Cambridge Essentials Mathematics Extension 8 N1.1 Answersessentials.cambridge.org/media/CEMKS3_E8_N1_WS_ANS.pdf · Cambridge Essentials Mathematics Extension 8 N1.1 Answers Original

Cambridge Essentials Mathematics Extension 8 N1.1 Answers

Original Material © Cambridge University Press 2009 1

N1.1 Answers

1 a –4 b 14 c –6 d 18

e 11 f –19 g 29 h –25

2 a 5 + –2 = 5 – 2 = 3 b –3 – –7 = –3 + 7 = 4

c 0 – –9 = 0 + 9 = 9 d –11 + –12 – –6 = –11 – 12 + 6 = –17

e 5 + –13 – –7 = 5 – 13 + 7 = –1 f 14 – –24 + –18 = 14 + 24 – 18 = 20

3 a 25 b –32 c 25 d –40

e –1 f 10 g –18 h –11

i 3 j –13 k –3 l –96

4

5 a i 3 × 3 = 9 ii 3 × –3 = –9

2 × 3 = 6 2 × –3 = –6

1 × 3 = 3 1 × –3 = –3

0 × 3 = 0 0 × –3 = 0

–1 × 3 = –3 –1 × –3 = 3

–2 × 3 = –6 –2 × –3 = 6

–3 × 3 = –9 –3 × –3 = 9

b In part i the answers decrease in 3s and in part ii the answers increase in 3s.

c The answer is a negative number.

d The answer is a positive number.

x 18 7 11 3 –4 –7 –12 –5 1 8 –10 –6

y 13 12 –2 2 –6 –7 –5 –9 –1 10 7 4

x – y 5 –5 13 1 2 0 –7 4 2 –2 –17 –10

x + y 31 19 9 5 –10 –14 –17 –14 0 18 –3 –2



6 a –12 b 28 c –21 d –72

e –60 f –120 g 230 h –120

i 64 j –3200 k –700 l 125

m 16 n 1 o 49 p 50

7

8 a 1 × –6; –1 × 6; –2 × 3; 2 × –3

b 1 × 18; –1 × –18; 2 × 9; –2 × –9; 3 × 6; –3 × –6

c 1 × –9; –1 × 9; –3 × 3

9 a The top number is the product of the middle two numbers.

b The bottom number is the sum of the middle two numbers.

c

10 a 4 b 9

11 a –9 b 9 c –10 d –5

e –8 f –9 g –15 h 8

i 15 j –8 k 16 l –4

× –2 5 3 –7

4 –8 20 12 –28

–9 18 –45 –27 63

–3 6 –15 –9 21

6 –12 30 18 –42



12 a Any correct questions with answer –16, e.g. –4 × 4 = –16.

b Any correct questions with answer 32, e.g. –64 ÷ –2 = 32.

c Any correct questions with answer –42, e.g. 3 × –14 = –42.

d Any correct questions with answer –60, e.g. 180 ÷ –3 = –60.

13

14 a 6 b 8 c –45

d 8 e 14 f 77

g –21 h –10 i –50

15 a 4 – 7 + 6 = 3, –8 – –6 = –2 10 ÷ –5 = –2, 2 × –4 = –8

5 + 3 × –2 = –1, (–1)2 = 1 –6 ÷ –2 = 3, (–2)2 – 1 = 3

–12 – –4 = –8, –3 – 8 = –11 –3 ÷ –3 = 1, 7 – –5 = 12

–22 ÷ 2 = –11, (–3)2 – 10 = –1

b

7 – –5 = 7 + 5 = 12

16

x 36 –84 –48 64 72 –65 –39 –17

y –9 12 8 –8 –2 –13 13 1

x ÷ y –4 –7 –6 –8 –36 5 –3 –17

x –8 –12 7 9 –7 15 –3 –0.5 10 –8

y –2 –4 –2 –2 –10 –5 –6 –2 –100 –4

x + y –10 –16 5 7 –17 10 –9 –2.5 –90 –12

x – y –6 –8 9 11 3 20 3 1.5 110 –4

x × y 16 48 – 14 –18 70 –75 18 1 –1000 32

x ÷ y 4 3 –3.5 –4.5 0.7 –3 0.5 0.25 –0.1 2



17 a i (–5)2 – –5 – 12 = 25 + 5 – 12 = 18

Equation must be equal to zero for –5 to be a solution.

ii x = –3 and x = 4

b i x = –1 and x = –5 ii x = –3 and x = 2



N1.2 Answers

1 Ben is right 20 × 20 = 400.

2 a

b 0

c If the last digit is 2 or 8 then the number squared ends in 4.

If the last digit is 3 or 7 then the number squared ends in 9.

If the last digit is 4 or 6 then the number squared ends in 6.

If the last digit is 5 then the number squared ends in 25.

d 121, 484, 676

e 62 + 82 = 102; 102 + 242 = 262; 182 + 242 = 302; 52 + 122 = 132; 122 + 162 = 202;

