Download Fractions 93

EXERCISE 1
ORAL EXAMPLES:
The following shapes are divided into equal areas. State the fraction of each shape which is
(a) shaded and (b) unshaded.
WRITTEN EXAMPLES:
The following circles are divided into equal areas. The shaded areas show
two equal fractions. Write down these equal fractions.
Copy down and complete these statements.
1
11. 2 = 4 = 6 = 8 = 10 = 12
12.
3
13. 4 = 8 = 12
14.
2
15. 3 = 6 = 9 = 12
16.
1
4 = 8 = 12
1
3 = 6 = 9 = 12
1 = 2 = 3 = 4 = 6 = 8 = 10
DS/GRB/JMH: June 20, 2017
Page 1 of 21
EXERCISE 2
WORKED EXAMPLES:
3 3´1 1
=
=
18 3´ 6 6
a.
3
a. 18
20 4 ´ 5 5
=
=
24 4 ´ 6 6
Simplify
b.
20
b. 24
WRITTEN EXAMPLES:
Simplify these fractions.
2
1. 6
2
5. 10
3
9. 9
3
13. 15
3
17. 24
5
21. 15
4
25. 12
4
29. 20
6
33. 18
6
37. 12
2
41. 2
4
45. 4
10
49. 10
50
53. 10
20
57. 10
4
6
4
10
6
9
6
15
9
24
10
15
8
12
8
20
12
18
8
16
6
2
12
4
10
100
50
100
20
100
2.
6.
10.
14.
18.
22.
26.
30.
34.
38.
42.
46.
50.
54.
58.
3.
7.
11.
15.
19.
23.
27.
31.
35.
39.
43.
47.
51.
55.
59.
2
8
6
10
3
12
9
15
15
24
5
20
4
16
12
20
6
24
8
24
3
3
6
6
10
1000
50
1000
20
1000
4.
8.
12.
16.
20.
24.
28.
32.
36.
40.
44.
48.
52.
56.
60.
6
8
8
10
9
12
12
15
21
24
15
20
12
16
16
20
18
24
16
24
6
3
12
6
10
10 000
50
10 000
20
10 000
Note: 25 ´ 4 =100 and 125 ´ 8 =1000
* 61.
* 65.
* 69.
* 73.
40
10
25
10
80
10
125
10
* 62.
* 66.
* 70.
* 74.
40
100
25
100
80
100
125
100
* 63.
* 67.
* 71.
* 75.
40
1000
25
1000
80
1000
125
1000
* 64.
* 68.
* 72.
* 76.
40
10 000
25
10 000
80
10 000
125
10 000
DS/GRB/JMH: June 20, 2017
Page 2 of 21
Note: Equal fractions can be displayed in a
2 8
table of multiples shown here e.g. = .
3 12
EXERCISE 3
ORAL OR WRITTEN EXAMPLES:
State or write down the fractions which represent each point on the number lines.
(Each number line is divided into the stated number of fractional parts – some of these parts have been
labelled for you.)
1.
2.
3.
4.
5.
6.
.
Note:
1.
2.
The number line becomes packed or "dense" with points which represent fractions.
1
1
12
=1+2
i.e. the + is understood.
DS/GRB/JMH: June 20, 2017
Page 3 of 21
EXERCISE 4
1 9
8
1
1 8 1 9
= ( 2 = and so 2 = 2 + = + = )
4 4
4
4
4 4 4 4
1
9
2 is called a mixed number while is called an improper fraction.
4
4
23 20 3
3
3
b.
= + =4+ = 4
5
5 5
5
5
a. The diagram shows that 2
WRITTEN EXAMPLES: Change these mixed numbers to proper fractions.
1
1
1
1
1. 12
2. 32
3. 52
4. 72
1
3
1
3
5. 14
6. 24
7. 34
8. 44
2
1
2
1
9. 13
10. 23
11. 33
12. 43
3
9
1
7
13. 110
14. 110
15. 210
16. 210
3
9
1
7
17. 310
18. 310
19. 410
20. 410
1
3
2
4
21. 15
22. 15
23. 25
24. 25
1
3
2
4
25. 35
26. 35
27. 45
28. 45
5
1
5
1
29. 16
30. 26
31. 36
32. 46
1
5
3
7
33. 18
34. 18
35. 28
36. 28
1
5
3
5
37. 38
38. 38
39. 48
40. 48
Change these improper fractions to mixed numbers.
5
9
13
17
41. 2
42. 2
43.
44. 2
2
7
9
15
17
45. 4
46. 4
47.
48.
4
4
4
8
10
14
49. 3
50. 3
51.
52. 3
3
11
17
23
29
53. 10
54. 10
55. 10
56. 10
31
37
43
49
57. 10
58. 10
59. 10
60. 10
7
9
11
13
61. 5
62. 5
63.
64.
5
5
17
19
21
23
65. 5
66. 5
67.
68. 5
5
7
17
19
29
69. 6
70. 6
71.
72. 6
6
11
15
17
21
73. 8
74. 8
75.
76. 8
8
27
31
33
39
77. 8
78. 8
79.
80. 8
8
DS/GRB/JMH: June 20, 2017
Page 4 of 21
EXERCISE 5
Copy down and complete this table by dividing the numbers in the top row by the numbers in the first
column. Write your answers as fractions in their simplest form. (Check the ones which have been done for
you.)
