Mathematics 5
AFAQ Mathematics Sun Series has been developed strictly in accordance with the
National Curriculum Guidelines of the Ministry of Education, Government of Pakistan.
About This Book
PREFACE
AFAQ Mathematics Sun Series comprises ten books from Beginner to
Grade 8. It covers all the basic concepts and skills necessary to build a solid
foundation for today’s young mathematicians.
The series focuses on five core areas of Mathematics including
Numeration System, Algebraic Concepts, Measurements, Geometry and
Spatial Sense, and Handling Data.
The series focuses on activity-based learning to ensure active
participation of the learners while nurturing mathematical communication,
thinking and problem solving skills. The content is developed in Islamic and
cultural context to ensure value-oriented learning connected with the real
world.
The structure of the series has been made meaningful with the
objective statements at the start of each concept, sufficient examples,
practice activities, important points, highlighted key words, open-ended
questions and notes for teachers to facilitate effective learning.
Appealing layout, colourful illustrations and familiar language
attract the learners as well as promote independent and self-directed
learning.
AFAQ Mathematics Sun Series will also be supported by the teacher
guides and support material i.e. flashcards, cut-outs, yearly planners,
assessment sheets and examination papers, etc. This attempt to enhance
interest in Mathematic learning can be improved by feedback of our
respectable teachers and other Maths learners. We will appreciate your
valuable feedback and suggestions to make the series more useful for
young learners.
May Allah guide and help us! (Amen)
AFAQ Mathematics Sun Series distributes the content in proper chunks and
reflect through interesting activities. This approach is consistent throughout the
book. The given sample pages can describe it in a better way.
Unit Number
Unit Heading
Objectives at the start of each
topic help teachers to focus
clearly on the concept.
Important points
regarding the
concept are given
under this heading.
Clearly written instructions direct the
learners to perform the given activities.
Sufficient practice through a variety of
activities strengthens understanding.
Director, Research
Open-ended questions and questions
involving mental strategies are given here
to promote independent learning.
Visit
to
a
Dentist
Contents
Unit
1
Whole Numbers
Objectives
Unit
No.
1
Topic
Page No.
By the end of this unit, the students will:
read and write whole numbers up to 1,00,00,00,000 ( 1 arab) with their names.
identify the place values of digits up to 1 arab in Pakistani way.
Whole Numbers
1
identify the place values of digits up to 1 billion in international way.
tell the relationship between the Pakistani and international place value systems.
read and write numbers in different forms.
2
Algebraic Operations on
18
Whole Numbers
compare whole numbers up to 1,00,00,00,000 by using terms (less than, equal to,
greater than) and symbols (<,=,>).
arrange the numbers in ascending and descending order.
identify the smallest and the greatest numbers up to 10-digits.
3
Factors and Multiples
26
round off whole numbers up to 1000.
analyze, extend and create arithmetic and geometric patterns.
4
Common Fractions
35
5
Decimal Fractions
45
6
Unitary Method
72
7
Average
79
8
Geometry
85
9
Tessellations
101
10
Perimeter, Area and Volume
105
11
Graphs
Answers
116
122
1.1 8 and 9-Digit Numbers
1.1.1 Crores and 10 Crores
Al-hamdu-lillah we can count , read and write the numbers up to one
crore “1,00,00,000” which is the smallest 8-digit number.
We know that:
100 thousands = 1 lac
10 lac
= 1 million
10 million
= 1 crore
Can you tell?
How many lacs are there in 1 crore?
What is the predecessor of 1 crore?
What is the successor of 1 crore?
The successor of 9 (the greatest 1-digit number ) is 10 (the smallest 2
digit-number).
Now complete the following pattern:
The successor of 99 is
The successor of 999 is
.
.
01
The successor of
9,99,99,999 is the greatest 8-digit number.
is 10,000.
The number next to 9,99,99,999 is 10,00,00,000.
The successor of 99,999 is
.
The successor of
We read it as ten crore.
Here is the place value chart for ten crore:
is 10,00,000.
The successor of 99,99,999 is
.