72 + 242 = 252; 82 + 152 = 172

3 a i 7 = 22 + 12 + 12 + 12

ii 20 = 32 + 32 + 12 + 12 or 20 = 42 + 22 + 02 + 02

iii 100 = 52 + 52 + 52 + 52 or 100 = 72 + 72 + 12 + 12 or 100 = 102 + 02 + 02 + 02

or 100 = 92 + 32 + 32 + 12 or 100 = 82 + 62 + 02 + 02 or 100 = 72 + 52 + 52 + 12

iv 18 = 32 + 22 + 22 + 12 or 18 = 32 + 32 + 02 + 02 or 18 = 42 + 12 + 12 + 02

v 35 = 42 + 32 + 32 + 12 or 35 = 52 + 32 + 12 + 02

12 = 1 112 = 121 212 = 441

22 = 4 122 = 144 222 = 484

32 = 9 132 = 169 232 = 529

42 = 16 142 = 196 242 = 576

52 = 25 152 = 225 252 = 625

62 = 36 162 = 256 262 = 676

72 = 49 172 = 289 272 = 729

82 = 64 182 = 324 282 = 784

92 = 81 192 = 361 292 = 841

102 = 100 202 = 400 302 = 900



b i Yes, because 102 + 42 + 22 + 22 = 100 + 16 + 4 + 4 = 124

ii 124 = 112 + 12 + 12 + 12

124 = 92 + 52 + 32 + 32

124 = 72 + 72 + 52 + 12

124 = 72 + 52 + 52 + 52

124 = 62 + 62 + 62 + 42

c 50 = 72 + 12 + 02 + 02

50 = 62 + 32 + 22 + 12

50 = 52 + 52 + 02 + 02

50 = 52 + 42 + 32 + 02

50 = 42 + 42 + 32 + 32

d Pupils choose their own number and find all the different ways of writing it as the sum

of four square numbers.

4 a 4 b 7 c 5 d 10

e 12 f 17 g 6 h 14

i 99 j 78 k 40 l 40

5 a 400 = 1004× = 4 × 100 = 2 × 10 = 20

b 2500 = 10025× = 25 × 100 = 5 × 10 = 50

c 6400 = 10064× = 64 × 100 = 8 × 10 = 80

d 900 = 1009× = 9 × 100 = 3 × 10 = 30

6 12 = 1 not possible 72 = 49 = 47 + 2

22 = 4 = 2 + 2 82 = 64 = 61 + 3

32 = 9 = 2 + 7 92 = 81 = 79 + 2

42 = 16 = 13 + 3 102 = 100 = 97 + 3

52 = 25 = 23 + 2 112 = 121 not possible

62 = 36 = 31 + 5 122 = 144 = 139 + 5



7 a

b 343, 1331

c 13 = 1 and 1 = 1; 83 = 512 and 5 + 1 + 2 = 8; 173 = 4913 and 4 + 9 + 1 + 3 = 17;

183 = 5832 and 5 + 8 + 3 + 2 =18

d i 20 = 23 + 23 + 13 + 13 + 13+ 13 ii 100 = 43 + 33 + 23 + 13

iii 19 = 23 + 23 + 13 + 13 + 13 iv 31 = 33 + 13 + 13 + 13 + 13

v 65 = 43 + 13

e 23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13.

8 a 21, 28

b i 1 ii 9 iii 36 iv 100

c Yes. The sum of the cube numbers is equal to the square of a triangular number, for

example, the sum of the first two cube numbers, 13 + 23 is equal to the square of

the 2nd triangular number. This is true for every triangular number.

d 282 = 784

9

10 a 1 b 4 c 6 d 6

e 7 f 16 g 3 h 20

n 1 2 3 4 5 6 7 8 9

n3 1 8 27 64 125 216 343 512 729

10 11 12 13 14 15 16 17 18

1000 1331 1728 2197 2744 3375 4096 4913 5832

Even Cube number Odd

Factors of 24 4 8 3

Square number 16 64 25

Multiples of 3 12 27 9



11 a 34 b 75 c 57 d 63

e 99 f 134 g 105 h 103

12 a 55, 91 b 385

c If you take blocks to make a square-based pyramid there will be 1 block in the top

layer, 4 in the second layer, 9 in the third layer and so on.

13 a (–2)3 = –8 b (–3)3 = –27 c (–5)3 = –125

d (–6)3 = –216 e (–9)3 = –729

14 a (–3)3 – (7 × –3) + 6 = –27 + 21 + 6 = 0. Yes, x = –3 is a solution.

b 33 – 7 × 3 + 6 = 27 – 21 + 6 = 12. No, x = 3 is not a solution.

c x = 1 and x = 2 d x = 1 and x = 3 and x = –4

15 a 324 b 729 c 4225 d 6561

e 14 400 f 2665 g 6565 h 1248

16 a 6.25 b 33.64 c 37.21 d 79.21

e 53.29 f 104.04 g 2197 h 8000

i 4096 j 15 625 k 5.832 l 1030.301

17 a 2 < 6 < 3 b 4 < 24 < 5 c 6 < 45 < 7

d 9 < 88 < 10 e 12 < 152 < 13 f 14 < 200 < 15

18 a Choose a bigger number if the answer is too small and a smaller number if the answer

is too big. This will get you closer and closer to the answer.

b 1.75 c 1.70 < x < 1.71

19 a The answer for 1 is too big and the answer for 2 is too small.

b 1.5 c 1.21 < x < 1.22

20 3.77 < x < 3.78



N1.3 Answers

1 a 8 b 104 c 12

d 24 e 120

2 a 10 b 28 c 18 d 21

e 60 f 30 g 60 h 40

3 a 1, 30; 2, 15; 3, 10; 5, 6

b Any three numbers whose lowest common multiple is 36.