-2
-1
÷
0
1
2
3
4 5
6
7
8
9 10 11 12
1
-2
2
-1
3
-2
4
5
6
3
-1
2
-1
-1
2
-1
3
-1
4
0
0
0
0
1
1
2
1
3
1
4
2
1
2
3
1
2
3
1
12
4
2
1
1
13
3
4
1
7
8
9
10
Note: 1. Q = {fractions} = {rational numbers}
ìï x
üï
= í : x Î I, y = N ý
îï y
þï
2. I Ì Q, because all integers are fractions
3
2
(e.g. 3 = , - 2 = ) but not all fractions are integers
1
1
3 2
1
(e.g. 2 ,
)
,
4 3 2
EXERCISE 6
ORAL EXAMPLES: State whether < or > replaces the circles.
1
2
3
1
4
2
1
1.
2.
3.
4.
3
3
4
4
5
5
5
3
2
5
6
1
3
7
5.
6.
7.
8.
7
7
7
7
8
8
8
3
1
7
9
5
1
7
9.
10.
11.
12.
10
10
10
10
12
12
12
1
1
3
3
1
1
2
13.
14.
15.
16.
4
5
4
5
5
3
5
1
1
2
2
3
3
4
17.
18.
19.
20.
7
5
7
5
7
5
7
WRITTEN EXAMPLES:
Write down and complete these statements by replacing the circles with < or > or =.
1
5
7
8
6
4
5
1.
2.
3.
4.
6
6
9
9
7
7
8
1
1
1
1
1
1
1
5.
6.
7.
8.
3
2
2
4
4
3
5
1
1
3
3
1
1
5
9.
10.
11.
12.
4
7
4
7
6
7
6
1
1
3
3
3
3
5
13.
14.
15.
16.
5
8
5
8
8
7
8
3
5
5
8
11
12
2
3
4
5
3
8
1
6
5
7
5
7
DS/GRB/JMH: June 20, 2017
Page 5 of 21
2
1
4
2
3
1
21.
8
2
6
3
25.
8
4
7
4
29.
10
5
2
1
33.
3
2
1
3
37. 1
2
2
1
5
41. 1
3
3
EXERCISE 7
17.
18.
22.
26.
30.
34.
38.
42.
c.
4 3
5 +5
4 2
5 -5
1
2
6
12
3
4
4
5
8
12
3
2
4
3
19.
23.
27.
31.
35.
39.
43.
4
8
1
2
7
8
2
3
9
12
1
3
2
1
2
3
1
2
7
12
3
4
4
6
3
4
9
2
8
3
20.
24.
28.
32.
36.
40.
44.
5
8
1
2
8
10
3
6
2
3
1
4
2
2
2
3
1
2
5
12
4
5
1
2
3
4
9
2
7
3
2 1 3
2 1 2 +1
)
+ = (i.e. + =
5 5 5
5 5
5
3 1 2
3 1 3- 1
And so - = (i.e. - =
)
5 5 5
5 5
5
Hence to add (or subtract) fractions with the same denominator we just
add (or subtract) the numerators.
4+3
7
2
4 3
4-3
1
= 5
= 5 = 15
b.
5 -5 = 5 =5
3
4-2+3
5
4 2
1 3
4-2+1-3
0
+5 =
=5 =1
d.
=5 =0
5
5 -5 +5 -5 =
5
WORKED EXAMPLES:
a.
3
4
1
2
5
8
9
10
2
3
1
2
2
2
1
3
The diagram shows that
WRITTEN EXAMPLES:
Find the following sums and differences.
Simplify your answers where possible.
2 1
1. 3 + 3
2.
3 1
5. 5 + 5
6.
4 1
9. 5 + 5
10.
5 1
13. 6 + 6
14.
7 1
17. 8 + 8
18.
7 3
21. 8 + 8
22.
5 3 1
25. 7 + 7 - 7
26.
8 5 1
29. 9 + 9 - 9
30.
4 3 2 1
33. 5 + 5 - 5 - 5
34.
9
7
3
1
37. 10 + 10 - 10 - 10
11
7
5
1
39. 12 + 12 - 12 - 12
2
3
3
5
4
5
5
6
7
8
7
8
5
7
7
9
4
5
1
-3
3.
1
-5
7.
1
-5
11.
1
-6
15.
1
-8
19.
3
-8
23.
3
1
-7 +7
4 2
+9 -9
3 2 1
-5 +5 -5
27.
31.
35.
38.
40.
3 1
4 +4
3 2
5 +5
4 2
5 +5
5 1
8 +8
5 3
8 +8
7 5
8 +8
6
4 2
+
7
7 -7
8 7
5
+
9 9
9
7 5 3 1
8 +8 -8 -8
9
7
3
1
+
10 10 10 10
11 7
5
1
+
12 12 12 12
4.
8.
12.
16.
20.
24.
28.
32.
36.
