................
Complete : ........... cans are completely filled
Complete : .................. sheets of 3 m
and .................. can to put the remaining .................. l
and .................. m remain.
.
Then .................. sheets can be made.
Then .................. cans are needed.
Remark
8 is the quotient to the nearest higher
number of the division of 23 by 3.
124
Remark
7 is the quotient to the nearest
lower number of the division of
23 by 3.
1
Give the quotients to the nearest higher number and lower number of the
following divisions :
351 divided by 10 : .................................................................................................................................................................................
908 divided by 9 : ....................................................................................................................................................................................
2
A florist has 213 tulips. He wishes to prepare bunches of 12 tulips each.
How many bunches will he prepare ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
3
The maximum load of a truck is 3,500 kg.
How many wheat bags of 45 kg can the truck carry ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
How many trips should it make to carry 165 of these bags ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
4
A rectangular field of dimensions 84 m and 35 m is fenced with a double row of barbed
wire. When fencing the field, an entrance of 4 m is left.
What is the length of wire used ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
The wire is sold in rolls of 45 m.
How many rolls should be bought ?
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
125
36
Dividing a number by a number
Decimal quotient
Let 17 be divided by 4.
Observe :
25
17 16 + 1 16 1
1
= = + = 4 + = 4 + = 4 + 0.25 = 4.25.
4
4
4
4
4
100
Ones
17
– 16
1
Tenths
.
Technique
Hundredths
0
0
10
–8
2
4.25
4
20
– 20
00
4.25
17.00
4
– 16
10
– 8
20
– 20
00
17 ÷ 4 = 4.25
4.25 is the decimal quotient of the division 17 by 4.
1
Find the decimal quotient.
2
37
.................................................
4
38
.................................................
8
126
115
.................................................
2
15
.................................................
4
106
.................................................
8
17
.................................................
2
47
.................................................
4
57
.................................................
8
234
.................................................
2
59
.................................................
4
109
.................................................
8
4,153
.................................................
2
Set up and calculate.
139 ÷ 5
3
378 ÷ 24
95 ÷ 4
513 ÷ 6
Set up and calculate.
301 ÷ 2
6,897 ÷ 125
47,643 ÷ 375
4,023 ÷ 12
24,051 ÷ 15
3,647 ÷ 14
127
4
With 62 liters of oil, we filled exactly 8 identical cans.
What is the content of each can in liters ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
5
A piece of cloth is 27 m long; it is divided into 6 sheets f the same length.
What is, in meters, the length of each sheet ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
6
On a chocolate box, containing 36 bars, we can read «net weight 918 g».
What is the weight of a bar in grams ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
7
An empty box weighs 3 kg. When filled with 24 identical cans, it weighs 33 kg.
What is the mass of a can in kg ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
128
Dividing a number by a number
Decimal quotient
37
The dividend is less than the divisor
Let 8 be divided by 25.
32
8
Observe : = = 0.32 .
25 100
Ones
8
8
Tenths
Technique
Hundredths
0.32
80
– 75
5
25
50
– 50
00
0.32
25 8
–0
80
– 75
50
– 50
00
8 ÷ 25 = 0.32
0.32 is the decimal quotient of the division of 8 by 25.
1
What is the half of 8 ? of 3 ? of 1 ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
2
Write the following fractions in the form of a decimal number.
1
= .......................................
2
2
= .......................................
4
1
= .......................................
8
4
= .......................................
8
1
= .......................................
4
3
= .......................................
4
3
= .......................................
8
2
= .......................................
8
129
3
4
Find the decimal quotient in each of the following divisions.
26
13
25
7
38
19
14
35
45
18
66
33
70
49
92
138
One kilogram of nuts is divided equally among 5 friends.
What is, in kg, the share of each ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
5
An oil can of 5 liters is emptied into 40 identical bottles.
What is, in liter, the content of each bottle ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
6
A dictionary of 1,790 pages has a thickness of 182 mm. The cover has 3 mm of
thickness.
What is the number of papers in this dictionary ?
Calculate, in mm, the thickness of a sheet of this dictionary.
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
................................................................................................................................................................................................................................
130
38
Dividing a decimal
by a number
Let 34.12 be divided by 4.
Observe :
Ones
Tenths
34 .
– 32
2
1
Technique
Thousandths
8.53
2
4
21
– 20
1
12
– 12
00
8.53
34.12
4
– 32
21
– 20
12
– 12
00
34.12 ÷ 4 = 8.53
8.53 is the decimal quotient of the division of 34.12 by 4.
1
2
3
What is the half
of 3.8 ?
of 0.64 ?
of 1.09 ?
........................................................................
........................................................................
........................................................................
of 0.3 ?
of 0.1 ?
of 10.4 ?
........................................................................
........................................................................
........................................................................
What is the quarter
Find the decimal quotient in each of the following divisions.
13
846.56
24
257.28
58
2 819.96
66
5 858.82
53
38.796
131
Problems (Solve on your copybook)
1
2
Set up and calculate.
74.4 ÷ 6 = .........................................................................
5 206.5 ÷ 225 = .........................................................
3 194.36 ÷ 13 = .........................................................
45 565.38 ÷ 3 654 = .............................................
817 949.4 ÷ 402 = ..................................................
1 811.48 ÷ 506 = ......................................................
Mom used a piece of 15 m of clothes to sew 4 large curtains of the same length and a
small curtain of 2.4 m length.
What is the length of a large curtain ?
3
Hadi covered 802.1 m making 1,234 footsteps.