4 a 1, 2, 13, 26 b 1, 2, 4, 8, 16, 32 c 1, 3, 9, 27

d 1, 2, 4, 5, 8, 10, 20, 40 e 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

5 a

b 1, 2 and 4 c 4

d i 30: 1, 2, 3, 5, 6, 10, 15, 30 18: 1, 2, 3, 6, 9, 18

Common factors are 1, 2, 3 and 6. Highest common factor is 6.

ii 45: 1, 3, 5, 9, 15, 45 63: 1, 3, 7, 9, 21, 63 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Common factors are 1, 3 and 9. Highest common factor is 9.

iii 52: 1, 2, 4, 13, 26, 52 65: 1, 5, 13, 65 91: 1, 7, 13, 91

Common factors are 1 and 13. Highest common factor is 13.

iv 64: 1, 2, 4, 8, 16, 32, 64 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

104: 1, 2, 4, 8, 13, 26, 52, 104

Common factors are 1, 2, and 4. Highest common factor is 4.



6

7 a 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

b 1, 3, 7 and 9

c If it ended in 0, 2, 4 or 8, it would be even and have a factor of 2.

If it ended in a 5, it would have a factor of 5.

8 a yes b no c no

d no e no f no

g yes h yes i no

9 The answers less than 100 are: 11, 13, 17, 37.

10 True for all values of n and p.

11 a

b Pupils’ tables.

12 a Yes.

b 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97

c 5 = 22 + 12 53 = 72 + 22

13 = 32 + 22 61 = 62 + 52

17 = 42 + 12 73 = 82 + 32

29 = 52 + 22 89 = 82 + 52

37 = 62 + 12 97 = 92 + 42

41 = 52 + 42

Factor of 30 Multiple of 8 Odd number

Multiple of 3 6 24 27

Factor of 40 10 40 5


Factor of 42 Multiple of 3 Prime number


Factor of 27 1 9 3




13

36 = 2 × 2 × 3 × 3 36 = 3 × 2 × 2 × 3

36 = 22 × 32 36 = 22 × 32

c It makes no difference to the final answer which factors you begin the tree with.

14 a 23 × 32 b 22 × 52 c 34 d 24 × 3

e 22 × 31 f 2 × 53 g 24 × 52 h 2 × 52 × 11

15 a 80 = 2 × 40

= 2 × 2 × 20

= 2 × 2 × 2 × 10

= 2 × 2 × 2 × 2 × 5

= 24 × 5

b i 22 × 32 × 5 ii 24 iii 5 × 32 × 11 iv 23 × 52 × 13

v 23 × 5 × 11 vi 22 × 32 × 52 vii 22 × 5 × 43 viii 24 × 3 × 11

ix 26 × 32 × 52 x 23 × 33 × 5

16 a 23 = 8 which means that 2904 is a multiple of 8, so 8 is a factor 2904.

b 1, 2, 3, 4, 6, 8, 11, 12, 22, 24.

17 a 22 × 7 × 11

b 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308



18 a 91 b 22 × 7 × 13 c 23 × 5 × 7 × 11

d Write numerator and denominator as products of prime factors and cancel common

factors.

e i 11013

117521372

3080364

3

2

=×××

××=

ii 2521

532732

150126

2

2

=××××

=

iii 65

7327532

252210

22 =×××××

=

19 a 80 = 2 × 2 × 2 × 2 × 5 24 = 2 × 2 × 2 × 3

b Common factors (circled) are 2, 2, 2 c Highest common factor is 8

20 a 22 × 3 × 5 b 23 × 32 c 23 × 32 × 5 d 360

21 a 12 = 22 × 3; 15 = 3 × 5

b 3 c 22 × 3 × 5 = 60

22 a i 5 ii 150

b i 16 ii 96

c i 9 ii 180

d i 11 ii 330

e i 12 ii 672

f i 5 ii 4290

g i 30 ii 180

h i 3 ii 180

i i 7 ii 588



N2.1 Answers

1 a 54 b

209 c

2518 d

5049

e 81 f

20051 g

12539 h

500391

2 a 432 b

20714 c

201155 d

8536

e 2004779 f

250113124 g

12516163 h

500201201

3 a 0.2 b 0.375 c 1.8 d 5.75 e 4.625

4 a 0.025 b 0.0625 c 0.090 909… d 0.151 515…

e 0.06 f 1.406 25 g 1.68 h 2.2

5 a 300 minutes or 5 hours

6 a 0.333 333…

b Goes on forever

c 0.666 666…

d 0.999 999…

e 33 = 1, so 0.999 999… = 1

f 34 = 1.333 333…,

35 = 1.666 666…,

36 = 2,

37 = 2.333 333…

b Maths English Science History Geography RS French Art Music

61

203

6011

101

101

201

607

151

151

c 0.166… 0.15 0.183… 0.1 0.1 0.05 0.116… 0.066… 0.066…



7 61 = 0.166...,

91 = 0.111...,

121 = 0.0833...

8 a 9920 b 0.111… + 0.0909… = 0.2020…

c The numerator is repeated as decimal part. d 0.171 717…

e 0.010 101… f 0.001 001… , 0.547 547… , 0.058 058…

9 a i 0.5 ii 0.666... iii 0.428 571… iv 0.777...

v 0.454 545... vi 0.4 vii 0.4375 viii 0.9166...

b i Parts i, vi and vii give terminating decimals.

ii Parts ii, iii, iv, v and viii give recurring decimals.

10 Pupils own results

11

Each recurring decimal consists of the cycle 142857, starting at different digits.