3
4
3
5
4
5
5
8
5
8
7
8
6
7
4
9
7
8
1
-4
2
-5
2
-5
1
-8
3
-8
5
-8
4
2
-7 +7
2
1
-9 +9
5 3 1
-8 +8 -8
DS/GRB/JMH: June 20, 2017
Page 6 of 21
EXERCISE 8
1 2 5
The diagram shows that 6 + 3 = 6
2
4
This is because 3 = 6 and so the steps are
1 2 1 4 1+ 4 5
+ = + =
=
6 3 6 6
6
6
5 2 5 4 5-4 1
and so - = - =
=
6 3 6 6
6
6
Hence to add (or subtract) fractions with different denominators we make sure each fraction has the same
denominator and then we add (or subtract) the numerators.
WORKED EXAMPLES:
a.
5 2
5 4
5+4
9
3
1
+
=
+
=
=
=
1
=
1
6 3
6 6
6
6
6
2
b.
5 2
5 4
5-4
1
6 -3 =6 -6 = 6 =6
c.
5 2
1
5 4
3
5-4+3
4
+
=
+
=
=
6 3
2
6 6
6
6
6
2
=3
WRITTEN EXAMPLES: Find the following sums and differences.
Simplify your answers where possible.
1 1
1. 2 + 4
1
1
5. 6 + 12
1
1
9. 3 + 12
2 1
13. 3 + 6
5 1
17. 8 + 4
3 5
21. 4 + 8
5
7
25. 6 + 12
7 1
29. 8 + 2
7 3 1
33. 8 + 4 - 2
7
4 1
37. 10 + 5 - 2
2.
6.
10.
14.
18.
22.
26.
30.
34.
38.
1 1
3 -6
1 1
2 -6
1
1
2 - 10
5 1
6 -3
7 1
8 -4
7 3
8 -4
11 5
12 - 6
2
5
3 - 12
5 3
1
+
8 4
2
3
4
1
+
10 5
2
3.
7.
11.
15.
19.
23.
27.
31.
35.
39.
1 1
4 +8
1
1
4 + 12
1
1
2 + 12
5 2
6 +3
3 1
4 +8
2
3
5 + 10
5 1
6 +2
7
1
10 + 2
5
5 1
+
12
6 -2
11
3 2
+
12
4 -3
4.
8.
12.
16.
20.
24.
28.
32.
36.
40.
1
1
5 10
1 1
2 -8
3 1
4 -2
3 1
8 -4
3 3
4 -8
9
4
10 - 5
11 3
12 - 4
11 1
12 - 2
7
5
1
+
12 6
2
7
3
2
+
12 4
3
DS/GRB/JMH: June 20, 2017
Page 7 of 21
EXERCISE 9
WORKED EXAMPLES:
5 3
10
9
10 + 9
19
7
a. 6 + 4 = 12 + 12 = 12
= 12 = 112
5 3
10
9
10 - 9
1
=
=
=
6 4
12 12
12
12
b.
-1
3 5
9
10
9 - 10
=
=
=
4 6
12 12
12
12
1
2 3
6
8
9
6+8-9
5
=
2 + 3 - 4 = 12 + 12 - 12 = 12
12
c.
d.
WRITTEN EXAMPLES: Find the following sums and differences.
Simplify your answers where possible.
1.
5.
9.
13.
17.
21.
25.
29.
33.
37.
41.
45.
1
2
3
5
2
3
2
3
3
4
5
6
5
8
1
4
1
2
3
4
5
6
5
6
1
+3
2.
1
+2
6.
1
+4
10.
3
+5
14.
3
+5
18.
1
+4
22.
1
+6
26.
3
-4
30.
3
-5
34.
2 1
+3 +2
1 3
+2 +4
3 2
+4 +3
38.
42.
46.
2
3
4
5
3
4
4
5
4
5
5
6
7
8
1
6
1
4
3
4
5
6
5
6
1
-2
3.
1
-2
7.
2
-3
11.
2
-3
15.
3
-4
19.
3
-4
23.
5
-6
27.
5
-6
31.
2
-3
35.
2 1
+3 -2
1 3
+2 -4
3 2
+4 -3
39.
43.
47.
1
2
1
3
1
3
1
4
1
4
1
6
1
3
1
2
1
5
3
4
5
6
5
6
1
+5
4.
1
+4
8.
1
+5
12.
1
+5
16.
1
+6
20.
1
+8
24.
2
-3
28.
3
-4
32.
2
-3
36.
2 1
-3 +2
1 3
-2 +4
3 2
-4 +3
40.
44.
48.
1
2
3
4
2
5
2
5
3
4
5
6
1
5
1
3
1
4
3
4
5
6
5
6
2
-5
1
-3
1
-3
1
-4
1
-6
3
-8
4
-5
5
-6
3
-5
2 1
-3 -2
1 3
-2 -4
3 2
-4 -3
DS/GRB/JMH: June 20, 2017
Page 8 of 21
EXERCISE 10
WORKED EXAMPLES:
1
1
1
/ 0 45 = 45
a.
2 of 90 is the same as dividing 90 by 2. i.e. 2 ´ 90 = 2/ ´ 9/
1
1
1
/ 0 30 = 30
b.
3 of 90 is the same as dividing 90 by 3. i.e. 3 ´ 90 = 3/ ´ 9/
2
2
2
/ 0 30 = 60
and so 3 of 90 = 2´ 30 = 60 i.e. ´ 90 = ´ 9/
3
3/
1
1
/ 015 = 15
c.