What is the length of his footstep in meters ?
4
A squared garden has a perimeter of 495.36 m.
What is, in meters, the length of its side ?
5
A rectangular garden has a perimeter of 103.8 m. Its width is equal to 21.5 m.
What is the length of its side ?
132
39
The divisor ending
with one or more zeros
Let us calculate the following divisions :
423,000 ÷ 1,800
468.7 ÷ 430
235
1,800
1.09
430
423,000
– 36
063
– 54
090
– 90
00
Delete
the
same
number of zeros from
the divisor and from the
dividend.
1
2
85,320 ÷ 36,000
2.37
36,000
46.8.7
– 43
0387
– 387
000
Delete one zero from
the divisor, move the
decimal point a digit to
the left.
85.320
– 72
133
–108
0252
– 252
000
Delete 3 zeros from the
divisor,
move
the
decimal point 3 digits to
the left.
Divide.
430
468.700
1,200
1,476,000
36,000 3,553,200
250
5,007.5
3,400
107,780
1,100
47,360
Set up and calculate (solve on the copybook).
10,500 ÷ 15,000
6,867 ÷ 2,100
36,400 ÷ 28,000
2,460.5 ÷ 350
133
40
Dividing a decimal
by a decimal
Activity
Observe and complete.
132 × 3 = 396 gives 396 ÷ 3 = ........................
13.2 × 0.3 = 3.96 gives 3.96 ÷ 0.3 = ........................
×10
×10
13.2 × 3 = 39.6 gives 39.6 ÷ 3 = ........................
When we multiply the divisor and the dividend by 10, 100
or 1,000 the quotient never changes.
4.8 12
1.25 2.875
120
125 287.5
To perform the divisions, we transform the divisor into a whole
number, we multiply the divisor and the dividend by the same
number 10, 100 or 1,000.
Remember
1
48
Find the decimal quotient.
0.3
18
18 ÷ 0.3 = ..........................................
0.4
2.5
.......................................................................
134
3.75
45
.......................................................................
0.25
0.5
.......................................................................
3.52
88
.......................................................................
7.35
13.23
.......................................................................
2
Calculate the quotient of 0.52 ÷ 1.3.
Multiply the dividend and the divisor by 7 and
calculate the new division. What did you notice for the
quotient ?
...............................................................................................................................................................
...............................................................................................................................................................
...............................................................................................................................................................
3
4
Set up and calculate.
108.3 ÷ 0.03
4.02 ÷ 0.5
316.827 ÷ 32.1
0.0585 ÷ 0.15
3.317 ÷ 1.07
7,176.23 ÷ 9.43
29.233 ÷ 0.943
35.5212 ÷ 2.53
1.268 ÷ 3.17
3.25 m of cloth are needed to sew a suit.
How many suits can me made with a 30 m cloth ?
.......................................................................................................................................................................................................................................
5
How many chocolate bars of 0.5 kg are there in a box containing 17.5 kg of
chocolate ?
.......................................................................................................................................................................................................................................
135
Problems (Solve on your copybook)
1
A chef bought a 50 m piece of cloth to cover 30 tables in a restaurant.
He used 1.85 m for the kitchen table.
a) How many tables in the restaurant can he cover if he needs 1.5 m for each
table ?
b) How much cloth does the still need to cover the rest of the tables ?
2
A car consumes 20 l of fuel in 250 km.
What is the distance covered per liter ?
3
The construction of a 300 m wall costs 9,360,000 L.L.
What is the cost of one meter ?
4
A person bought a second hand car for 15,300,000 L.L.
On delivery, he paid 4,500,000 L.L. and the rest in 30 monthly payments.
What was the amount of each monthly payment ?
5
I bought 3.5 kg of apples for 6,475 L.L. and 1.250 kg of lemon for 4,750 L.L.
What was the price of a kg of apples ? of a kg of lemon ?
6
A pan and a dozen of plates weigh 3.692 kg.
What is the mass of one plate if the mass of the pan is 3,002 g ?
7
136
How many bottles of 0.5 l can you fill with 45 l of oil ?
8
A barrel contains 162.75 l of oil.
How many buckets of 4.65 l can be filled to empty the barrel ?
9
A race of 42.3 km took place on a circuit that is 2.35 km long.
How many laps did the motorists do to finish the race ?
10
a) A barrel of milk weighs 30.280 kg when full; it weighs 3.5 kg when empty.
What is the weight of the milk in this barrel ?
b) One liter of milk weighs 1.03 kg.
How many liters of milk does the barrel contain ?
c) A truck carries 69 full barrels.
Calculate the weight load.
11
A barrel of capacity 96.9 l is filled with 9 buckets of 9.5 l each and 6 identical
bottles.
What is the content of each bottle ?
12
0.005 kg of coffee is needed to prepare a cup.
How many cups are prepared with one kg of coffee ?
13
To fence a rectangular field 97.8 m wide, we paid 705,375 L.L. The meter cost
1,650 L.L. .
a) Wha tis the perimeter of the field ?
b) What is the length of the field ?
14
A barrel of perfume contained 5 l. Hadi emptied its contents in flasks. He filled 5 flasks
of 15 cl each and 9 flasks of 0.2 l each.
a) What quantity of perfume did Hadi use ?
b) How many flasks of 12 cll each did he still need ?
137
41
Division - Decimal quotient
to the nearest lower or higher
Observe :
2.8
7
20
– 14
060
– 56
04
2.85
7
20
– 14
060
– 56
040
– 35
05
Since the division is not terminating,
Since the division is not terminating,
then the quotient is included between
then the quotient is included between
2.8 and 2.9.