12 a 71 ,

61 ,

51 ,

41 b

21 ,

43 ,

65 ,

87

c 74 ,

85 ,

32 d

169 ,

117 ,

2013 ,

13

14 a 0.21 b 154 c 0.36 d

3427

15 a 4519 b

3523 c

2011 d

4837

16 Pupil’s answer.

Fraction 71

72

73

74

75

76

Decimal 0.142 857… 0.285 714… 0.428 571… 0.571 428… 0.714 285... 0.857 142…



N2.2 Answers

1 a 76 b

52 c

137 d

21

e 31 f

71 g

2521 h

154

2 32 +

41 =

128 +

123 =

1211

3 a 109 b

85 c

1411 d

43

e 4213 f 1 g

3914 h

8164

4 a 1217 or 1

125 b

2031 or 1

2011 c

1823 or 1

185 d

2129 or 1

218

e 5531 f

2625 g

2431 or 1

247 h

8867

5 43 –

52 =

2015 –

208 =

207

6 a 81 b

85 c

154 d

187

e 91 f

327 g

125 h

211

7 a 187 b

207 c

245 d

121

e 367 f

3516 g

6310 h

5714



8

9

10 There are many ways these can be written. The following are examples.

a 85 =

21 +

81 b

127 =

21 +

121 c

1513 =

21 +

31 +

301

d 209 =

41 +

51 e

3017 =

21 +

151

11 a 1032 b £96 c 56

41 g

d 2874 kg e 56

41 f 16

32

g 4132 cm h 33

31 p i 122

21 m

12 a ii 117 × 77 b i

85 of 48

+ 21

32

83

51

107

1513

4023

41

43

1211

85

72

1411

2120

5637

+ 131 2

31

1514

52 1

1511 2

1511 1

31

165 3

61 4

61 2

3023

241 3

127 4

127 3

6011



13 a 20 b 30 c 8012 =

203 d 18

e 8018 =

409 f Silver 90°, black 54°, red 135° and blue 81°

14 a 553 b 10

32 c 9

53 d 21

31

e 541 f 2

1613 g 28 h

52

15 a 35 kg b 6125 kg c 43

31 cm d 216 g

e 1276 m f 16

31 seconds g 4

117 ml h £67

51 (£67.20)

16

17 a 51 b

71 c

101

d 4 e 8 f 25

18 a 7 b 12 c 18

19 Dividing by 51 is the same as multiplying by 5.

20 a 20 b 27 c 60 d 140

e 32 f 90 g 30 h 12

52 × 80 = 32

× × ×

105 × 103 = 31

21

= = =

42 × 24 = 1008



21 a 30 × 51 = 6 therefore 6 ÷

51 = 30 and 6 ÷ 30 =

51

b 15 × 52 = 6 therefore 6 ÷

52 = 15 and 6 ÷ 15 =

52

c 10 × 53 = 6 therefore 6 ÷

53 = 10 and 6 ÷ 10 =

53

d 721 ×

54 = 6 therefore 6 ÷

54 = 7

21 and 6 ÷ 7

21 =

54

22 a Answer is smaller b Answer is larger

23 a 75 b

43 c

2815 d

75 ×

43 =

2815

24 a 8021 b

5536 c

4516

25 a 400216 =

5027 b

39688 =

92 c

350105 =

103

26 a 43 ×

32 =

126 =

21 b Pupils who walk to school but do not have school dinners.

27 a F = 65 b

92

28 24 ÷ 2512 = 24 ×

1225 = 50

29 a 36 b 421 c 3

31

30 80

31 a 278 b

72 c

52

32 a 75000050000 =

151 b 15 months



N2.3 Answers

1

2 a 2518 b

5049 c

209 d

1003 e

81

3 a 69% b 5% c 48.5% d 164% e 351.6%

4

5 a 3.6 b 3.96 c 14

d 31.5 e 33 f 45.6

6 a 31.08 kg b £12.96 c 1.479 km

d 9.128 m e £38.40 f 49.35 kg

g 33.21 litres h 0.052 tonnes i 73.125 cm

Fraction Decimal Percentage

a 41 0.25 25%

b 5027 0.54 54%

c 254 0.16 16%

d 31 0.333… 33

31 %

e 85 0.625 62.5%

f 10081 0.81 81%

g 251 0.04 4%

Plain Cheese and onion Salt and vinegar BBQ Prawn Cocktail

a

151

41

152

207

51

b 6

32 % 25% 13

31 % 35% 20%



7 a £9.66 b £17.16 c £11.86

8 a £9.00 b £209

9 1300 cm2

10 54

11 a 75% b 20% c 25% d 37.5%

12 25.2% (to 1 decimal place)

13 a

.

b History was Steve’s best result.

14 a 8% b 8.3% c 2.3% d 22.1%

15 3.5%

16 a £34.50 b £72.80 c £84.46

d £23.30 e £287.68 f £241.20

17 a £77.70 b £399.60 c £521.70 d £610.50

18 a This year’s council tax bill will be 104% of last year’s bill.

104% is equivalent to 1.04

b i £832 ii £1092 iii £1788.80

19 a £2181.60 b £101

20 a £156 750 b £163 803.75

21 a Yes, it is now 68.9%, which is an increase of 1.4%.

b 189.

22 a £22.75 b £22.54 c £360.24

Subject English Maths Geography History Science

Score 4035

9080

3528

6054

6456

Percentage 87.5% 88.9% 80% 90% 87.5%



23 £744.80

24 £80

25 £239 616

26 a 6.65 kg b 4.89 kg c 5 days

27 a No, in 15 years it will generate 15.4% of the world’s electricity.

b No, in 15 years it will generate 5.4% of world’s electricity.