´ 90 = ´ 9/
6
6/
5
5
/ 015 = 5 ´15 = 75
d.
´ 90 = ´ 9/
6
6/
ORAL EXAMPLES: Find these products.
1
1
1.
2.
´ 24
´ 24
2
4
2
1
5.
6.
´ 24
´ 24
3
6
1
3
9.
10.
´100
´100
4
4
WRITTEN EXAMPLES:
1
1.
2.
´1000
5
1
5. 2 x 1000
6.
1
9. 8 x 1000
10.
3.
7.
11.
Find these products.
2
3.
´1000
5
1
7.
4 x 1000
3
11.
8 x 1000
3
´ 24
4
5
´ 24
6
1
´100
5
3
´1000
5
3
4 x 1000
5
8 x 1000
4.
8.
12.
4.
8.
12.
13. Find (in minutes) the following fractions of 1 hour (60 minutes).
1
1
3
a.
b.
c.
2
4
4
2
1
5
e.
f.
g.
3
6
6
1
´ 24
3
1
´100
2
4
4
´100 5
5
4
´1000
5
1
2 x 1 million
7
8 x 1000
d.
h.
1
3
1
10
14. Find (in degrees) the following fractions of 1 full turn (360 degrees).
1
1
3
a.
b.
c.
d.
2
4
4
1
2
1
e.
f.
g.
h.
3
3
6
1
1
1
i.
j.
k.
l.
12
10
5
1
8
5
6
1
9
15. Find (in metres) the following fractions of 1 kilometre (1000m).
1
1
3
a.
b.
c.
2
4
4
1
2
4
e.
f.
g.
5
5
5
1
10
8
5
d.
h.
16. Find (in square metres) the following fractions of 1 hectare (10 000m2).
1
1
3
a.
b.
c.
d.
2
4
4
1
100
DS/GRB/JMH: June 20, 2017
Page 9 of 21
1
10
1
i.
5
1
m. 8
EXERCISE 11
e.
f.
j.
n.
3
10
2
5
3
8
g.
k.
o.
7
10
3
5
5
8
h.
l.
p.
9
10
4
5
7
8
1
3
3
1
3
1x3
a. The diagram shows that 2 of 4 = 8 i.e. 2 x 4 = 2 x 4
Hence to multiply fractions we multiply the numerators and multiply
the denominators.
1
2
1 x 2/
1
1
2 1
b. Thus 2 x 3 = 2/ x 3 = 3 (and clearly 2 of 3 = 3)
æ 4 ö2 4 4
4x4
16
16
4
c. Also ç ÷ = 5 x 5 = 5 x 5 = 25 and so
=
25
5
è 5ø
ORAL EXAMPLES: Find the following products.
1
1
1
1
1
3
1
1
1. 2 x 2
2. 2 x 4
3. 2 x 4
4. 2 x 3
1
1
1
5
1
1
1
3
5. 2 x 6
6. 2 x 6
7. 2 x 5
8. 2 x 5
1
1
1
1
1
2
1 4
9. 3 x 4
10. 3 x 5
11. 3 x 5
12. 3 x 5
1
1
1
5
2
1
2 2
13. 3 x 6
14. 3 x 6
15. 3 x 5
16. 3 x 5
WRITTEN EXAMPLES: Find the following products. Simply your answers where possible.
2
4
1
1
1
3
1
1
1. 3 x 5
2. 4 x 5
3. 4 x 5
4. 4 x 6
1
5
3
1
3
3
1
1
5. 4 x 6
6. 4 x 5
7. 4 x 5
8. 5 x 6
1
2
1
2
1
3
1
3
9. 2 x 3
10. 2 x 5
11. 3 x 4
12. 3 x 5
2
3
1
4
3
4
1
5
13. 3 x 5
14. 4 x 5
15. 4 x 5
16. 5 x 6
1
4
2
1
1
2
3
2
17. 2 x 5
18. 3 x 4
19. 4 x 5
20. 4 x 5
2
1
2
5
2
1
3
1
21. 3 x 6
22. 3 x 6
23. 5 x 6
24. 4 x 6
3
5
3
1
4
1
2
1
25. 4 x 6
26. 5 x 6
27. 5 x 6
28. 3 x 4
2
5
3
5
4
5
3
2
29. 5 x 6
30. 5 x 6
31. 5 x 6
32. 2 x 3
æ1 ö2
æ2 ö2
æ 1 ö2
æ 3 ö2
33. ç ÷
34. ç ÷
35. ç ÷
36. ç ÷
è 3ø
è 3ø
è4ø
è4ø
2
2
2
æ1 ö
æ2 ö
æ 3ö
æ 4 ö2
37. ç ÷
38. ç ÷
39. ç ÷
40. ç ÷
è5 ø
è5 ø
è5 ø
è5ø
2
2
2
æ1 ö
æ5 ö
æ1 ö
æ 3ö 2
41. ç ÷
42. ç ÷
43. ç ÷
44. ç ÷
è6 ø
è6 ø
è2 ø
è2 ø
1
4
9
16
45.
46.
47.
48.
81
49
64
81
WORKED EXAMPLES:
DS/GRB/JMH: June 20, 2017
Page 10 of 21
49.
