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Math in Action (2nd Edition) 2B
11
Full Solutions
Rational and Irrational
Numbers
2
2.
Review Exercise 11 (p. 11.3)
(c)
1
1
   ,
2
4
 
(a)
49  7
(b)
64  8
(c)
1 1

9 3
1 1

4 2
1.
2.
(a)
20  2  2  5
 22  5
Classwork (p. 11.5)
(∵
4  2)
(e) No

∵


1
1
 
16 4 
(f) No

∵


4
2
 
25 5 
(a) No
(b) 162  2  3  3  3  3
(b) Yes
 2  34
(c) Yes
(c)
315  3  3  5  7
(d) Yes
 32  5  7
Activity
Activity 11.1 (p. 11.10)
1.
4  9  36 ,
6
(a) (i)
(ii)
(iii)
4  9  23
6
4  25  100 ,
 10
4  25  2  5
9  16  144 ,
 12
9  16  3  4
 10
2
4
2
2
    ,
9
3
3
 
(b) (i)
2
(ii)
4
2
2
    ,
25
5
5
 
2
(iii)
2.
9
3
3
    ,
16
4
4
 
(a)
ab  a  b
(b)
a

b
Classwork (p. 11.7)
(a)
 12
4

9
4
16
(c)

2
5

3
4
25
9
(b) 6 
2
3
4
(d) 
6
1
2 14

3 3
3 3 
3 

 or

5
5 
5
15
3

100 20
(e)
0.15 
(f)
 3.5  
35
7 7 
7 
 
 or

10
2 2 
2
a
Classwork (p. 11.8)
b
Classwork
(a) No
(∵ 5 is an integer.)
(b) Yes
Classwork (p. 11.4)
1.
1 
1 
 or

1 
1
1 
9 3)
42  16 ,
16  4
(c) No
(∵
(b) 92  81 ,
81  9
(d) No
(∵ 0.16 3 is a recurring decimal. )
(a)

(e) Yes
(f) Yes
68
11 Rational and Irrational Numbers
Quick Practice
(c)
0.5  100
2.42  100
0 .5

2.42
Quick Practice 11.1 (p. 11.6)
(a) ∵
∴
(b) ∵
∴
82
< 80 <
92
80 lies between 8 and 9.
12 < 2.59 < 22
2.59 lies between 1 and 2.
∴
∴
∴
∴
5 is 2.
The integral part of

6
142 < 199 < 152

25
1
6

6
6
6

6
199 lies between 14 and 15.
199 is 14.
The integral part of
(b)
484  2  11
2
1
2
5

6
5

5
6
6

6
 22  112
 2  11
 22
(b)
25
121
Quick Practice 11.5 (p. 11.14)
5 lies between 2 and 3.
Quick Practice 11.3 (p. 11.11)
(a)

22 < 5 < 32
(a)
(b) ∵
50
242
121
5

11
Quick Practice 11.2 (p. 11.6)
(a) ∵

3600  22  22  32  52
6

5 6
6

30
6
 22  22  32  52
 2  2  3 5
Quick Practice 11.6 (p. 11.15)
 60
(a)
63  32  7
3 7
(c)
 0.25   25  0.01
 ( 25  0.01 )
(b)
 ( 5  0.1 )
2
2
 (5  0.1)

  0 .5

Quick Practice 11.4 (p. 11.12)
(a)
64

9

(b)
6

8
3


25
27
22  112
33
2  11
22

3 3
9
25
4
484
3 3
64
1

4
3
3
22 3
9
Quick Practice 11.7 (p. 11.18)
(a)
50  18  5 2  3 2
2 2
4

484

27
5
2
(b)
80  112  20  4 5  4 7  2 5
6 5 4 7
69
Math in Action (2nd Edition) 2B
(c)
Full Solutions
48  75  44  4 3  5 3  2 11
9  16  25
(b)
 9 3  2 11
 32  42  52
 32  42  52
(d)
7
 84 
3
28