N2.4 Answers

1 a 0.2 b 0.75 c 0.35

d 0.24 e 0.375 f 0.8

g 0.625 h 0.9 i 0.14

2 a i 0.375 ii Pupil’s check iii 0.875, methods include 0.125 × 7 or 1 – 0.125

b i 0.0125 ii 0.0625 iii 0.025 iv 0.1875

3 a 52 b

2017 c

2521

d 85 e 1

109 f 2

83

g 32518 h 4

2511 i 10

254

4

Fractions Decimals Percentage

21 0.5 50%

1003 0.03 3%

43 0.75 75%

54 0.8 80%

103 0.3 30%

203 0.15 15%

31 0.333… 33

31 %

52 0.4 40%

251 0.04 4%



5 a The chocolate digestive is Britain’s favourite biscuit. The chocolate hobnob came

second in a poll of 5000 people. About 33% of people in the UK eat biscuits as a mid-

morning snack but the largest proportion, 40%, eat them as they watch TV.

b 90% of people say that parents should be told if their children are obese. More than

70% said the government should abandon its plan to allow parents to opt out of being

told. The poll used a research panel of a hundred families, each of two adults and up

to four children.

6 a £5.70 b 25.2 kg c £13.02 d £16.80

e 24.96 km f 23 m g £1 h 0.45 km

7

8 a £35, £96.25, £21 b £235, £646.25, £141

9 a £38.25 b £32.30 c £13.60

d £46.75 e £136 f 51p

10 a i £20 ii £20

b i £20 ii £20

c i £20 ii £20

d It does not matter which percentage you find first, the answers are the same.

11

Year 2008 2007 2006

% of total TV audience 19.9% 24.8% 33.4%

Estimated total TV audience in millions 20.5 20.0 21.6

Estimated programme audience in millions 4.1 5.0 7.2

3 × 4 = 12

2 × 4 = 8

1 × 4 = 4

0.1 × 4 = 0.4

0.2 × 4 = 0.8

0.3 × 4 = 1.2



12 a 3.2 b 0.32 c 32

d 0.32 e 3200 f 320

13 a 6.3 b 6.3 c 630

d 63 e 0.063 f 630

14 a 5 b 0.5 c 0.05 d 0.005

15 a 9 b 0.9 c 0.09 d 0.009

16 Many different answers are possible.

17 (5 + 1) × (3 + 3) = 36 or (5 – 1) × 3 × 3 = 36

18 Numbers from 1 to 30 except 21, 22, 29 and 30.

19 × 5 × 5 = × 25

× 100 ÷ 4 = × 25

÷ 2 ÷ 2 = ÷ 4, moving 2 places left = × 100, ÷ 4 × 100 = × 25

20

21 a 80 = 82 + 42 86 = 92 + 22 + 12

81 = 82 + 42 + 12 87 = 92 + 22 + 12 + 12

82 = 92 + 12 88 = 82 + 42 + 22 + 22

83 = 92 + 12 + 12 89 = 82 + 52

84 = 92 + 12 + 12 + 12 90 = 82 + 52 + 12

85 = 92 + 22

b All of them.