In the above pattern, all the numbers on the left are the "predecessors" of
the numbers on the right side.
Try Yourself
C
TL
L
T Th
Th
H
T
U
1
0
0
0
0
0
0
0
0
Ten crore is the smallest 9-digit number.
1. Read the following numbers and write their successors:
(b) Can you find a relationship between the successor of one row and
the predecessor of the next row?
(c) Continue the pattern to the next row to find the greatest 8-digit
number. What is its name?
Write the following number names in numerals in the given place value
chart. Also write them in standard form.
Number Name
(1) Three crore, fifteen
lac, thirty-eight thousand
and thirty
(2) Six crore, eight lac,
five thousand, two
hundred and fifteen
(3) Five crore, twenty -one
lac, forty thousand and
forty
(4) Nine crore, seven lac,
one thousand and five
hundred
(5) Four lac, seventy -six
thousand, two hundred
and fifty-eight
02
Units
Thousands
TC
(a) Observe the number of digits of each predecessor and successor.
What do you notice?
Lacs
Crores
C
TL
L T Th Th
H
T
U
Standard
Form
(i)
10,00,00,000
(ii)
10,00,95,999
(iii)
12,12,12,121
(iv)
27,35,00,029
(v)
80,99,99,999
(vi)
64,08,18,309
(vii) 53,21,73,400
(viii) 79,44,00,291
(ix)
33,08,11,111
(x)
40,98,00,198
2. Write the number names of the following numbers:
(i) 1,24,63,500
(ii) 20,17,36,377
03
(iii)
52,12,589
1.2 Introduction to 1 Arab
The greatest 9-digit number is 99,99,99,999.
Its successor is 1,00,00,00,000, which is the smallest 10-digit number.
(iv)
3,40,67,709
Its name is one arab.
Look at the place value chart of one arab:
(v)
(vi)
71,24,80,100
6,50,44,009
Arabs
Lacs
Crores
Units
Thousands
A
TC
C
TL
L
T Th
Th
H
T
U
1
0
0
0
0
0
0
0
0
0
We write one arab in standard form as "1,00,00,00,000". The 4 commas
(vii) 9,23,51,251
separate the periods of units, thousands, lacs, crores and arabs.
Can you tell?
(viii) 2,16,74,502
How many crores are there in 1 arab?
1.2.1 International Place Value System
(ix)
88,23,40,030
We are familiar with the numbers up to 8-digits in international place value
system also.
(x)
6,15,000
We know that the smallest 8-digit number in international place value
system is called "10 million".
Can you tell?
How do we write 10 million in standard form?
Try Yourself
Using the digits 4,2,7,9,5,1and 6 make the numbers that are:
(i) greater than 50,00,000 (any 3 numbers).
(ii) greater than sixty lac and has the even numbers at units place.
(iii) between fifty lac and sixty lac having the digit at thousands place
double than the digit at hundreds place.
For
Teacher
04
Write different numbers up to ten crore on board, and ask the students to read the
numbers efficiently.
1. Place commas to write the numbers in international place value
system. Also write the number names.
(i)
1215396
(ii)
33371580
(iii)
5302325
(iv)
67888980
(v)
1512171
(vi)
2309745
(vii) 80572441
(viii) 13994067
(ix)
(x)
4282169
35499917
05
We are familiar with the periods of units, thousands and millions in
international place value system.
The period of millions completes on the 9-digit numbers.
The smallest 9-digit number is called 100 million.
Look at the place value chart for 100 million:
Billion
HM
TM
M
HTh
TTh
Th
H
1
0
0
0
0
0
0
T
0
HM
TM
M
1
0
0
0
0
(v)
Th
H
T
U
0
0
0
0
0
0
Commas help us to distinguish the place values of digits in numbers.
For example:
We read
One hundred and thirteen million
Thirty million
Nine million
Four hundred and forty-five
million and forty-five
Ninety-eight million and two
TTh
Remember: One billion = One arab
One hundred million is written as 100,000,000 in standard form.
(i)
(ii)
(iii)
(iv)
HT h
We write one billion in standard form as 1,000,000,000.