3
7

3
3
 22  3  7 
3
2 7
2
3

3
 3 4  5
3
 60
21
2 21

 2 21 
3
3


(c)
21  6 21  2 21
3
 1.44
  144  0.01
 ( 144  0.01 )
 3 21
3
 ( 12 2  0.12 )
 (12  0.1)
  21
  1 .2
Quick Practice 11.8 (p. 11.18)
(d)
6  81  (  24 )   6  81  24
5.76
 576  0.01
  11 664
 9  64  0.01
  2 4  36
 32  82  0.12
 2 2  33
 32  82  0.12
  108
 3  8  0 .1
 2.4
Alternative Solution
6  81  ( 24 )   2  3  9  23  3
  2  3 9 2 2  3
2.
18
50
(a)
 9  2  2  3
9
25

  108
9

Quick Practice 11.9 (p. 11.19)
25
48  6  42  24  3  2  3  2  3  7

 4 3  ( 2  3)  2  3  7

4 3 2 3 7
2 3
3
5
32
800
(b)
4 3 7
1
25

 4 3 7
 4 21
1

25
Further Practice

1
5
Further Practice (p. 11.12)
1.
(a)
441
(c)
 32  7 2
 32  7 2
 3 7
 21
 5
1
16

81
16

81
16
9

4
70
11 Rational and Irrational Numbers
0.45
1.25
(d)
(d)
4 11
0.45  100
1.25  100

3


4 11
11
11
3  11

4  11
45

125
9
25

3

33
44

9

2.
25
3

5
(a) Yes
(b) Yes
(c) No
Further Practice (p. 11.15)
1.
3 8
 3  23
10
3
(a)

 3  22  2
 3 2  2
10
6 2
3

10

3
3
(d) No
3

10  3
3
 20

30
3
2 5
  22  5
2
(e) No
5
2 3
2
(b)
2


2 3
3

23
23

6
6


(f)

2
5 2
2
No

7
33

7
32  3

1
7

3
3

1
7
3


3
3
3

1
73

3
3
7
7

10

2
7
27
10


2
7
10
(c)
5

3
10
10
7  10
10
70
10

71
21
9
Math in Action (2nd Edition) 2B
Full Solutions
Further Practice (p. 11.19)
1.
 5  25  3  3  5 2
18  72
(a)
5  96  75
(c)
 5  4 2  3 5 3
 32  2  6 2  2
3 2 6 2
9 2
45  80
(b)
5  4 2  3

 32  5  42  5
5 3

4 5  2
5

4 10
5
3 5 4 5
 5
12  28  ( 7 )
(d)
  22  3  22  7  7
(c)
5 2
 2 3  2 7  7
72
2 8
3

62  2
5 2 
 2 22  2
3

6 2
5 2 
4 2
3
5 2 2 2 4 2
2 3
2 7 7
3
7
Exercise
3 2
Exercise 11A (p. 11.9)
2
 54 
3
(d)
2


3
3
27
2
Level 1
33
 33  2 
3

2

23
3  2
 3 3 2 
3
2

6
3 6
3 6 
3
2
1.
2

(b)
 0.4  
2
(c) 1.75 
7 6
6
2.
7
4
 3 14  2 2
 6 28
 6 22  7
 12 7
567
105
21

10 000 2000
(a)

2
 0 .0 4
45
(b)

(c)
 
34
 0.3 4
99
(d)

 32  14  23
(b)
4
2

10
5
(d) 0.0105 
126  8
(a)
2
3
2 6  18 6  9 6

6
2.
3 11

4 4
(a)

28
  3.1
9
 
89
  3. 2 9 6
27
7
92  7

3.
(a)
2
7

4  22
9 7
7
(b)
9
 81   92
 9
72
11 Rational and Irrational Numbers
144  122
(c)
2
(d) ∵
 12
 25
4.
∴
(a) ∵  169   132  13
∴ The square roots of 169 are 13.
∴
(a) ∵
∴
12 lies between 3 and 4.
∴
∴
10 is 3.
12. (a)
17 lies between 4 and 5.
∴ The integral part of
∴
(b)
83 lies between 9 and 10.
∴ The integral part of
83 is 9.
5
 1.25 and 7  2.645...
4
(c)
 8  2.828... and 12  3.464...
∴
Level 2
9.
(a) ∵  0.04   0.22  0.2
∴ The square roots of 0.04 are 0.2.
13. ∵
∴
2