Number (x) 1 2 3 4 5 6 7 8 9 10

Square (x2) 1 4 9 16 25 36 49 64 81 100

Cube (x3) 1 8 27 64 125 216 343 512 729 1000



22 5 = 32 – 22 55 = 82 – 32

10 = 32 + 12 60 = 82 – 22

15 = 42 – 12 65 = 82 + 12

20 = 42 + 22 70 can not be done

25 = 42 + 32 75 = 102 – 52

30 can not be done 80 = 92 – 12

35 = 62 – 12 85 = 92 + 42

40 = 72 – 32 90 = 92 + 32

45 = 62 + 32 95 = 122 – 72

50 = 52 + 52 100 = 82 + 62

23 a 400 b 2100 c 280

d 120 000 e 17 f 33

24 a 18 b 30 c 27

d 14 e 50 f 35

25 a 567 or 387 or 369 b 42 345 or 15 345 or 12 645 or 12 375 or 12 348

c 92 628 or 65 628 or 62 028 or 62 928 or 62 658 or 62 622

d 12 222 or 21 222 or 22 122 or 22 212 or 22 221

26 a 765 or 315 or 360 b 17 280 or 26 280 or 27 180 or 27 270

c 92 025 or 65 025 or 62 325 or 62 055 d 48 105

27 a 35 575, 24 255, 160 600

b 4 283 604, 24 255, 5 377 779



N3.1 Answers

1 a 102 b 103 c 101 d 104

e 106 f 105 g 100 h 109

2 a 10–2 b 10–3 c 10–6

3 a 1 m = 102 cm b 1 cm = 10–2 m c 1 cm = 101 mm

d 1 mm = 10–1 cm e 1 m = 103 mm f 1 mm = 10–3 m

4 a 1 nm = 10–9 m b 1 nm = 10–6 mm c 1 cm = 107 nm

5 a 200 b 4000 c 90 000 d 700 000

e 8 000 000 f 2100 g 350 h 125

6 a 6 × 102 b 5 × 103 c 8 × 104 d 1 × 105

e 3 × 106 f 2 × 108 g 7 × 10–2 h 19 × 10–3

7 a 2.3 b 9.9 c 1.49

d 0.08 e 0.765 f 0.055

g 0.09 h 6.581 i 0.0062

8 a 30 b 200 c 1690

d 1000 e 200 f 1400

g 12 800 h 5000 i 850 000

9 a 1.4 b 3.6 c 0.01

d 0.1 e 2000 f 16

g 30 h 0.1 i 0.27

10 a b



11 a 0.6 × 0.4 = 0.24 b 0.2 × 0.9 = 0.18 c 0.5 × 0.5 = 0.25 d 0.6 × 0.8 = 0.48

12 a 0.24 ÷ 0.4 = 0.6 or 0.24 ÷ 0.6 = 0.4 b 0.18 ÷ 0.2 = 0.9 or 0.18 ÷ 0.9 = 0.2

c 0.25 ÷ 0.5 = 0.5 d 0.48 ÷ 0.8 = 0.6 or 0.48 ÷ 0.6 = 0.8

13 a 0.06 b 0.32 c 0.45

d 2 e 3 f 9

14 a 0.12 b 0.48 c 0.015

d 25 e 360 f 24

15 a 0.04 b 0.002 c 0.3 d 0.8

e 3 f 0.6 g 0.2 h 0.4

16 a 800 b 100 c 2000

d 2600 e 1000 f 21 000

17 a i 67 190 ii 67 200 iii 67 000 iv 70 000

b i 2 000 000 ii 1 600 000 iii 1 630 000

iv 1 629 000 v 1 628 600 vi 1 628 600

18 a 3 cm b 38 cm c 10 litres

d 4 m e 18 kg f 10 cm2

19 a 23.7 b 1.8 c 9.9 d 7.0

e 19.9 f 20.0 g 20.0 h 100.0

20 a 41.67 b 80.05 c 1.01 d 30.00

e 3.33 f 6.67 g 10.00 h 100.00

21 a 0.7 b 1.5 c 1.55

d 0.8 e 0.20 f 0.78

22 a 74 cm2 b 15 cm2 c 10 cm2 d 166 cm2

23 a 8 cm b 31 cm



N3.2 Answers

1 a 97 b 97 c 97 d 179

e 379 f 513 g 1803 h 4011

2 a 22 b 57 c 134 d 69

e 474 f 478 g 217 h 2301

3 a £1.64 b 9p c £2.67

d £3.57 e £11.28 f £20.75

4 a 5.4 b 16.5 c 28.8 d 2.5

e 12.8 f 20.5 g 4.56 h 32.8

5 a £7.20 b £24

c i 9.6 m ii 76.8 m

d 42 m

6 a 253 b 176 c 384 d 855

e 1235 f 3625 g 7140 h 2277

7 a 682 pupils b 608 cm2 c 841 m2 d 435 chairs

8 a 334 b 99 c 2401

9 £1179.41

10 a 62 b 51 c 28 d 32

e 6 f 55 g 28 h 112

i 115 j 62 k 340 l 141

11 a 54 b 9640

12 Four and a half hours.



13

14 a 20 b 20 c 12 d 36

e 7 f 28 g 42 h 62

i 31 j 155 k 60 l 144

m 144 n 175 o 2.2 p 6.3

15 a 144 b £450 c 162 d £199.80 e £16.60

16 a 5 ≈ 2.2 b 24 ≈ 4.9 c 84 ≈ 9.2

d 108 ≈ 10.4 e 77 ≈ 8.8 f 2 ≈ 1.4

g 150 ≈ 12.2 h 410 ≈ 20.2 i 34 ≈ 5.8

17 a 5.8 cm b 10.4 cm c 2.2 cm

Decimal Fraction Percentage

0.5 21 50%

0.25 41 25%

0.333… 31 33

31 %

0.1 101 10%

0.2 51 20%

0.666… 32 66

32 %

0.75 43 75%

Decimal Fraction Percentage

0.125 81 12.5%

0.625 85 62.5%

0.8 54 80%

1.5 23 150%

1.2 56 120%

1.45 2029 145%

2.5 25 250%



18 a 62 × 39 ≈ 60 × 40 = 2400 b 71 × 48 ≈ 70 × 50 = 3500

c 321 × 148 ≈ 320 × 150 = 48 000 d 242 ÷ 62 ≈ 240 ÷ 60 = 4

e 389 ÷ 47 ≈ 400 ÷ 50 = 8 f 5

3112× ≈ 5

3012× = 72

g 8

211189× ≈ 8

200200× = 5000 h 8 × 17 ≈ 8 × 16 = 32

i 52 × 14 ≈ 49 × 16 = 28 j 4

8915 × ≈ 4

8916 × = 89

k 15

67125 × ≈ 16

64121× = 22 l 232 × 6 ≈ 252 × 4 = 1250

m 1059

312

× ≈

1009302

× = 10

n 192 × 412 × 10 ≈ 202 × 402 × 9 = 1 920 000

o 2

31⎟⎠⎞

⎜⎝⎛ × 385 ≈

91 × 387 = 43

19 a 5.03040× = 2400 which is larger than the true value.

b 7.02936× ≈

6.03036× = 1800 or

7.02936× ≈

7.02835× = 1400

c 1491



N3.3 Answers

1 a 38.7 b 157.2 c 830.3

d 441.05 e 37.96 f 488.06

g 906.26 h 641.28 i 72.109

2 a 63.8 b 39.2 c 122.03

d 1431.14 e 1427.143 f 158.209

3 a 22.2 b 85 c 21.9

d 125.4 e 818.6 f 25.7

g 314.67 h 90.29 i 5.4268

4 a 398.64 b 275.52 c 561.89

d 159.56 e 57.26 f 91.714

5 a 5.25 m b 0.511 kg c x = 1.58 m

d d = 2.37 m e l = 1.57 m f Packet X has mass 3.94 kg

6 a i 90 ii 80

b i 3200 ii 3362

c i 470 ii 460.6

d i 500 ii 452.5

e i 2000 ii 2207.2

f i 2000 ii 2452.8

g i 175 ii 165.92

h i 900 ii 973.52

i i 250 ii 257.244



7 a 3496.8 g b £15.48 c £29.04 d 203.58 m2

8 a 5.8 b 8.1 c 9.3

d 4.7 e 1.21 f 0.153

9 Get original number back.

7 × 11 × 13 = 1001

Dividing by 1001 is the same as dividing by (1000 + 1)

abcabc ÷ 1001 = abc since abc × 1001 = abc × (1000 + 1)