U
Remember: 100 million=10 crore
2. Write the numerals for the following number names:
Units
Thousands
Millions
B
Units
Thousands
Millions
The place value chart for 1 billion is as follows:
6,371,533,780
as:
Six billion, three hundred and seventy-one million, five hundred and thirtythree thousand, seven hundred and eighty.
Whereas we read
6,37,15,33,780
as:
Six arab, thirty-seven crore, fifteen lac, thirty-three thousand, seven
3. Write the number names for the following numerals after placing
hundred and eighty.
commas in international way:
Place commas to write the given numbers in Pakistani and international
(i)
42519800
(ii)
629015279
(iii)
65028040
(iv)
1187650
(v)
526791538
(vi)
23019850
(vii) 778413990
(viii) 54962212
Try Yourself
Add 1 to the greatest 9-digit number i.e. 999,999,999. Which number
do you get?
How many digits are in it? -------------------------------------We say it one billion. One billion adds the fourth "period of billions" to the
international place value chart.
06
place value systems respectively. Also write the number names.
(i)
62315805
(ii)
541729606
(iii)
3600835911
(iv)
7605888
(v)
92354266
(vi)
3216458
(vii) 8598030667
(viii) 517499803
(ix)
(x)
For
Teacher
45501609
51247653
Write different numbers up to one billion on board, and ask the students to read the
numbers efficiently.
07
1.2.2 Place Value of Numbers
3. Write the number that has:
1. Observe the commas carefully, then write the value of the bold digits
in words.
Example
650,389,260
(i)
91,756,523
(ii)
13,28,60,500
(iii)
1,76,00,251
(iv)
92,523,005
(v)
120,456,651
(vi)
6,247,329,508
(vii)
25,67,068
Fifty million
(i)
6 arabs, 26 crores, 3 lacs, 15 thousands and 6 hundreds.
(ii)
5 crores, 35 lacs, 6 thousands, 1 ten and nine units.
(iii)
40 crores, 7 lacs, 4 hundreds and 8 units.
(iv)
2 arabs, 16 crores, 53 thousands, 5 hundreds and 2 units.
(v)
1 crore, 1 lac and 2 tens.
Try Yourself
Find a 9-digit number with all odd identical digits having a sum of 45.
(viii)
8,63,25,10,999
(ix)
32,15,80,176
(x)
92,523,005
1.2.3 Comparing Whole Numbers
We can use the concept of place value to compare and order big numbers.
2. Write the following in the expanded form:
(i)
51,31,02,957
(ii)
63,529,702
(iii) 17,25,69,832
(iv) 2,91,50,75,291
(v)
08
12,223,560
1. Think and write "your own numbers" to make the given statements true.
(i)
4,24,60,550 <
(ii)
7,10,75,506 =
(iii)
83,067,129 <
(iv)
4,62,81,920 >
(v)
8,15,08,360 <
(vi)
951,260,355 <
(vii) 17,600,089 <
(viii) 1,29,86,53,211>
(ix)
(x)
326,589,201 <
6,25,398 <
09
(ii) 67767762, 67672776, 67777662, 66277777
2. Complete the following statements:
(i)
10 more than 1,52,65,238 is
(iii) 8592689, 8568929, 5896982, 8925689
(ii) 10,000 less than 6,00,66,050 is
(iv) 34534555, 54334555, 55554433, 33445555
(iii) 100 less than 58,03,275 is
(v)
632857910, 6091753248, 896417523, 889461752
(iv) One lac more than 61,129,590 is
(v) 1,000 more than 5,23,20,761 is
3. Arrange the following numbers in descending order:
(i)
52190256, 50912652, 51965201, 59012561, 5695102
The place value concept is also helpful to form the smallest and the greatest
numbers by using the given digits.
Example:
Form the smallest and the greatest numbers from the digits 1,0,5,9,3,5,7,
and 1.
To form the smallest number we simply arrange the numbers in ascending
(ii)
72516333, 72351633, 75633312, 67521333
order as: 011,35,579.