2.85 and 2.86.
In the division of 20 by 7 :
In the division of 20 by 7 :
2.8 is the quotient to the nearest
2.85 is the quotient to the nearest
lower 0.1 or to the nearest lower
lower 0.01 or to the nearest lower
tenth.
hundredth.
2.9 is the quotient to the nearest
2.86 is the quotient to the nearest
higher 0.1 or to the nearest higher
higher 0.01 or to the nearest
tenth.
higher hundredth.
The quotient is said to be to the nearest tenth, if we limit the
division to the first digit after the decimal point.
Remember
The quotient is said to be to the nearest hundredth, if we limit the
division to the second digit after the decimal point.
138
1
Give the quotients to the nearest tenths by default and then by excess.
3
2
1,230
17
1,382
15,900
244,220
........................................................................
........................................................................
........................................................................
........................................................................
........................................................................
........................................................................
Divide using the calculator. Give the result to the nearest 0.01.
To the nearest lower
To the nearest higher
716,000 ÷ 2,800
936.4 ÷ 750
81,275 ÷ 46,000
4,758,000 ÷ 890,000
3,050.4 ÷ 730
5,721.35 ÷ 8,100
139
Problems (Solve on your copybook)
1
Give the quotient to the nearest lower or higher hundredths.
1,544 23,488.87
2
158,700 801,435
160
1,464.5
Calculate to the nearest 0.01.
3,800
3
287,500
94,000
760,000
4,720
5,148.3
A ribbon is 1 m long; it is cut into 3 equal parts.
What is the length of each part to the nearest hundredth ?
4
An oil barrel of 200 l is emptied into 15 identical cans.
What is, to the nearest hundredth, the content of each can ?
5
In a box, there are 4 congruent layers of chocolate pieces. In each layer there are
6 rows of 12 pieces each.
If the box weighs 3.5 kg, calculate the mass of one piece to the nearest lower
hundredth.
140
42
Surfaces and their area
The area of a square whose side is a unit of length, is often chosen as a unit of area.
1 cm
1 cm
1
cm2
The area of a square with one
centimeter as a side is the
centimeter squared (cm2).
1 dm
The area of a square with one
decimeter as a side is the
decimeter squared (dm2).
The area of a square with one
meter as a side is the meter
squared (m2).
1 dm
The meter squared (m2), its multiples and its submultiples are the units of area.
The multiples of square meters are : the kilometer squared (km2); the hectometer squared (hm2);
the decameter squared (dam2).
The submultiples of square meters are : the decimeter squared (dm2); the centimeter squared (cm2);
the millimeter squared (mm2).
1
What is, in cm2 , the area of each of the following surfaces ?
B
C
D
.........................................................
..........................................
.......................................................................
1 cm
1 cm
A
141
2
Observe the adjacent square ABCD.
A
B
Its side measures 3 cm.
How many squares of side 1 cm can you count inside square
3 cm
ABCD ? ......................................................................................................................................................................
D
3 cm
C
Complete :
The area of square ABCD = ........................ cm2 .
= 3 cm × 3 cm .
Area of a square = side × side.
Remember
3
Observe the adjacent rectangle EFGH.
E
F
Its lenth is 4 cm and its width is 3 cm.
How many squares of side 1 cm can you count
3 cm
inside rectangle EFGH ? ...........................................................................................
H
Complete :
The area of rectangle EFGH = ........................ cm2 .
= 3 cm × 4 cm .
Area of a rectangle = length × width.
Remember
4
Draw a right angle xAy .
Place a point B on [Ax) and a point C
on [Ay).
Join B to C.
The obtained triangle ABC is right-angled
at A. [AB] and [AC] are the sides of the right
angle of the right triangle ABC.
142
4 cm
G
5
The adjacent quadrilateral ABCD is a rectangle of dimensions
4 cm and 3 cm.
Draw its diagonal [AC].
Name the two right-angled triangles thus obtained.
A
4 cm
B
3 cm
3 cm
.......................................................................................................................................................................
D
C
4 cm
Calculate, in cm2, the area of rectangle ABCD. ......................................................................................
ABC and ACD are two triangles having the same area; calculate their areas.
.......................................................................................................................................................................................................................................
Area of a triangle ABC, right at B = (AB × BC) ÷ 2
Remember
6
(Half of the product of the sides of the right angle).
Calculate the perimeter and the area of rectangle MNPQ, of square ABEF, and of the right
triangle IJK below.
M
6 cm
A
N
4 cm
3.5 cm
2 cm
Q
P
F
7
I
B
E
K
J
3 cm
......................................................................................
.......................................................................
............................................................
......................................................................................
.......................................................................
............................................................
Observe the adjacent figure (solve on the copybook).
Find, in cm2, the area of each of surfaces A, B and C.
What fraction does the area of each of
the surfaces A, B and C represent
with respect to the total area ?
1 cm
1 cm
A
C
B
143
Problems (Solve on your copybook)
1
On a piece of land of 3,540 m2, the buildings cover a surface area of 1,640 m2.
Calculate the area of the part of land that can be planted.
2
The area of a rectangular land is 600 m2. One of its dimensions (length or width)
is 75 m.
What is the length of the other dimension ?
Calculate its perimeter.
3
Calculate the area of quadrilateral ABCD below.
