20140918 Chapter 0 – Basic Mathematics; Chapter 1 – Directed Numbers and the Number Line
Revision
1.
Fractions
(a) Different types of fractions
Type
Proper fraction
Improper fraction
Mixed number
Example
2 3
,
7 5
8 7
,
5 3
1
2
2 , 4
3
5
(b) Operations of fractions
Operations
Example
Addition and Subtraction
3 2 1 5 1 10 7
3
+ − = − = − =
7 7 2 7 2 14 14 14
Multiplication and
7  1 3 7 7 3 7 7 7 4 1
÷2 ×  = ÷ ×  = ÷ = × =
8  3 4 8 3 4 8 4 8 7 2
Division
2.
Directed numbers
Numbers that carry either a positive sign ‘ + ’ or a negative sign ‘ − ’ are called
directed numbers. They include:
(i)
1
positive numbers, e.g. +1 , +2.3 , +4 .
2
1
(ii) negative numbers, e.g. − , −3 , −4.6 .
2
Note:
3.
0 is not a directed number. It is neither positive nor negative.
The number line
positive
negative
directioin
−3
−2
−1
0
+1
+2
+3
directioin
(a) For any two numbers on a number line, the one lying on the right is always
greater than that lying on the left.
(b) All positive numbers lie on the right of ‘0’, and all negative numbers lie on
the left of ‘0’.
1
20140918 Chapter 0 – Basic Mathematics; Chapter 1 – Directed Numbers and the Number Line
Exercise (Chapter 0)
Find the result of each of the following.
1.
4 6
+
13 13
2.
3 1
+
8 5
3.
11 9
−
15 15
4.
2 1
−
7 8
5.
2 −1
2
5
6.
4 9 1
+ −
7 14 2
7.
4 2 7
− +
5 3 10
8.
7 3
×
12 14
9.
8 5
×
25 16
10. 5 × 2
11.
1 4
×1
3 5
12.
4 16
÷
7 21
7
13
14.
11
7
÷6
12
8
28 3 5
÷ ×
35 7 14
16.
1
1
5
+2 +7
8
6
12
13. 9 ÷ 5
15.
5
7
1
17. 4 − 3 − 1
6 12 4
13
20
1
3
13
18. 8 − 5 + 2
3 10
15
19.
5
1 2
×3 ÷
11 7 3
3
1 9
20. 2 ÷ 9 ÷
4
3 14
21.
13 1 7
+ ×
16 4 8
22. 1
1  4 2
4
23. 11 ÷ 1 −  − 5
3  5 3
7
17 5
2
− ÷2
24 6
5
1  9
 7

24.  6 − 5  ×  3 − 2 
4   11 
 10
25. William buys a box of apple juice of capacity 1
morning,
1
1
L . He drinks
L in the
2
5
1
5
L in the afternoon and
L at night. How much apple juice is
4
8
left?
26. Mrs Wong has a bog of flour of weight 3
make a birthday cake and then 1
2
2
kg . She uses half of the flour to
5
1
kg to make pancakes. Later Mrs Wong buys
4
1
kg of flour, how much flour does she have now?
2
2
20140918 Chapter 0 – Basic Mathematics; Chapter 1 – Directed Numbers and the Number Line
Exercise (Chapter 1)
1.
Assume that Level +1 means one floor above the ground level in a shopping
mall.
(a) Use a directed number to describe the floor where each of the following
stores is located.
(i)
The fast food shop is seven floors above the ground level.
(ii) The electronic store is one floor below the ground level.
2.
(b) What is the mean of Level −2 ?
Suppose that +1 kg means a person gains 1 kg of weight.
(a) Express each of the following using a directed number.
(i) A person gains 5 kg of weight.
(ii) A person loses 3 kg of weight.
(b) What is the meaning of −2 kg ?
3.
Assume that +100 visitors means an arrival of 100 visitors in an airport.
(a) Express each of the following using a directed number.
(i)
The arrival of 200 visitors.
(ii) The departure of 150 visitors.
(b) What is the meaning of −100 visitors?
4.
Suppose that +10 m means 10 m to the east of a point Q.
(a) Express each of the following using a directed number.
(i)
15 m to the east of the point Q.
(ii) 25 m to the west of the point Q.
(b) What is the meaning of −20 m ?
5.
Find the directed numbers represented by the letters on the numer line below.
D
A
C
0
6.
B
+1 +2
Write down the following numbers on the number line below.
−1 , +4.5 , −3.5 , −5
0
7.
+1 +2
In each of the following, compare the values of the directed numbers and use the
inequality sign ‘ < ’ or ‘ > ’ to show their relationship.
(a)
+12 , +5
(b) −4 , 0
(c)
−17 , −15
(d) −
3
3
11
, −
23
23
20140918 Chapter 0 – Basic Mathematics; Chapter 1 – Directed Numbers and the Number Line
8.
In each of the following, arrange the numbers in ascending order.
(a)
9.
−6 , +4 , −2 , +1
(b) −3.2 , +3.4 , −2.1 , +2.6
In each of the following, arrange the numbers in descending order.
(a)
−3 , +7 , 0, −11
(b) −6.3 , +7.4 , −6.43 , +6.52
10. Suppose that −100 m means 100 m to the south of a school.
(a) What is the meaning of +100 m ?
(b) The location of a church is represented by +80 m while that of a library is
represented by −120 m . Which building, the church or the library, is
farther from the school?
11. Assume that +$500 means a profit of $500 and −$500 means a loss of
$500 in business.
(a) What is the meaning of +$800 ?
(b) The following table shows the profits or losses of three companies.
Company A: +$28 000
Company B:
−$40 000
Company C:
+$34 000
Which company makes the greatest profit?
(c) The profit of company D lies between the profits of companies A and C.
Which of the following may be the profit of company D?
A. +$38 000
B. +$26 000
C.
+$32 000
D.
−$30 000
12. Arrange te following numbers in ascending order.
2
1
−3 , +3 , −3 , +3
5
5
13. Arrange the following numbers in descending order.
2
4
−7 , −7 , +7 , −7
9
9
4