= abc 000 + abc = abcabc

10 a 40 b 5.2 c 255 d 6.9

e 34.6 f 36.95 g 17 700 h 14.82

i 54.5 j 42.6 k 1.75 l 42.6

11 a x = 3.8 cm b x = 6.4 m c x = 6.4 cm d x = 16 mm

12 a 0.3125 b 0.6875 c 0.36

d 0.875 e 0.425 f 0.65625

13 a 58% = 0.58, 4023 = 0.575,

169 = 0.5625 so in order:

169 , 0.57,

4023 , 58%

b 258 = 0.32,

329 = 0.28125, 30.7% = 0.307 so in order:

329 , 30.7%, 0.31,

258

14 a x = 0.955 b x = 2.642



N3.4 Answers

1 a 5 b 7 c 8 d 3

e –3 f 18 g 4 h 17.5

i 26.5 j 8 k 40 l 40

2 a 8.9 b 18.7824 c –26.85

d 152.561 e 184.1575 f –3.56

g 6.45 h 3 i 11

3 a i 86 °F ii 64.4 °F iii 32 °F iv 23 °F v –40 °F

b i 90 °F ii 66 °F iii 30 °F iv 20 °F v –50 °F

4 a

b C = 95 (F – 32)

c i 37.8 °C ii 21.1 °C iii –17.8 °C

iv 0 °C v –40.0 °C vi –60

d i 35 °C ii 20 °C iii –15 °C

iv 1 °C v –35 °C vi –53

5 a £709.22 b £242.72 c £43.77

d £39.45 e £318.70 f £267.93

6 a 730.50 Dollars b 43.27 Yuan c 0.62 Euros d 1.38 Dollars

7 a £32.52 b £42.74 c 125.65 Dollars

8 a 1.62 Euros, 2.37 US Dollars, 262.76 Japanese Yen and 17.49 Chinese Yuan.

b 1.31 Euros, 1.92 US Dollars, 212.50 Japanese Yen and 14.15 Chinese Yuan.



9 19.04 Euros less

10 a 12 hours b 18 hours c 15 hours

11 a 12 minutes b 45 minutes c 50 minutes

d 40 minutes e 21 minutes f 22 minutes

12 a 3 hours 30 minutes b 4 hours 45 minutes c 1 hour 48 minutes

d 1 hour 6 minutes e 7 hours 39 minutes f 12 hours 21 minutes

13 a 10 hours 24 minutes b 3 hours 24 minutes

14 a 1 hour 15 minutes b 1 hour 40 minutes c 37.5 minutes

d 247 e 1 hour 27.5 minutes

15 a 182 days 12 hours b 91 days 6 hours c 10 days 3 hours 20 minutes

d 1 day 20 minutes e 17 days 5 hours 40 minutes

16 a 4 hours b 3 hours c 1 hour 20 minutes

d 4 minutes e 1 hour 16 minutes

17 a 2.2 b 313.5 c 370

18 a 12 mm b 24

19 a 3 cm b 11 cm

20 a 5 cm b 29 cm2

21 a 20 Mexican pesos b 14 998 Mexican pesos



N4.1 Answers

1 a 29 b 81 c 41 d 0.8 e 20

f 4 g 8 h 56 i 13 j 3

k 4.5 l 4 m 9 n 0.5 o 9

2 a 6 b 18 c 57

d 25 e 208 f –235

g –27 h –43 i –12

3 a 6 + 24 ÷ 6 + 4 = 14 b (6 + 24) ÷ 6 + 4 = 9

c (6 + 24) ÷ (6 + 4) = 3 d 6 + 24 ÷ (6 + 4) = 8.4

e (16 + 4)2 × 8 – 3 = 3197 f 16 + 42 × 8 – 3 = 141

g 16 + 42 × (8 – 3) = 96 h (16 + 42) × (8 – 3) = 160

4 a P = 2(x + y) b P = 19 cm

5 P = 2(a + b + c)

6 a P = 4(m + n) b A = (m + n)2 c 2(m + n) d 4

)( 2nm +

7 a 7 × 0.1 < 7 b 6 × 0.4 < 6 c 8 × 0.2 > 0.2

d 15 × 0.3 < 15 e 12 × 0.1 = 1.2 f 0.3 × 0.5 < 0.3

g 25 × 0.2 = 5 h 0.2 × 30 < 30 i 0.4 × 0.6 < 0.6

8 a 0.5 b 1.5 c 4 d 0.8

e 2.4 f 7.2 g 0.2 h 0.4

i 1 j 0.55 k 1.1 l 3.3

9 a 8 ÷ 0.2 > 8 b 12 ÷ 0.5 > 12 c 20 ÷ 0.8 > 20

d 0.5 ÷ 0.1 > 0.5 e 5 ÷ 0.2 = 25 f 0.5 ÷ 5 < 0.5



10 a 50 b 25 c 10

d 20 e 200 f 2000

g 1000 h 1000 i 1000

11 a The answer is less than the original number.

b The answer is more than the original number.

12 a The answer is more than the original number.

b The answer is less than the original number.