25
64
50.
36
49
51.
49
100
52.
9
4
DS/GRB/JMH: June 20, 2017
Page 11 of 21
EXERCISE 12
WORKED EXAMPLES:
a. These diagrams show that
1
3
2
3 ÷ 2 = 6 (= 3 x 2 = 1 x 1 )
1
3 1
3
2
i.e. 3 ÷ 2 = 1 ÷ 2 = 1 x 1
b. This diagram shows that
1
1
1
1 1
2 ÷ 3 = 6 (= 2 ´ 3 = 2 ´ 3)
1
1
3
1 1
i.e. 2 ÷ 3 = 2 ÷ 1 = ´
2 3
1
a
a
1
Hence to divide a fraction by a we multiply it by 1 and to divide a fraction by 1 we multiply it by a .
a
1
1
Note:
1 (or a) is called the reciprocal of a (e.g. 2 is the reciprocal of 2) .
1
a
1
Likewise a is called the reciprocal of 1 (or a) (e.g. 3 is the reciprocal of 3) .
Hence, instead of dividing by a number, we multiply by its reciprocal.
ORAL EXAMPLES: State the answers to the quotients in (a) without using the rule.
For each question check that the answers in (b) are the same as in (a) (i.e. check the rule).
1
2
1
2
1. a. 1¸
b. 1´
2. a. 2 ¸
b. 2 ´
2
1
2
1
1
4
1
4
3. a. 1¸
b. 1´
4. a. 2 ¸
b. 2 ´
4
1
4
1
1
1 1
1
1 1
5. a.
b.
6. a
b.
¸2
´
¸2
´
2
2 2
4
4 2
2
2 1
3
3 1
7. a.
b.
8. a.
b.
¸2
´
¸3
´
3
3 2
4
4 3
WRITTEN EXAMPLES: Find the following quotients. Simplify your answers.
3 1
¸
4 2
3 1
5.
¸
4 4
4
9.
¸2
5
2
13.
¸4
3
1.
2.
6.
10.
14.
5 1
¸
6 2
5 1
¸
6 4
2
¸4
5
5
¸5
6
3.
7.
11.
15.
2 1
¸
3 3
2 1
¸
3 6
2
¸6
5
2
¸6
3
4.
8.
12.
16.
5 1
¸
6 3
3 1
¸
4 6
4
¸6
5
3
¸6
4
DS/GRB/JMH: June 20, 2017
Page 12 of 21
EXERCISE 13
WORKED EXAMPLES:
1
1
1
5
a. 4 ÷ 5 = 4 x 1
3
2
3
5
b. 4 ÷ 5 = 4 x 2
1
1
1
6
c. 4 ÷ 6 = 4 x 1
3
5
3
6
d. 4 ÷ 6 = 4 x 5
1x5
5
1
1
1
5
1
5x1
5
1
= 4 x 1 = 4 = 14
(Check: 14 x 5 = 4 x 5 = 4 x 5 = 20 = 4 )
3x5
15
7
7
2
15
2
15 x 2
30
3
= 4 x 2 = 8 = 18
(Check: 18 x 5 = 8 x 5 = 8 x 5 = 40 = 4 )
1x6
6
3
1
1
1
3
1
3x1
3
1
= 4 x 1 = 4 = 2 = 12
(Check: 12 x 6 = 2 x 6 = 2 x 6 = 12 = 4 )
3x6
18
9
9
5
9x5
45
3
= 4 x 5 = 20 = 10
(Check: 10 x 6 = 10 x 6 = 60 = 4 )
a
b
Hence to divide a fraction by b we multiply it by a .
b
a
4
5
Note: a is called the reciprocal of b . (e.g. 5 is the reciprocal of 4 )
Hence, instead of dividing by a fraction, we multiply by its reciprocal.
ORAL EXAMPLES:
1
1
1
1
1
1
3
1
3
1
3
1
4 ÷ 3 = 4 x ? = ? 2. 3 ÷ 4 = 3 x ? = ? 3. 4 ÷ 3 = 4 x ? = ? 4. 3 ÷ 4 = 3 x ? = ?
2
1
2
1
2
1
2
3
2
3
2
3
5. 3 ÷ 4 = 3 x ? = ? 6. 4 ÷ 3 = 4 x ? = ? 7. 3 ÷ 4 = 3 x ? = ? 8. 4 ÷ 3 = 4 x ? = ?
WRITTEN EXAMPLES: Find the following quotients. Simplify your answers where possible.
1.
1
1. 3
1
5. 5
1
9. 3
1
13. 5
3
17. 4
1
21. 4
1
25. 3
1
29. 6
1
÷2
2.
1
÷2
6.
1
÷5
10.
2
÷3
14.
1
÷5
18.
1
÷2
22.
1
÷6
26.
1
÷4
30.
1
2
2
5
1
3
2
5
3
4
3
4
1
3
1
6
1
÷3
3.
1
÷2
7.
2
÷5
11.
2
÷3
15.
2
÷5
19.
1
÷2
23.
5
÷6
27.
3
÷4
31.
2
3
3
5
1
3
3
5
3
4
1
6
2
3
5
6
1
÷2
4.
1
÷2
8.
3
÷5
12.