A
9 cm
B
3 cm
D
C
3 cm
4
A garden has the form of a right triangle. The measure of one side of the right angle is
37 m and the other is 12 m less.
What is the area of the garden ?
5
A square field of side 39.45 m is fenced with 3 rows of iron wire.
a) What is the length of wire used ?
b) Calculate the area of the field.
6
A rectangular field of 125.3 m long and 80 m wide is fenced with a wire worth 1,800 L.L.
the meter.
a) How much money is paid for the wire ?
b) Calculate the area of this field.
7
836 m is the perimeter of a squared field.
Calculate its side, then its area.
144
8
Calculate the area of a rectangular land of perimeter 315 m and of width 68 m.
9
A rectangular carpet is 3.2 m long and 2.6 m wide.
Each m2 is sold for 7,450 L.L.
What is the price of the carpet ?
10
A squared garden is fenced with a wire. Dad paid 172,800 L.L. for a rate of 1,200 L.L.
the meter.
What is the area of this garden ?
11
A rectangular table of dimensions 1.2 m and 0.8 m is covered with a tablecloth that is
30 cm untucked from each side.
The m2 of this tablecloth is worth 37,250 L.L.
How much money has been paid for this tablecloth ?
12
What is the width of a rectangular field 112 m long and whose area is equal to
that of a square of side 56 m ?
13
Observe the drawing below.
32 m
x
3m
2m
10 m
y
5m
8m
8m
Calculate the measure of x and that of y.
Calculate, in m, the perimeter of this drawing.
Calculate, in m2, the area of the colored surface.
145
Problems (Solve on your copybook)
14
15
Two fields, one squared, the other rectangular, have the same perimeter. The area of
the rectangular field is 4,800 m2 and the length is 120 m.
a) Calculate the width of the rectangular field.
b) Calculate the side of the squared field.
The perimeter of a rectangle is 24.88 m; one of its dimensions is 5.50 m.
Calculate its area.
16
Calculate the area of the colored surface.
0.8 m
0.8 m
0.8 m
0.8 m
2.3 m
0.8 m
0.8 m
0.8 m
5.7 m
17
The dimensions of a rectangular field are 36 m and 10 m.
Each m2 is sold for 12,350 L.L.
Calculate its price.
18
19
Calculate the area of a rectangle of dimensions 16 cm and 4 cm.
Construct a square having the same area as this rectangle.
Calculate the perimeter of a square of side 5 cm. Cacule its area.
If we double the length of its side, what would its perimeter and its area be ?
20
2
The length of a rectangular field measures 63 m. Its width is of its length.
3
a) Calculate its width.
b) Calculate its perimeter.
c) Calculate its area.
146
0.8 m
43
Length of a circle (perimeter)
Area of a disc
3 cm
2c
m
(C1)
(C2)
6.28 cm
9.42 cm
Length of (C1) : 6.28 cm
Length of (C2) : 9.42 cm
Diameter of (C1) : 2 cm
Diameter of (C2) : 3 cm
6.28 ÷ 2 = 3.14
9.42 ÷ 3 = 3.14
4 cm
(C3)
Length of (C3) : 12.56 cm
Diameter of (C3) : 4 cm
12.56 ÷ 4 = 3.14
12.56 cm
Remark
The length of a circle
= 3.14 that is a value representd by a Greek
Its diameter
letter π (paï).
147
Observe the adjacent circle of radius
OA = 3 cm and of center O.
A
3
Color in yellow the surface limited by this
cm
circle. You will obtain a disc of center O and
of radius OA = 3 cm.
O
The area of this disc is : π × 3 × 3 = 28.26
28.26 cm2.
Length of a circle = π × diameter.
Length of a circle = π × 2 × radius.
area of a disc = π × r × r , where r is its radius.
Remember
1
Complete the table.
Circle C1
Diameter
in m
Circle C2
Circle C3
30
Length
in m
Circle C4
2.5
37.68
Radius
in m
Area of the
disc in m2
2
Observe and complete.
K is a point in ..................................................................
12.56
Circle (A)
U is a point in ..................................................................
148
Circle (B)
S
disc (A)
A
S is a point in ..................................................................
T is a point in ...................................................................
K
U
B
disc (B)
T
3
Draw a circle of center A and of diameter 4 cm.
Calculate its perimeter. Calculate the area of the
disc of center A and of diameter 4 cm.
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
.............................................................................................................................................
4
81.64 cm is the perimeter of a circle.
Calculate its radius then the area of the disc limited by this circle.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
5
Draw the circle of diameter [AB]. Calculate its perimeter
and the area of the disc limited by this circle.
B
..........................................................................................................................................................................
..........................................................................................................................................................................
A
..........................................................................................................................................................................
6
Calculate the area of the colored surface.
..................................................................
..................................................................
28 m
..................................................................
..................................................................
28 m
56 m
28 m
..................................................................
.......................................................................................................................................................................................................................................
149
7
Calculate the perimeter of each of the following figures.
6m
4 cm
4 cm
.........................................................
.........................................................
.........................................................
.........................................................
.........................................................
.........................................................
...................................................................................................................
...................................................................................................................
...................................................................................................................
2 hm
4 hm
150
8 dam
8 hm
...................................................................................................................
...................................................................................................................
...................................................................................................................
...................................................................................................................
...................................................................................................................
...................................................................................................................
8
A
Complete the circle of diameter [AB].
B
9
Calculate the area of the colored part.
......................................................................................................
......................................................................................................
......................................................................................................
20 m
10 m
......................................................................................................
......................................................................................................