Repeating numbers are written consecutively. Note that '0' in the start has
(iii) 111248427, 11184247, 11188424, 111788422
no value, so the number actually consists of 7-digits.
The greatest number is 9,75,53,110 that is obtained by arranging the
(iv) 300265391, 303126539, 29999999, 300623591
digits in descending order.
Make the smallest and the greatest numbers using the given digits:
(v)
586432197, 591326847, 865432197, 612345789
Try Yourself
Write five 10-digit numbers and arrange them in descending order.
4. Arrange the following numbers in ascending order:
(i)
10
2060212, 2062021, 16902102, 2200616
Digits
Smallest Number
Greatest Number
(a)
6,2,5,7,1,2,8
(a)
(a)
(b)
3,1,0,2,9,6,5
(b)
(b)
(c)
3,2,4,9,8,4,0,2
(c)
(c)
(d)
1,2,1,5,2,1,5, 1, 2
(d)
(d)
(e)
5, 6,7,2,8,4,1,3,9,0
(e)
(e)
11
1.3 Rounding Off Numbers
Round the numbers to the nearest 10, the number line can help you.
Read the following sentences:
(a) Amna reads the book in about one month.
(b) The time was about half past 3.
(c) Ayesha is about ten years old.
(d) Gujranwala is about 100 kilometres from Lahore.
When someone asks us the time, we often tell an estimate.
For example, rather than saying 13 minutes and 15 seconds past six, we
might say that "the time is about quarter past six".
Another example is of age. We never say that our age is 12 years 1 month
and 5 days. Rather, we tell people our age nearest the years.
This is called rounding off numbers.
1.3.1 Rounding Off to the Nearest 10
To understand the rules of rounding off, see the following examples:
(i) Round off 38 to the nearest 10.
We locate 38 on a number line between 30 and 40 as:
30
38
38 is more nearer to forty than 30, so we round off 38 to 40.
(ii)
(i)
6
(ii)
113
(iii)
21
(iv)
259
(v)
57
(vi)
525
(vii)
45
(viii)
888
(ix)
84
(x)
995
40
Round off 23 to the nearest 10.
20
23
30
We round off 23 to twenty and not to 30 because 23 is nearer to 20.
(iii) Round off 15 to the nearest 10.
1.3.2 Rounding Off to the Nearest 100
Example: Round 157 to the nearest 100.
157
100 110 120 130 140 150 160 170 180 190 200
Since 157 is more than half way between 100 and 200, so it is rounded off
to 200.
The following are the rules for rounding off to the nearest 100.
10
15
20
15 is equally distant to 10 and 20. In such cases, we round the number
upwards. So 15 is rounded off to 20.
We summarize these rules as follows:
1. If the unit digit is less than 5, round it back to the nearest 10.
2. If the unit digit is exactly 5, round it next to the nearest 10.
3. Similarly, if the unit digit is greater than 5, round the number to
the next 10.
12
1. If the number formed by the last two digits is less than fifty, the n u m b e r
is rounded back to the nearest 100.
2. If the number is equal to or greater than fifty, the number is
rounded up to the next hundred.
Round these numbers to the nearest 100.
(i)
68
(ii)
251
13
(iii)
318
(iv)
389
(v)
150
(vi)
404
(vii)
677
(viii)
1090
(ix)
548
(x)
1125
1.4 Patterns
Patterns make our world beautiful. We see many patterns in our
surroundings. Repeating patterns, growing patterns, shrinking patterns,
rotating patterns and so on.
Patterns play an important role in Mathematics. Patterns with numbers are
not only interesting, but are also useful specially for mental calculations and
help us to understand properties of numbers better. Number patterns
(sequence) are of different types:
1.3.3 Rounding Off to the Nearest 1000
Let us study two major types of patterns:
Example: Round off 1350 to the nearest 1000.
We draw the number line,
1.
2.
1000
1350
1500
2000
Here we have to see the number formed by the last three digits. Since 350 is
less than 500, we round off the number 1350 back to the nearest thousand i.e
"1000".