2
÷3
16.
3
÷5
20.
1
÷2
24.
1
÷6
28.
1
÷4
32.
1
2
4
5
1
3
4
5
3
4
5
6
2
3
5
6
2
÷3
1
÷2
4
÷5
2
÷3
4
÷5
1
÷2
5
÷6
3
÷4
EXERCISE 14
WORKED EXAMPLE:
/3 ´ 2/ ´ 5 5
3
1
5
3
2
5
1
÷
x
=
x
x
=
= =1
4
2
6
4
1
6
4 ´1´ 6/ 4
4
WRITTEN EXAMPLES:
1.
5.
3
2
1
4 x3 x2
5
1 3
x
6
2 x4
2.
6.
Find the following products and quotients.
Simplify your answers where possible.
3
2
1
3
2
1
3.
4.
4 x3 ÷2
4 ÷3 x2
5
1 3
5
1
3
x
÷
7.
÷
x
8.
6
2 4
6
2
4
3
2
1
4 ÷3 ÷2
5
1 3
÷
6
2 ÷4
DS/GRB/JMH: June 20, 2017
Page 13 of 21
5
3 2
x
6
4 x3
* EXERCISE 15
Simplify
3
2 1
1. 4 x 3 - 2
3 2 1
5. 4 ÷ 3 + 2
5
3 1
9. 6 x 4 - 2
5
3 1
13. 6 ÷ 4 + 2
5
3 2
17. 6 x 4 - 3
5
3 2
21. 6 ÷ 4 + 3
9.
10.
2.
6.
10.
14.
18.
22.
5
3 2
x
6
4 ÷3
3
4
3
4
5
6
5
6
5
6
5
6
2 1
x (3 - 2 )
2 1
÷ (3 + 2 )
3 1
x (4 - 2 )
3 1
÷ (4 + 2 )
3 2
x (4 - 3 )
3
2
÷ (4 + 3 )
11.
3.
7.
11.
15.
19.
23.
5
3 2
÷
6
4 x3
3
4
3
4
5
6
5
6
5
6
5
6
2
1
-3 x2
2
1
+3 ÷2
3
1
-4 x2
3
1
+4 ÷2
3 2
-4 x3
3 2
+4 ÷3
12.
5
3 2
÷
6
4 ÷3
4.
(34
8.
(34
12.
(56
16.
(56
20.
(56
24.
(56
2
1
-3) x2
2
1
+3)÷2
3
1
-4)x2
3
1
+4) ÷2
3
2
-4) x3
3
2
+4) ÷3
*EXERCISE 16
1.
a.
Find (in its simplest form) the fraction which is halfway between
2
3
1
3
1
2
2
3
(i)
and
(ii)
and
(iii)
and
(iv)
and
5
5
2
4
2
3
3
4
p
b.
Find (in its simplest form) the fraction
if
q
2
3
1
p
p
(i)
is halfway between
and
(ii)
is halfway between
and
q
q
5
5
2
1
2
2
p
p
(iii)
is halfway between
and
(iv)
is halfway between
and
q
q
2
3
3
3
4
3
4
2.
A City Council has 3 sources of income 11
 its rates revenue which is
of its income.
15
1
 its operations revenue which is
of its income.
5
 its miscellaneous revenue which is made up of dividends, petrol tax and interest from
investments.
a.
If the Council’s operations revenue is $90 million find
(i) its total income.
(ii) its rates revenue.
b.
(i) Find the fraction of the Council’s total income which is miscellaneous revenue.
1
(ii) If dividends are of the miscellaneous revenue, what fraction are they of the total income?
2
1
(iii) If petrol tax is
of the total income, what fraction is it of the miscellaneous revenue?
150
What fraction is interest from investments of
(i) the miscellaneous revenue?
(ii) the total income?
c.
DS/GRB/JMH: June 20, 2017
Page 14 of 21
3.
When 2 dice are thrown and the numbers which turn up are added then the
outcomes range from 2 to 12 ( inclusive) as shown in the table to the right. Each + 1
outcome in the table is equally likely and so the table can be used to find the 1 2
2 3
proportion expected for each outcome.
3 4
a.
Copy down the table below and
(i) complete the row of the proportions expected for each outcome. 4 5
(Do not simplify your answers ).
5 6
(ii) complete the row of the numbers expected for each outcome
6 7
1
after 180 throws of both dice (e.g.
´180 = 5)
36
Outcomes
2
3 4 5 6 7 8 9 10 11 12
2
Proportion Expected 1
for each outcome
36 36
Number Expected
5
for each outcome
b.
(i)
(ii)
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Check that the expected proportions add up to 1.
Check that the sum of the expected proportions for the odd outcomes (i.e. 3, 5, 7, 9, 11) is
equal to that of the even outcomes.
4.
Repeat Question 3 for when 2 dice are thrown and the numbers which turn
up are subtracted as shown in the table to the right (where~ means “larger
– smaller” and the outcomes range from 0 to 5 inclusive).
Note: For the purposes of b (ii) zero is to regarded as an even number.
5.
The following table shows some information on the votes which each party
got at a General Election.
Note: 1. There were no other parties contesting the Election.