16
4
4
   
49
7
7
 
∴ The square roots of
36 4 6 4
  
9
7 9 7
42  36

63
6

63
2

21
It is a rational number.
(b) ∵  0.64   0.82  0.8
∴ The square roots of 0.64 are 0.8.
(c) ∵
1
1 3
 1 .5  
3
3 2
29

6
11

6
It is a rational number.
∴
∵
1 1 1 2
  
4 2 4 4
1

4
It is a rational number.
17 is 4.
(c) ∵ 92 < 83 < 102
8.
200 is 14.
10 lies between 3 and 4.
∴

200 lies between 14 and 15.
∴ The integral part of
(b) ∵ 42 < 17 < 52
∵
179 is 13.
(c) ∵ 142 < 200 < 152
42
∴ The integral part of
7.
179 lies between 13 and 14.
∴ The integral part of
39 lies between 6 and 7.
< 10 <
150 is 12.
(b) ∵ 132 < 179 < 142
(b) ∵ 62 < 39 < 72
6.
150 lies between 12 and 13.
∴ The integral part of
(a) ∵ 32 < 12 < 42
32
4.37 lies between 2 and 3.
11. (a) ∵ 122 < 150 < 132
∴
∴
185 lies between 13 and 14.
(b) ∵ 22 < 4.37 < 32
(c) ∵  324   182  18
∴ The square roots of 324 are 18.
∴
64
8
are 
.
625
25
10. (a) ∵ 132 < 185 < 142
(b) ∵  225   152  15
∴ The square roots of 225 are 15.
5.
64
8
 8 
   
625
25
 25 
∴ The square roots of
625  252
(d)

16
4
are  .
49
7
73
8 16
 22   
 19
3 9
8
16
, S  19
P   22 , Q   , R 
9
3
Math in Action (2nd Edition) 2B
Full Solutions
Exercise 11B (p. 11.13)
Level 1
1.
10.
36

49

52  112  52  112
 5  11
36
49
6
7
 55
18
9

98
49
11. 
2.
 36  81   62  92
49
3

7
  54
3.
144  121  122  112
12.
2
 122  112
 12  11
1

4
9
4

9
4
 132
4.
3

2
 2 2  4 2  7 2  ( 2 2  4 2  7 2 )
 ( 2  4  7 )
13.  5
  56
5.
9

 ( 6 2  9 2 )
 (6  9)
4
49

9
9
49

16  25  49  42  52  7 2
9
 4 2  52  7 2
 45 7
7

3
 140
6.
14.
 1.96   196  0.01
1.44  100
0.49  100
1.44

0.49
 ( 196  0.01 )

144
49
 (14  0.1)

144
  1.4
49
12

7
 ( 142  0.12 )
7.
25  2.89  25  289  0.01
 52  289  0.01
2
1.21
1.21  100

2.25
2.25  100
15. 
 5  17  0.1
2
2
 5  17  0.1
 8.5
8.
0.01 0.09  0.01 0.01 9
121
225

121
225
11

15
 0.01  0.01  9
 0.01 32
 0.01 3
Level 2
 0.03
16.
9.

25  16  0.09  25  16  9  0.01
784  22  22  7 2
 25  16  9  0.01
 22  22  7 2
 2 27
 52  42  32  0.12
 28
 5  4  3  0.1
6
74
11 Rational and Irrational Numbers
17.  2304   22  22  22  22  32
2.
16
16

2
 ( 22  22  22  22  32 )
 (2  2  2  2  3)
2

2
2
16 2

2
  48
8 2
15 876  22  32  32  7 2
18.
2

7
3.
 22  32  32  7 2
 2  3 3 7
 126
2
7
2

7

7
1.25
1.25  100

1 .8
1.8  100
19.

125
180

25
36

0.63  100
0.28  100

63
28


9
4

9
5.

1 .5
1.5  100

0.54
0.54  100

6.
150
54

25
9

25
7.
Exercise 11C (p. 11.16)
Level 1

5

5

65
13
11
11

2
2
11
2


11  2
2

22
2
5
5

3
3
5


53
3

15
3
28  2 2  7
2 7
5
5
8.
50  2  52
5 2
5 5
5
 5
75
2

3
9
5
13
5  13
13

5

3
5
13

2
4
1.
5

3

2
21. 
14
7
13
5
6
0.63

0.28
20.