......................................................................................................
......................................................................................................
......................................................................................................
10
The diameter of a wheel is 84 cm.
What is the distance covered by one turn ? by ten turns ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
11
The perimeter of a wheel is 251.2 cm. Calculate its diameter.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
151
12
The length of a circle is 753.6 cm.
Calculate its radius.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
13
Construct a square ABCD of side 8 cm.
Draw in the exterior of the square 4 semi-circles of diameters [AB], [BC], [CD]
and [DA].
Calculate the perimeter of the obtained figure.
Calculate the area of this figure.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
152
44
1
The day (d) ;
The hour (h)
;
the minute (min)
1 d = 24 h
1 h = 60 min
;
1 min = 60 s
;
;
the second (s).
1 h = 60 × 60 s = 3,600 s.
;
Convert.
2 h = 60 min × 2 = ......................................... min
6 h = ........................................ = ........................................ min
3 h = 60 min × 3 = ......................................... min
10 h = ...................................... = ...................................... min
240 min = 60 min × 4 = ............................ h
24 h = ...................................... = ...................................... min
...............
2
Time - Duration (1)
h = 60 min × ................................ = 300 min.
....................
h = ................................................... = 420 min.
Convert.
2 min = 60 s × 2 = .......................................... s
5 h = ..................................... min = ...................................... s
3 min = ........................................ = ........................................ s
30 min 25 s = .......................................... s
....................................
min = ..................................... = 300 s
2 h 50 s = .......................................... s
....................................
min = ..................................... = 600 s
3 h 20 min = .......................................... s
10 h = .................................. min = ..................................... s 6 h 15 min 45 s = ............................................... s
3
Count by 30 min from 2 h 30 min to 8 h 30 min.
.......................................................................................................................................................................................................................................
4
Count by 60 min from 3 h 25 min to 10 h 25 min.
.......................................................................................................................................................................................................................................
153
5
Calculate the following.
8 h 15 min 42 s + 3 h 25 min 13 s = ..................................................................................................................................
2 h 36 min 58 s + 4 h 29 min 32 s = ..................................................................................................................................
3 h 50 min + 4 h 3 min 10 s = ....................................................................................................................................................
8 h 52 min 35 s – 6 h 38 min 17 s = ...................................................................................................................................
17 h 45 min – 7 h 25 s = .................................................................................................................................................................
48 min 13 s – 15 min 20 s = ........................................................................................................................................................
6
Express in hours (h) and minutes (min).
1
1
3
7
h
3h
;
7h
;
6h
;
2
4
4
10
;
0.5 h.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
7
Complete.
35 min + .......................................... = 1 h
240 min = .......................................... h
27 min + .......................................... = 1 h 27 min
..........................................
..........................................
s + 40 s = 1 min
3,650 s = ................................... min .................................. s
8
min = 8 h
2 h 15 min = .......................................... min
516 min = ................................. h .................................. min
A cyclist left his village at 7 h 25 min in the morning and arrived at Tripoli at 2 h 10 min
in the afternoon. Calculate the duration of his journey.
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
9
A rocket moves at a speed of 12 km per second.
What is the distance covered in 1 h 30 min ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
154
45
Time - Duration (2)
Let us calculate (5 h 28 min 16 s) × 4.
×
5h
28 min
16 s
4
64 s = 60 s + 4 s
20 h
112 min
64 s
= 1 min + 4 s
20 h
112 min
1 min
60 s + 4 s
20 h
113 min
4s
20 h
1h
60 min + 53 min
4s
21 h
53 min
4s
113 min = 60 min + 53 min
= 1 h + 53 min
Then : 5 h 28 min 16 s × 4 = 21 h 53 min 4 s.
1
Complete.
(3 h 27 min 15 s) × 7 = ....................................
3h
27 min
×
105 s = 60 s + 55 s
15 s
7
............
h
..............
min
105 s
............
h
..............
min
60 s + ............ s
= 1 min + 55 s
1 min
2
............
h
190 min
............
s
............
h
+ 10 min
............
s
............
h
..............
min
............
s
Carry out the multiplication process in each of the following cases (solve on the
copybook).
(3 h 42 min 18 s) × 3
(12 h 45 min 28 s) × 6
(6 h 12 min 40 s) × 10
(10 h 48 s) × 5
155
Problems (Solve on your copybook)
1
A car takes 1 h 28 min 56 s to go from Beirut to Tripoli.
What time is needed to make three non-stop trips to and from ?
2
A bus leaves Tyre and arrives to Tripoli at 1 h 5 min in the afternoon after 4 h 38 min
on the road.
What was the departure time ?
3
A theater play consists of 3 acts.
The duration of each act is 25-minute interval. Two acts are interrupted by a 10-minute
interval.
The play starts at 18:15.
At what time does it end ?
4
The moon rotates around Earth in 29 days 12 hours 44 minutes 3 seconds.
The lunar year (the HEGIRA) is equivalent to 12 rotations of the moon around Earth.
What is the duration of the lunar year ?
What is the difference of time between the solar year (365 days and 6 h) and
the lunar year ?
5
Jupiter, the greatest planet, turns around itself in 9 h 51 min 10 s.
What is the duration of 6 rotations around itself ?
6
The planet Mars turns around the sun in 687 days and around itself in 24 h 37 min 15 s.
How many days does it need to turn 5 times around the sun ?
156
What is the duration of 5 rotations around itself ?
7
A car covers 250 km in 4 hours.
How many km are covered in one hour ?