1.4.1 Arithmetic Patterns
Round off these numbers to the nearest 1000.
(i)
1351
(ii)
Arithmetic patterns
Geometric patterns
An arithmetic pattern is a sequence of numbers in which the difference
between any two consecutive numbers is the same.
Example:
Extend the following pattern to the next two terms:
5, 9, 13, 17, 21, _ _ _ _ _ _ _ , _ _ _ _ _ _ _ _
2800
We need to find the rule of the pattern to extend it. So we proceed as follows:
(iii)
(v)
6590
2325
(iv)
(vi)
3575
5620
Step 1
The numbers in the pattern are getting larger indicating that we have to
add something to find the next term.
Step 2
(vii)
5291
(viii)
8590
Finding the difference of some consecutive pairs of numbers, we have
9 5=4,
13 9 = 4, 17 13 = 4
(ix)
6875
(x)
6500
The difference of any two consecutive numbers is 4. This means, we have to
add 4 to a number to obtain the next number in the pattern. So the rule of the
pattern is add 4 to each term indicating that it is an arithmetic pattern.
Try Yourself
(i) Write the numbers that are rounded off by 1000 to 3000 and the sum
of their digits is 15. (Hint: Consider the number 2571.
Sum of its digits = 2+5+7+1 = 15
And 2571 is 3000 when rounded off to the nearest 1000.)
(ii) Round off these numbers to the nearest 10, 100 and 1000.
(a) 1755
(b) 998
14
Step 3
Adding 4 to the last given term of the pattern.
21 + 4 = 25 and 25 + 4 = 29
Therefore, the last 2 terms of the pattern are 25 and 29.
15
Complete the missing terms in the following arithmetic sequence by
finding their rules:
(i)
1, 3, 5, 7, 9, _ _ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _
(ii)
12, 16, 20,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ , 40
1.4.2 Geometric Patterns
A geometric pattern is a sequence of numbers in which each term is
obtained by multiplying the preceding term by a fix number.
Example:
Extend the following sequence to the next three terms:
4, 8, 16, 32, _ _ _ _ _ , _ _ _ _ _ , _ _ _ _ _
(iii) 46, 49, 52, 55, _ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _, 67
(iv) 15, 22, 29,_ _ _ _ _ ,_ _ _ _ _,_ _ _ _ _, 57
We find that it is not an arithmetic pattern, because the difference of each
consecutive pair of terms is not the same.
(v)
51, 54,_ _ _ _ _, 60, 63,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _
If we observe the pattern carefully, we can break the pattern as:
(vi) 76, 87, 98,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _
(vii) 118, 127, 136,_ _ _ _ _ ,154, _ _ _ _ _, 172, _ _ _ _ _
(viii) 515, 605, 695,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _ ,_ _ _ _ _
(ix) 1039, 1056, 1073,_ _ _ _ _ , 1107, _ _ _ _ _,_ _ _ _ _
(x)
4, (2x4), (2x8), (2x16), ...........
So the rule of the pattern is;
Multiply each term by 2
Therefore, we find the next three terms of the geometric pattern by
multiplying each previous term by 2 as:
2 x 32 = 64
2125, 2150,_ _ _ _ _ ,_ _ _ _ _, 2225, _ _ _ _ _ , _ _ _ _ _
2 x 64 = 128
Try Yourself
2 x 128 = 256
Extend the following pattern. State the rule and type of the pattern also.
So the pattern is 4,8,16,32,64,128,256.
Extend the following geometric patterns by writing the missing terms.
3
Rule:
Type:
16
9
15
(i)
2, 6, 18, _ _ _ _ _ , _ _ _ _ _ , _ _ _ _ _
(ii)
5, 10, 20 , _ _ _ _ _ , _ _ _ _ _ ,_ _ _ _ _
(iii) 2, 4, 8, 16, _ _ _ _ _ ,_ _ _ _ _ , 128
(iv) 3, 6, 12, _ _ _ _ _ , _ _ _ _ _ , 96 ,_ _ _ _ _
(v)
1, 3, 9, _ _ _ _ _ 81, _ _ _ _ _
17