36


5
5

13
13
4.
25

7
27
7
3
3
Math in Action (2nd Edition) 2B
9.
Full Solutions
Level 2
 98   2  7 2
 7 2
10.
15.
27

8
8
23

3
3

2 2
3


27
8
33
23
3 3
2 2
3 3


18
18

8
8

16.
2  32

2
3 3 2

4
3 3
4

12.
2 2
27
33

4
4
11. 
2

3 6
4
175
175

12
12

23
3 2

2 2
3

2
52  7
22  3
5 7
2 3

5 7
3

2 3
13.
315

20


315
22  5
3 5 7
17.  5
2 5

14.
1
3 7
2


5 21
6
1
36

7
7

36

6
7

27
6
7

7
16

5 73
23
7
11
27

16
16


20
32  5  7
3
7
6 7

7
33
42
3 3
4
18. 
99
99

28
28

32  11
22  7

3 11

3 11
2 7

2 7
76

3 7  11
27

3 77
14
7
7
11 Rational and Irrational Numbers
19.
448

363
448
2 2 7 2 3 2  2 2
6.
6 2 5 3  3  6 2 6 3
7.
9 5  6 3  5  10 5  6 3
363
26  7


5.
3  11
2
23 7
11 3

8 7
3

11 3
8.
4 3
8 73

11  3

9.
8  18  2 2  3 2
 2
8 21
33
10.
20.
3  27  3  3 3
3
150  24  2  3  52  23  3
5 3.6
5 3.6  10

6 150 6 150  10
5 6 2 6
7 6
5 36

6 1500
11.
5 9

6 375
1

2 15
21.
3

2 5
15
2  15

15
30
2 5


15
15

3
12. 
5
5
 20  
2 5
7
7

 5  14 5
7

13 5
7
13. 2 2  6  2 2  6
 2 2 23
5
 2 2 3
5
4 3
3 5
25
14. 2 3  3 21  2  3  3  21
3 5
10
3  2.236 07

10
 0.6708 (cor. to 4 sig. fig.)

 6 3 3 7
 6  3 7
 18 7
15.
Exercise 11D (p. 11.19)
Level 1
1.
6
2
6
2

5
3

6 5 3 5

3
3
2
 6

2
2
2
 6
5
9

6
375
5
3
 
6
3  53


6
35  2 5  5  7  2 5
 5 7 2 5
2 3 5 3 7 3

2.
7 5 4 5 3 5
3.
5 7 3 7  2 7
4.
6 3  7 3  3  12 3
5 7
2 5
7

2
77
Math in Action (2nd Edition) 2B
16.
10 27

50

10 33
25
2
242
 6 3
3
3
22.
2
10  3  3
2

5 2

3
3
2  112
 6  3
3
6
11  2
3
 6  3

3
3
3
6 6

2

6
3  11  6
 6
3
3
3 6

6  3 6  33 6
3

29 6
3
6 3
2

2
2
36
75 6
3  52

 
5
3
5
3

6 5 3

5
3

3


17.
Full Solutions
3
3
23.  15  8  10   15  8  10
  1200
2 3
  2 4  3  52
18.
 22  5  3
49
108 7
2 2  33

 
3
6
3
6
  20 3
7 2 3 3
 
3
6
7
6
 
3 6 3

7

3 3

Alternative Solution
 15  8  10   3  5  2 2  2  5
 2  2  2  3  5  5
3
 2  2  5  3
3
  20 3
7 3
9
24.
28  35  20  28  35  20
 19 600
Level 2
19.
75  27  147  5 3  3 3  7 3
 2 4  52  7 2
 15 3
 22  5  7
 140
20.
486  96  24  2  3  2  3  2  3
5
5
3
Alternative Solution
 32  6  22  6  2 6
28  35  20  2 7  5  7  2 5
9 6 4 6 2 6
 2 2 55 7  7
7 6
21.
 2 25 7
 140
245  363  45  5  7 2  3  112  32  5
 7 5  11 3  3 5
25.
 4 5  11 3
12  ( 50 )  24  22  3  ( 2  52 )  23  3
 2 3  (5 2 )  (2  2  3 )