Remark
8
The distance covered in one hour is called the average
speed km/h.
What fraction of an hour do 20 min represent ?
A cyclist covers 10.5 km in 20 min.
What is his average speed in km/h ?
9
A driver should cover 70 km. He leaves at 10 h 35 min, but the car breaks down a little
bit later.
He succeeds to restart his car 35 minutes later and arrives at his destination at 12h25min.
Calculate the time of the distance covered.
What fraction of the hour does this time represent ?
Calculate the average speed.
10
Lara uses her car to go to the movies 5 km far.
She drives at an average speed of 25 km per hour.
The session starts at 15 h 10 min.
At what time should she leave if she needs 3 minutes more to park her car ?
11
A man takes 30 minutes with an average speed of 50 km per hour to go to his
work.
How many kilometers does he cover ?
12
An airplane flies at an average speed of 900 km per hour.
What distance does the airplane cover in 1 min ? in 1 h 15 min ?
157
Problems (Solve on your copybook)
13
Two bicyclists leave at the same hour and move towards each other.
The first leaves Beirut at an average speed of 18 km per hour and the second leaves
Jbeil at 24 km per hour.
a) What distance does each one cover in 5 min ?
b) Knowing that the distance between the two cities is 40 km, what is the
distance between the two bicyclists 5 min after their departure ?
14
A truck, moving at an average speed of 40 km per hour, covers a distance in 5 hours.
What is the time needed for a car to cover the same distance at an average speed
of 90 km per hour ?
15
Sound propagates in the air at an average speed of 340 m per second.
What is, in km/h, the speed of an airplane having the speed of sound ?
What is the time needed by this airplane to cover 6,120 km ?
16
A driver leaves Beirut at 7 h 30 min.
During the trip, he stops 45 minutes to have lunch and 3 times, 15 minutes each,
to rest.
He gets to his destination at 13 h 30 min.
Knowing that his average speed is 70km per hour, what is the distance covered?
17
To cover a distance at an average speed of 40 km/h, a truck takes over 8 hours.
What time does a sports car need to cover the same distance, knowing that its
average speed is 160 km/h ?
158
46
1
Statistics
The following list shows the number of children in 50 families.
1
2
0
1
3
2
4
0
1
3
0
5
2
2
1
3
4
5
0
3
3
4
0
1
1
2
3
1
4
5
0
2
3
2
2
0
2
4
3
3
3
5
1
2
2
a) Complete the following table.
The number of children
0
1
2
2
1
2
1
3
3
4
5
The number of families
b) What is the number of families having :
4 children ? ............................................................................................................................................................................................
2 children at least ? ......................................................................................................................................................................
3 children at the most ? ............................................................................................................................................................
2
Out of 100 students in a school, there are 48 that are at least 11 years old, 34 that are
between 11 and 15 years and 18 that are more than 15.
Represent the numbers of this text in the table below.
The age
The number of students
3
a) Write the list of the grades of the last mathematics test your classmates did
(each grade is rounded to the nearest whole number.
...............................................................................................................................................................................................................................
...............................................................................................................................................................................................................................
...............................................................................................................................................................................................................................
b) Complete the table.
The grade
from 0 to 5
from 6 to 10
from 11 to 15
from 16 to 20
The number of students
159
4
Among the following television programs : Cartoons, sports, live broadcasting, which
is the most preferable ?
Ask this question to 10 of your friends and write down the results in a table.
5
The following figure is a histogram. It is formed of vertical rectangles where the height
indicates a quantity (here the average temperature).
the temperatures
25
20
15
10
December
November
October
September
August
July
June
May
April
March
February
January
5
the months
a) Indicate, for each month, the average temperature.
...........................................................
....................................................................
.................................................................
...........................................................
....................................................................
.................................................................
...........................................................
....................................................................
.................................................................
...........................................................
....................................................................
.................................................................
b) What are the six warmest months in the year ?
...............................................................................................................................................................................................................................
c) What is the coldest month in the year ?
160
...............................................................................................................................................................................................................................
6
a) Observe the table.
Months of year 1995
November
December
The quantity of rain in
Beirut (in mm)
19
24
January February
27
March
April
May
21
5
0
18
b) Represent the quantity of water in a histogram (solve on copybook).
c) What month did it rain the most ? the least ?
d) Write the months in decreasing order according to the quantity of rain.
7
A teacher gives the following grades upon correcting an assignment.
12
18
15
10
9
11
10
16
12
14
7
17
13
11
14
8
12
11
5
14
9
8
12
15
16
a) What is the lowest grade ? ...............................................................................................................................................
b) What is the highest grade ?
..........................................................................................................................................
c) Complete the table.
The grade
The number of
students
d) Draw the histogram corresponding to the table.
the number of students
5
4
3
2
1
the grade
e) What is the most frequent grade ?
.......................................................................................................................
161
8
Here are the mathematics grades obtained by Hadi, Diana and Zahi in 10 Exams in
the year 1997.
Hadi
11
7
12
11
13
12
11
11
15
12.
Diana
9
11
8
6
11
9
14
13
8
10.
Zahi
17
20
16
14
16
17
17
16
15
15.
Draw in your copybook.
a) three tables to write down the grades and the number of times thus obtained
by the three students.
b) a histogram to represent the grades of each of the three children.
9
1. Here are the temperatures listed
20°°
at noon during the last week of
15°°
March.
10°°
5°°
Mon Tues Wed Thurs Fri
Complete the table.