13 a i False ii –2 × 3 = –6 (smaller)

b i True ii –4 × 0.5 = –2

c i False ii –6 ÷ 2 = –3 (bigger)



N4.2 Answers

1 a 56 + 8 ÷ 8 = 8 is incorrect. BIDMAS dictates that 8 ÷ 8 is done first.

b 25 – 10 × 2 = 30 is incorrect. 10 × 2 must be done first.

c (14.1 – 3.8)2 = 20.09 is incorrect. Estimate using (14 – 4)2 = 100.

d 87.01.9

9.23.27=

× is incorrect. Estimate using 99

327=

× .

e 4

36100 − = 20 is incorrect. Work out 2464

= .

f (20 – 15)2 – 10 ÷ 5 = 3 is incorrect. 10 ÷ 5 must be done after the brackets.

g 5 × 4.92 = 600.25 is incorrect. Estimate 5 × 52 = 5 × 25 = 125

h 8.3552 2÷ = 28.46 is incorrect. Estimate

42550 ÷ = 0.5

2 a 58 × 60 i 3480 Must be slightly less than 60 × 60 = 3600.

b 4.98 ÷ 0.33 ii 15.09 Estimate using 5 ÷ 31 = 15.

c 327 ÷ 1.1 i 297.27 Answer must be slightly less than 327 ÷ 1.

d 1001 × 2.1 ii 2102.1 2 × 1000 = 2000, 0.1 × 1000 = 100.

3 a ii x = 22.52 b iii p = 2116 c i m = 15 × 5 + 25

4 a The total of the percentages is 110% also 30% of 25 = 7.5, 25% of 25 = 6.25 and 15%

of 25 = 3.75, you can not have 0.5, 0.25 or 0.75 of a pupil.

b 30% of 25 = 7.5 and 14% of 25 = 3.5, you can not have 0.5 of a pupil.

c All percentages give whole numbers of children and the number of children totals 30.

5 Mean must lie within the range of results. 56 kg is bigger than the heaviest mass.



N4.3 Answers

1 a 21 b

2417 c

4843

d 127 e

97 f

32

2 a 1 : 2 b 17 : 24 c 43 : 48

d 7 : 12 e 7 : 9 f 2 : 3

3 a 61 b

65 c 1 : 5 d 5 : 1

4 a 21 b 1 : 1 c

121 d 1 : 11

5 a 1 : 5 b 1 : 9 c Brown to Blue

6 a 2 : 1 b 4 : 3 c 2 : 3 d 1 : 3

e 2 : 3 f 2 : 3 g 5 : 12 h 11 : 2

i 3 : 8 j 5 : 3 k 3 : 4 : 5 l 7 : 9 : 2

7 a x = 6 b p = 14 c n = 10

d y = 6 e a = 5 f b = 15

g m = 10, n = 15 h p = 18, q = 36 i d = 4, e = 28

8 a 5 m b 3.5 cm

9 a 1 : 3400 b 255 cm (2 m 55 cm)

10 a 1500 m b 16 cm

11 a 32 cm b 11.25 m



12 a 1 : 250 b 1 : 15 c 1 : 120 d 3 : 2500

e 1 : 250 000 f 3 : 400 g 1 : 8000 h 1 : 15

i 1 : 1440 j 1 : 100 000 k 1 : 432 l 1 : 250

13 a 3 cm, 21 cm b 4 cm, 20 cm c 6 cm, 18 cm

d 14 cm, 10 cm e 5 cm, 19 cm f 20.5 cm, 3.5 cm

g 4 cm, 8 cm, 12 cm h 6 cm, 10 cm, 8 cm i 19 cm, 3.5 cm, 1.5 cm

j 1132 cm, 11

32 cm,

32 cm

14 a 3 cm, 45 cm b 40 cm, 8 cm c 0.5 cm, 47.5 cm

d 3 cm, 3cm, 42 cm e 4 cm, 8 cm, 12 cm, 24 cm

f 1 cm, 5 cm, 12.5 cm, 29.5 cm g 12 cm, 16 cm, 10 cm, 6 cm, 4 cm

h 632 cm, 8 cm, 10

32 cm, 22

32 cm

15 a 4 litres b 600 ml or 0.6 litres c 400 ml or 0.4 litres

16 a 250 ml b 187.5 ml c 78.125 ml

17 a 875 b 142 c P 360; Q 1260

18 a length 45 cm and width 20 cm

b area = 900 cm2 c square side = 30 cm

d perimeter square : perimeter rectangle is 120 : 130 = 12 : 13

19 a Jar X = 36 sweets, jar Y = 13 sweets b Jar X = 30 sweets, jar Y = 26 sweets

c Jar X = 24 sweets, jar Y = 39 sweets d Jar X = 18 sweets, jar Y = 52 sweets

e Jar X = 12 sweets, jar Y = 65 sweets

f Not possible as the ratio is greater than the ratios of both jars.



N4.4 Answers

1 a b

2 a

b c

d 45 Euros less

Pounds (£) 0 5 10 15 20

Japanese Yen (¥) 0 1050 2100 3150 4200

Pounds (£) 0 10 20 50 100

Euros (€) 0 15 30 75 150



3 a 90 km/h

b

c

d 337.5 km

4 a 160 km b 1.6 km c 70 × 1.6 = 112 km

d 100 km = 62 miles, 200 km = 124 miles, 50 km = 31 miles so 250 km = 155 miles.

5 a 1 : 2.5 b 1 : 313 c 1 : 1.25 d 1 :

321

e 1 : 0.4 f 1 : 0.24 g 1 : 32 h 1 :

95

6 a 125 cm b 2.5 cm c 1 : 2.5 d 212.5 cm

e 40 inches f 0.4 inches g 1 : 0.4 h 70 inches

7 a 88 pints b 1.76 pints c 1 : 1.76

d 126.72 pints e 0.58 litres f 1 : 0.58

g L = 0.58P or P = 1.76L h 6380 litres

Time (hours) 0 1.5 3 4.5 6

Distance (km) 0 135 270 405 540



8 a

b For every 10 g of mass the extension is 2 cm. c M = 5E

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