 10  2  3
2 2  3
 5
78
11 Rational and Irrational Numbers
26.
48  75  32  24  3  3  52  25


( a )2  (3 b )2
a  9b
∴ The two pairs of possible values of a and b:
a  18, b  2 or a  27, b  3
(or any other reasonable answers)
2
4 3
5 34 2
1

5 2

27.
a 3 b
34. ∵
 (2  3 )  5 3  (2  2 )
2
2
2
Revision Exercise 11 (p. 11. 23)
Level 1
2
10
1.
(a)
289  17 2
 17
2( 8  6)  2  8  2  6
2
(b)  256   16
  16
 16  12
42 3
(c)
28. 2 3 ( 15  3 5 )  2 3  15  2 3  3 5
 1.69   169  0.01
 ( 169  0.01 )
 2 45  6 15
 ( 132  0.12 )
 2  3 5  6 15
 (13  0.1)
 6 5  6 15
29.
  1.3
15 ( 3  5  6 )  15  3  15  5  15  6
(d)
 45  75  90
3.24
3.24  100

0.25
0.25  100
 3 5  5 3  3 25

324
25

324
 3 5  5 3  3 10
25
30. 3 7 ( 14  2 2  3 3 )

 3 7  14  3 7  2 2  3 7  3 3
18
5
 3 98  6 14  9 21
 3  7  2  6 14  9 21
2.
  21 2  6 14  9 21
(a) ∵
9  32  3
∴ The positive square root of 9 is 3.
(b) ∵
100  102  10
∴ The positive square root of 100 is 10.
2
2
31. ( 3  2 )( 3  2 )  ( 3 )  ( 2 )
 32
1
144  122  12
(c) ∵
∴ The positive square root of 144 is 12.
32. ( 2 6  3 3 ) 2  ( 2 6 ) 2  2  ( 2 6 )  (3 3 )  (3 3 ) 2
(d) ∵
196  142  14
∴ The positive square root of 196 is 14.
 2 2 ( 6 ) 2  12 18  32 ( 3 ) 2
 24  12  3  2  27
 51  36 2
3.
33. ∵
 2.5 4 is a recurring decimal.
8
64  82  8 
1
3.1415 is a terminating decimal.
∴
 2.5 4 ,
6  12  a  2  3  2 3  a
 2  32 3 a
 2  3 2  a
 6 2a
∴ a = 2 or a = 2  22 = 8
4.
 
∵
 
(a) ∵
∴
(or any other reasonable answers)
(b) ∵
∴
79
64 , 3.1415 are rational numbers.
82  71  92
71 lies between 8 and 9.
132  172  142
172 lies between 13 and 14.
Math in Action (2nd Edition) 2B
22  5.16  32
(c) ∵
∴
5.
∵
Full Solutions
5.16 lies between 2 and 3.
 11  3.316... and
2
2

5
5
12. 
2 2  52  2 2  52
 10
13.
25
5

10
5
7

6
7

7
16  81  42  92
 42  92
 49
36  4  121  6  2  11
2
8.
2
2
 62  22  112
 6  2  11
14.
15.
6

76
6

42
6
112  2 4  7
 22 7
16

6
3 5
48
16

147
49


45  32  5
 132
9.
6
6
 36
5
5

 25
7.

5
∴
6.
2

14  3.741...
4 7
49
42

7
16.
2
252  2 2  32  7
 2 3 7
4

7
10.
3
6 7
13
121

36
36

17.
27

12
121

36

112
6
11

6
11.
1

3
1
3



2
27
12
33
22  3
3 3
2 3
3

2
3
18.
3
3
4
31

9
9

3
3
31
9

31
3
19. 3 8  50  3 2 3  2  5 2
 3 2  2  5 2
 6 2 5 2
 2
80
11 Rational and Irrational Numbers
20. 2

2 53
 2 15
3

2 15  6 15
3

21.
1 1 2 1
 
3 6
6
1

6
∴ It is a rational number.
5
5
3
 60  2 

 22  3  5
3
3
3
(b) ∵
534
100
∴ It is a rational number.
8 15
3
2
2 17
 0.34  
7
7 50
100  119