The day
Mon
Tues
Wed
Thurs
Fri
Sat
Sat
Sun
Sun
The temperature
20°°
2. Here is a bar diagram.
The days are indicated on the
horizontal line.
15°°
are
10°°
indicated on the vertical line.
5°°
The
temperatures
Mon Tues Wed Thurs Fri
Sat
Sun
a) What is the warmest day of the week ? ............................................................................................................
b) What is the coldest day of the week ? ...............................................................................................................
c) What are the days where the temperature is more than 15°° ? .............................................
...............................................................................................................................................................................................................................
162
10
Hadi got the following grades on
20
mathematics.
15
Monday: 12
;
Tuesday: 15
;
Wednesday: 10 ; Thursday: 11
10
and Friday: 18.
5
Mon Tues Wed Thurs Fri
Complete the following table and construct the corresponding bar diagram.
The day
Mon
Tues
Wed
Thurs
Fri
The grade
11
During the second week of February, the
abscences in a class were listed as follows :
Monday: 5 absences ; Tuesday: 3 absences ;
Wednesday: 4 absences ; Thursday: 7 absences
and Friday: 5 absences.
9
8
7
6
5
4
3
2
1
Mon Tues Wed Thurs Fri
Complete the following table and construct the corresponding bar diagram.
The day
Mon
Tues
Wed
Thurs
Fri
The number of
absences
163
12
In a school, 360 students of the second cycle are
F
divided into 6 groups according to their favorite game.
P Ping pong
B Basketball
T Tennis
F Football
G Gymnastics
V
T
This is shown in the following disc (circular Diagram).
V Volleyball
B
P
G
a) What fraction of the disc does each of the preceding games represent ?
V
.....................................................
F
.....................................................
T
.....................................................
B
.....................................................
P
.....................................................
G
.....................................................
b) Calculate the number of students in each group.
Volleyball: ..................................
Football: .......................................
Ping pong: .................................
Basketball: ................................
Tennis: ..........................................
Gymnastics: ............................
c) These 360 students are divided in the following way :
180 students in Gr 4, 90 students in Gr 5, and 90 students in Gr 6.
1. Complete the table.
The class
The number of students
2. What fraction, of all the students, does the number of students in each class
represent ?
..................................................................
..................................................................
3. Represent these results in a circular
diagram.
164
.................................................................
13
Observe the adjacent circular diagram that represents the expenses and the
savings of a family in a month.
Housing
Food
25%
30%
5%
Leisure
10%
30%
Clothes
Savings
30
for example 30% means 100
a) Knowing that the monthly income of that family is 2,150,000 L.L., complete
the table.
Housing
Food
Clothes
Leisure
The
expenses
b) What are their expected savings per month ?
.......................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................
165
47
1
Homothecy
Redraw the adjacent figure (squared sheet) :
a) Multiply all the dimensions by 2,
b) divide all the dimensions by 2.
a)
2
Find the rule of construction and
construct the following house.
166
b)
3
Observe point O and the triangle ABC below.
Draw the semi-straight line [OA) and take, on this semi-straight line, point E
such that OE = 2 OA (E is on the right of O).
Draw the semi-straight line [OB) and take, on this semi-straight line, point F
such that OF = 2 OB (F is on the right of O).
Draw the semi-straight line [OC) and take, on this semi-straight line, point
G such that OG = 2 OC (G is on the right of O).
B
O
5
4
A
3
C
EFG is an expansion of ABC at the scale 2.
1
ABC is a reduction of EFG at the scale .
2
Complete the table.
AB = .........................
AC = .........................
BC = .........................
× ......
............
EF = .........................
EG = .........................
FG = .........................
Use you protractor to measure the angles of triangles ABC and EGF and complete.
BAC = ...................................
ABC = ...................................
ACB = ...................................
EFG = ...................................
FGE = ...................................
FEG = ...................................
What do you notice ? ..............................................................................................................................................................
...............................................................................................................................................................................................................................
167
4
Redraw.
a) At the scale 3.
2 cm
O
2 cm
2 cm
2 cm
O
1
b) At the scale .
3
6 cm
6 cm
6 cm
c) At the scale 1.5.
4 cm
O
3 cm
168
5
O
C
A
K
P
M
B
J
N
MNJP is an expansion of ABKC at the scale 3.
K is the midpoint of [BC], verify with a compass that J is the midpoint of [NP].
169
Problems (Solve on your copybook)
1
The measures of the sides of a triangle ABC are 17 mm, 35 mm and 24 mm. Expand
it at the scale 5.
Calculate the sides of the obtained triangle EFG.
Calculate and compare the perimeters of these two triangles.
2
1
A rectangle has as dimensions 68 m and 56 m. Reduce it by .
4
Calculate and compare the perimeters of these two rectangles.
3
Observe the adjacent rectangle.
Calculate its area.
A
Place point E on [AB) such that AE = 2AB.
3 cm
B
2 cm
Place point G on [AD) such that AG = 2AD.
Complete the rectangle AEFG.
Calculate the area of the new rectangle.
Compare the two areas.
170
D
C
48
1
Cube
Is the sum of numbers of the dots on the two
opposite faces of a dice 7 ?
................................................................................................................................................
2
Here is the pattern of a dice. Redraw it on a squared sheet of paper.
Make this dice and mark the dots on each face. (Use an adhesive ribbon for the
final sticking).
3
The dice has the form of a cube.
A, B, C, D, E, F, G and H are the vertices of the cube.
a) The cube has 12 edges, [AB] is an edge.
H
G
Name the other edges. .......................................................................