2. Not all of those eligible to vote actually voted .
Name of Party Fraction of votes obtained from those who
actually voted
were eligible to vote
1
3
Conservation
10
40
Fair Go
~
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
5
4
3
2
1
0
1
6
5
4
3
2
1
0
1
25
3
50
Get Real
3
10
Global
3
80
Paradise
People
a.
b.
c.
The table shows that (for the Conservation Party) –
1
3
of the number who actually voted =
of the number who were eligible to vote.
10
40
Use this information to show that
3
the number who actually voted = of the number who were eligible to vote
4
Use the result in (a) to find the fraction of those eligible to vote who supported
(i) the Fair Go Party
(ii) the Get Real party
Use the result in (a) to find the fraction of actual voters who supported
DS/GRB/JMH: June 20, 2017
Page 15 of 21
d.
6. a.
(i) the Global Party
(ii) the Paradise Party
Find the fraction of support for the People’s Party from
(i) the actual voters
(ii) those eligible to vote
This table shows what fraction each North Island region is of the North Island’s area. It also
3
shows that the North Island is of New Zealand’s area. Copy down and complete the table.
7
Fraction of N.I’s area Fraction of N.Z’s area
North Island Region
Northland – Auckland
1
6
Waikato – Bay of Plenty
1
3
Gisborne – Hawkes Bay
1
5
1
4
Taranaki – Wanganui – Manawatu
Wellington
North Island
b.
3
7
1
This table shows what fraction each South Island region is of New Zealand’s area. It also shows
4
that the South Island is of New Zealand’s area. Copy down and complete the table.
7
South Island Region
Fraction of S.I’s area Fraction of N.Z’s area
Nelson – Marlborough - Tasman
1
12
West Coast
1
12
Canterbury
1
6
Otago – Southland
South Island
1
4
7
DS/GRB/JMH: June 20, 2017
Page 16 of 21
REVISION EXERCISE:
1. State the fraction of this square which is a. shaded
b. unshaded
2. State the value of x and y if
3
6
y
=
=
4
x
20
2
11
3. a. Convert 43 to an improper fraction. b. Convert 4 to a mixed number.
4. Write down and complete these statements by replacing the circles with < or > or =.
1
1
3
3
5
24
a.
b.
c.
4
5
4
5
6
25
7
21
5
7
4
19
d.
e.
f. 3
8
24
6
8
5
5
In questions 5 - 56 simplify your answers as far as is possible.
3
3
3
3
5. 2 + 4
6. 2 - 4
7. 2 x 4
8. 2 ÷ 4
2
2
2
2
9. 3 + 4
10. 3 - 4
11. 3 x 4
12. 3 ÷ 4
4 3
4 3
4
3
4 3
13. 5 + 5
14. 5 - 5
15. 5 x 5
16. 5 ÷ 5
7 5
7 5
7
5
7 5
17. 8 + 8
18. 8 - 8
19. 8 x 8
20. 8 ÷ 8
1 1
1 1
1
1
1 1
21. 2 + 6
22. 2 - 6
23. 2 x 6
24. 2 ÷ 6
3 3
3 3
3
3
3 3
25. 4 + 8
26. 4 - 8
27. 4 x 8
28. 4 ÷ 8
1 1
1 1
1
1
1 1
29. 6 + 8
30. 6 - 8
31. 6 x 8
32. 6 ÷ 8
5 3
5 3
5
3
5 3
33. 6 + 4
34. 6 - 4
35. 6 x 4
36. 6 ÷ 4
æ5 ö2
æ 4 ö2
1
4
37. ç ÷
38.
39. ç ÷
40.
100
9
è8 ø
è9ø
5 2 1
5 2 1
5 2 1
5 2 1
41. 6 + 3 + 2
42. 6 + 3 - 2
43. 6 - 3 + 2
44. 6 - 3 - 2
5
2
1
5
2
1
5 2
1
5 2 1
45. 6 x 3 x 2
46. 6 x 3 ÷ 2
47. 6 ÷ 3 x 2
48. 6 ÷ 3 ÷ 2
5
2
1
5
2
1
5 2
1
5 2
1
*49. (6 + 3 ) x 2
*50. 6 + 3 x 2
*51. (6 - 3 ) x 2
*52. 6 - 3 x 2
5
2 1
5 2 1
5
2 1
5
2 1
*53. 6 ÷ (3 + 2 )
*54. 6 ÷ 3 + 2
*55. 6 ÷ (3 - 2 )
*56. 6 ÷ 3 - 2
DS/GRB/JMH: June 20, 2017
Page 17 of 21
EXTRA EXERCISE
1.
a.
Find the value of
2.
1
1
1
1
1
1
1
(1 - 2 )(1 - 3 )(1 - 4 )(1 - 5 )
ii.
(1 - 2 )(1 - 3 )(1 - 4 )(1 - 5 ) ... (1 - 10 )
1
1
1
1
1
Find, in terms of n, the value of (1 - 2 )(1 - 3 )(1 - 4 )(1 - 5 ) ... (1 - n )
a.
Find the value of
1
1
1
1
1
1
1
1
i.
(1 + 2 )(1 + 3 )(1 + 4 )(1 + 5 )
ii.