350
219

350
∴ It is a rational number.
(d) ∵
15  6  90
 2  32  5
 3 10
22. 3 7  2 14  3  2  7  14
 6 98
 6 27
(e) It is not a rational number.
2
(f) ∵ ( 5 ) 3  5 5
∴ It is not a rational number.
 67 2
 42 2
23. 4 21  3 
27.
4 3 7
54
54

125
125
3

2.2  3.14  5.34 
(c) ∵
3

4 7
24.
14  2 28 


Level 2
25. (a) ∵
∴

27
2 7
2 2 7
2
4
28.
112  122  12 2
∴
6

3 30
25
13  181  14
5
32
5
25
5

2  2
2
5
166 is 12.

4 2

5
2
4 25

5
181 lies between 13 and 14.
181 is 13.
 

2. 0 5 is a rational number.
 
∴
5
2
32

5
5
122 is 11.
166 lies between 12 and 13.
2

3 65
55
122  166  132
∴ The integral part of
26. (a) ∵
3 6


122 lies between 11 and 12.
∴ The integral part of
(c) ∵
3 23
5 5
2 22  7

∴
53
5 5
∴ The integral part of
(b) ∵
2  33

4 3  7
2. 0 5
is also a rational number.
3
81
4 10
5
5
5
Math in Action (2nd Edition) 2B
29.
Full Solutions
4 350 4 350  10

5 6.4
5 6.4  10
35.
 ( 2 5  3 )(2 5 )
4 3500

5 64

( 2 5  3 )( 5  3 5 )
 ( 2 5 ) 2  3  2 5
  20  2 15
4 875
5 16
36. ∵

4
875

5
16

4
5 7

5
4
3
27
33
 3
a
a
33
 3 
3
a
 3 
45 5 7

5 4
3 3
a

 35
9 3
a
∴ a  3 or a  3  22  12
30.  108  4 27  5 5   22  33  4 33  5 5
(or any other reasonable answers)
 2  3  3  4  3  3  5 5
37. Each side of the square of area 27 m 2  27 m
 6 3  12 3  5 5
3 3 m
 6 3 5 5
31.
Each side of the square of area 48 m  48 m
2
4 3m
147  196  245  3  7 2  2 2  7 2  5  7 2
∴ Perimeter  [3(3 3 )  3(4 3 )  4 3  3 3 ] m
 7 3  2 7  7 5
 (9 3  12 3  3 ) m
 7 3  14  7 5
 22 3 m
32.  49  12  2 3  2 1764
Challenging Questions (p. 11.24)
 2 2 2  32  7 2
 ( 2  2  3  7 )
1.
  84
1
3 2

1
3 2


Alternative Solution
 49  12  2 3  7  2 3  2 3

 7  2  2  3
33. 2 12  45  75  2 2 2  3  32  5  3  5 2
2.
 2(2 3 )  3 5  5 3
2 2 3 3 5
( 3 )  ( 2 )2
2 3
32
Let
x  a 2 and
y b 2 .
a 2  b 2  98
a 2  b 2  2  72
2  2  3 5

5
34.
2 3
2
x  y  98
5 3

( 3  2 )( 3  2 )
2 3
  84

( 3  2)  ( 3  2)
(a  b)( 2 )  7 2
ab  7
∵ x and y are positive integers and x < y.
∴ a and b are positive integers and a < b.
∴ The values of a and b can be:
a  1, b  6
a  2, b  5
a  3, b  4
For a  1 and b  6, we have
x  1 2 and y  6 2
12 5
5
2 (4 2  8 )  2 (4 2  2 2 )
 2 (6 2 )
 6 2
 12
x 2
x2
y  36  2
y  72
y  72
82
11 Rational and Irrational Numbers
For a  2 and b  5, we have
x 2 2
and
y 5 2
x  22  2
y  52  2
x 8
y  50
x 8
y  50
For a  3 and b  4, we have
x 3 2
and
y 4 2
x  32  2
y  42  2
x  18
y  32
x  18
∴
y  32
All the pairs of values of x and y are (2, 72), (8, 50),
(18, 32).
83