E
F
...........................................................................................................................................
b) The cube has 6 faces which are squares. ABCD
D
C
is a face.
Name the other faces. .........................................................................
...........................................................................................................................................
A
B
171
4
a) This is a pattern of a cube.
Redraw it on a squared sheet of paper.
Form it (use adhesive ribbon).
b) Do two opposite faces have the same color ?
...............................................................................................................................................................................................................................
5
a) Without constructing the solid of this pattern, color with the same color the
opposite faces.
b) The unit of area is the square.
Calculate the total area of the faces of this solid.
...............................................................................................................................................................................................................................
...............................................................................................................................................................................................................................
172
49
Pyramid
Observe the solids set down on the board.
S
S
S
S
All these solids are pyramids.
A pyramid is a solid whose :
base is a polygon: a triangle, a quadrilateral...
faces are triangles having a common vertex called the vertex of the pyramid.
1
Draw the following pyramids.
S is the vertex, square ABCD is the base.
S is the vertex, triangle ABC is the base.
S
S
A
D
A
B
C
C
B
173
50
Pyramid of squared base
S
1
This solid is a pyramid of vertex S and of square
ABCD as base.
How many faces are there ? .............................................
Name them. .............................................................................................
D
A
..................................................................................................................................
B
2
C
Here are two patterns of two pyramids having a square for base.
Redraw them on a squared sheet of paper.
Form these pyramids (use adhesive ribbon).
5
4
5
4
4
4
3
174
5
51
Pyramid of triangular base
(tetrahedron)
1
S
This solid is a pyramid of base the
triangle ABC and of vertex S.
Name its faces.
.............................................................................................
A
C
.............................................................................................
B
2
a) Redraw this pattern. Use it to form a pyramid of a triangular base.
b) Point out in the obtained solid the vertex of the pyramid.
c) Measure the length of its sides.
................................................................................................................................................................................................................................
175
52
Right prism
A
B
This solid is a right prism having the
triangles ABC and DEF as bases.
C
The bases are congruent.
A right prism is a solid whose faces are
D
two rectangles.
E
The bases of the two polygons are
F
congruent.
1
Here is a pattern of a
right prism of a
triangular base.
a) Redraw it on a
squared paper.
b) Form it.
(Use adhesive ribbon
for the final sticking).
c) Name the vertices of this right prism. ................................................................................................................
d) Name the bases. .............................................................................................................................................................................
176
Problems (revision)
1
The difference of two numbers is 7.5 .
One of these two number is 13 .
What is the other number ? (Give the two solutions).
2
A farmer collects 80 eggs in the morning and 56 eggs at noon.
At night, he sold 8 dozens.
Complete the following text with the suitable numbers :
In the morning, the farmer collects …............ tens of eggs. At noon, he collects …............ eggs
less than the morning. He collects ….............. eggs in all. At night, he sold ….............. eggs.
After selling, …............ tens of eggs. He put then the rest in …............ packets of 8 eggs each.
3
In the adjacent figure, the side of the largest
equilateral triangle is 4 cm.
Each of its sides is divided in four equal parts.
What is the total number of equilateral triangles
of this figure ?
4
A merchant bought 8 dozen of plates for 2,500 L.L. each.
He sold each dozen for 36,000 L.L. .
What profit did he realize after this selling ?
5
Six biscuits cost 2.4 $.
How much costs one dozen and a half of these biscuits ?
177
Problems (revision)
6
A father distributed a sum of money between his three sons.
1
2
He give of this sum to the eldest, to the younger and the rest to the middle.
5
3
a) What fraction of this sum of money did the middle son receive ?
b) Calculate the part of each of them if the sum distributed ascends to 60 $.
7
In a school, the courses begin at 7h 50 min in the morning.
During a day’s study, there are 7 periods of 50 minutes each, and two recesses : one
of which is 20 minutes and the other is 25 minutes.
At what time did the students leave the school ?
8
A full gallon of oil weighs 4.5 kg.
Half full, it weighs 2.6 kg.
How much did the empty gallon weigh ?
Olive
oil
Olive
oil
Olive
oil
4.5 kg
2.6 kg
?
9
Complete the following table :
Angle
Measure in degree
c
Nature
b
aOb
aOc
aOd
178
bOd
d
O
a
Problems (revision)
10
A can contains 28 liters of water.
We fill 7 gallons of 2.5 liters each.
a) What quantity of water does remain in the can ?
b) We filled the rest in bottles of 75 cl each.
How many bottles could we fill ?
11
In the adjacent figure, we have :
A
B
ABDE is a square.
5
BDC is a triangle rectangle in D.
cm
a) Calculate the perimeter of this figure.
b) Calculate the area of this figure.
E
12
C
D
3 cm
4 cm
At the same pastry :
Aline bought one tart and 2 cakes : she paid 4,250 L.L.
Noura bought one tart and 5 cakes : she paid 8,000 L.L.
a) Find the price of one cake ?
b) Find the price of one tart ?
13
A circular table has a radius of 1.50 m.
a) Calculate the perimeter of this table.
b) Calculate the area of this table.
c) How many persons could be sit around this table if each person needs 78.50 cm ?
179
Problems (revision)
14
The adjacent rectangle is formed by five identical rectangular labels.
a) Calculate the perimeter of each label.
b) Calculate, by two different ways, the area of each label.
24 cm
20 cm
15
Using the digits 1; 2; 3 and 4, without repetition, How many numbers of four
digits can we obtain ?
180