(1 + 2 )(1 + 3 )(1 + 4 )(1 + 5 ) ... (1 + 10 )
1
1
1
1
1
1
Find, in terms of n, the value of (1 + 2 )(1 + 3 )(1 + 4 )(1 + 5 ) ... (1 + n )
Copy down and check the first four statements. Copy down and complete the next four statements.
1
2
1
2
3 +3
1
2
3
4 +4 +4
1
2
3
4
5 +5 +5 +5
1
2
3
4
5
6 +6 +6 +6 +6
1
2
3
4
5
6
7 +7 +7 +7 +7 +7
1
2
3
4
9
10 + 10 + 10 + 10 + ....... + 10
1
2
3
4
5
n-1
n + n + n + n + n + .......... + n
4.
1
b.
b.
3.
1
i.
1
=2
2
=2
3
=2
4
=2
=
=
=
=
Copy down and check the first four statements. Copy down and complete the next four statements.
1
2
1+3
2+4
1+3+5
2+4+6
1+3+5+7
2+4+6+8
1+3+5+7+9
2 + 4 + 6 + 8 + 10
1 + 3 + 5 + 7 + 9 + 11
2 + 4 + 6 + 8 + 10 + 12
1 + 3 + 5 + 7 + 9 + 11 + ... + 99
2 + 4 + 6 + 8 + 10 + 12 + ... + 100
1 + 3 + 5 + 7 + 9 + 11 + ... + 2n-1
2 + 4 + 6 + 8 + 10 + 12 + ... + 2n
1
=2
2
=3
3
=4
4
=5
=
=
=
=
DS/GRB/JMH: June 20, 2017
Page 18 of 21
5. a.
Copy down and check the first four statements.
statements.
1
Copy down and complete the next four
1
1-2 =1x2
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
1
1
1
1
1
-3 =2x3
-4 =3x4
-5 =4x5
1
-6 =
1
-7 =
1
-8 =
1
-9 =
1
1
1
1
1
1
1
1
1
1
1
1
1
b.
Hence find the sum of the series 1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + 5 x 6 + ...
6. a.
Copy down and check the first four statements. Copy down and complete the next four
statements.
1
2
1-3 =1x3
1
3
1
5
1
7
1
9
1
11
1
13
1
15
1
2
1
2
1
2
-5 =3x5
-7 =5x7
-9 =7x9
1
- 11 =
1
- 13 =
1
- 15 =
1
- 17 =
b.
Hence find the sum of the series 1 x 3 + 3 x 5 + 5 x 7 + 7 x 9 + 9 x 11 + ...
7. a.
Copy down and check the first four statements. Copy down and complete the next four
statements.
1
3
1-4 =1x4
1
4
1
7
1
10
1
13
1
16
1
19
1
22
b.
1
3
-7 =4x7
1
3
- 10 = 7 x 10
1
3
- 13 = 10 x 13
1
- 16 =
1
- 19 =
1
- 22 =
1
- 25 =
1
1
Hence find the sum of the series 1 x 4 + 4 x 7 + 7 x 10 + 10 x 13 + 13 x 16 + ...
DS/GRB/JMH: June 20, 2017
Page 19 of 21
8.
a.
b.
9.
Copy down and check the first four statements. Copy down and complete the next four statements.
2 1
2
1- + =
2 3 1´ 2 ´ 3
1 2 1
2
- + =
2 3 4 2´3´4
1 2 1
2
- + =
3 4 5 3´ 4 ´5
1 2 1
2
- + =
4 5 6 4 ´ 5 ´6
1 2 1
- + =
5 6 7
1 2 1
- + =
6 7 8
1 2 1
- + =
7 8 9
1 2 1
- + =
8 9 10
1
1
1
1
1
Hence find the sum of the series
+
+
+
+
+…
1´ 2 ´ 3 2 ´ 3 ´ 4 3´ 4 ´5 4 ´ 5 ´ 6 5 ´ 6 ´ 7
2
of its games at home (H) and
5
3
the rest away from home (A). It also shows that the team won (W) of its home games and that it
4
1
lost (L) of all its games when playing away. No games were drawn or defaulted.
5
The “tree” diagram below shows that in one season a team played
W
H
2
5
A
3
4
b
a
c
L
W
f
g
L
e
1
5
d
Find
a.
b.
c.
d.
e.
f.
g.
h.
i.
the fraction of home games which the team lost.
the fraction of all games which the team won at home.
the fraction of all games which the team lost at home.
the fraction of games which were played away.
the fraction of away games which the team lost.
the fraction of away games which the team won.
the fraction of all games which the team won away.
the fraction of all games which the team won (whether home or away).
the fraction of all games which the team lost (whether home or away).
DS/GRB/JMH: June 20, 2017
Page 20 of 21
10.
Note: If the population of N.Z. is P and its area ia A km2 then the average population density of N.Z. is
P
(people/km2).
A
Auckland North Island
3
3
Fraction of N.Z’s population
10
4
Fraction of N.Z’s area
1
50
3
7
Use this table and the definition above to show that
a.
the average population density of Auckland is 15 times that of N.Z.
b.
the average population density of Auckland is 21 times that of the rest of N.Z..
c.
the average population density of the North Island is 4 times that of the South Island. (Ignore the
off shore islands).
DS/GRB/JMH: June 20, 2017
Page 21 of 21