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Pass Publications GCSE Mathematics Higher 1 Number EXERCISES Topic 1 Bodmas 1. Evaluate the following: (a) 5 + 9 × 3 (b) 15 2 × 3 (c) 7 + 8 × 7 (d) 9 9 × 0.1 (e) 30 (13 × 3) (f) 15 +[17 × (3)]. 2. Find the following results: (a) 5(7 × 3 + 4) (b) 3[3 × (2) + 8] (c) 7(6 × 9 60) (d) 9(2 × 8 9) (e) [(3) × 7(5) (f) 6 ×[(9) × (1)]. 3. (a) (16 + 1)(8 ÷ 4) (b) ( 3 5 + 5 )( 3 2 ÷ 3 ) (c) ( 7 2 3 )( 4 2 ÷ 4 ) (d) ( 13 3 × 5 )( 9 2 ÷ 9 ) (e) ( 12 2 44 ) ÷ 10 (f) ( 14 2 96 ) × 1 10 . 4. (a) ( 1 5 + 1 10 ) ÷ 3 5 (b) 1 1 10 ÷ ( 1 3 4 + 2 1 4 ) (c) 2 1 7 ÷ 2 3 16 + 3 4 (d) ( 3 4 × 1 3 4 ) × 5 (e) ( 2 3 4 + 1 1 4 ) × ( 1 1 5 2 10 ) (f) 1 3 4 2 3 4 1 1 2 . 5. (a) Multiply by 2 3 4 the sum of 1 1 4 and 2 3 4 . (b) Find the product of 3 1 5 and 5 8 . (c) Find the difference between 5 2 and 3 2 . (d) Find the quotient of 4 1 5 divided by the sum of 3 5 and 5 2 5 . (e) Subtract 1 3 4 from the product of 1 7 8 and 4 5 . (f) Divide the difference between 2 1 2 and 3 4 by 4 1 8 .

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Page 1: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 1

Number

EXERCISES

Topic 1 ∼ Bodmas

1. Evaluate the following:

(a) 5 + 9 × 3 (b) 15 − 2 × 3 (c) 7 + 8 × 7

(d) 9 − 9 × 0.1 (e) 30 − (13 × 3) (f) −15 + [17 × (−3)].2. Find the following results:

(a) 5(7 × 3 + 4) (b) 3[3 × (−2) + 8] (c) −7(6 × 9 − 60)

(d) 9(2 × 8 − 9) (e) [(−3) × 7] × (−5) (f) −6 × [(−9) × (−1)].3. (a) (16 + 1)(8 ÷ 4) (b)

(35 + 5

) (32 ÷ 3

)(c)

(72 − 3

) (42 ÷ 4

)(d)

(133 × 5

) (92 ÷ 9

)(e)

(122 − 44

) ÷ 10 (f)(142 − 96

) × 110.

4. (a)(1

5 + 110

) ÷ 35 (b) 1 1

10 ÷ (13

4 + 214

)(c) 21

7 ÷ 2 316 + 3

4

(d)(3

4 × 134

) × 5 (e)(23

4 + 114

) × (11

5 − 210

)(f)

134

234 − 11

2

.

5. (a) Multiply by 234 the sum of 11

4 and 234.

(b) Find the product of 315 and 5

8.

(c) Find the difference between 52 and 32.

(d) Find the quotient of 415 divided by the sum of 3

5 and 525.

(e) Subtract 134 from the product of 17

8 and 45.

(f) Divide the difference between 212 and 3

4 by 418.

Page 2: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

2 Pass Publications GCSE Mathematics Higher

SolutionsTopic 1 Bodmas

1. (a) 5 + 9 × 3 = 5 + 27 = 32 (b) 15 − 2 × 3 = 15 − 6 = 9

(c) 7 + 8 × 7 = 7 + 56 = 63 (d) 9 − 9 × 0.1 = 9 − 0.9 = 8.1

(e) 30 − (13 × 3) = 30 − 39 = −9 (f) −15 +[17 × (−3)] = −15 − 51 = −66.

2. (a) 5(7 × 3 + 4) = 5(21 + 4) = 5(25) = 125

(b) 3 [3 × (−2) + 8] = 3{−6 + 8} = 3(2) = 6

(c) −7(6 × 9 − 60) = −7(54 − 60) = −7(−6) = 42

(d) 9(2 × 8 − 9) = 9(16 − 9) = 9(7) = 63

(e) [(−3) × 7] × (−5) = {−21} × (−5) = 105

(f) −6 × [(−9) × (−1)] = −6 × {9} = −54.

3. (a) (16 + 1)(8 ÷ 4) = (17)(2) = 34

(b) (35 + 5)(32 ÷ 3) = (243 + 5)(3) = 248 × 3 = 744

(c) (72 − 3)(42 ÷ 4) = (49 − 3)(4) = 46 × 4 = 184

(d) (132 × 5)(92 ÷ 9) = (169 × 5)(9) = 845 × 9 = 7605

(e) (122 − 44) ÷ 10 = (144 − 44) ÷ 10 = 100 ÷ 10 = 10

(f) (142 − 96) × 1

10= (196 − 96) × 1

10= 100 × 1

10= 10.

4. (a)

(1

5+ 1

10

)÷ 3

5=

(2

10+ 1

10

)÷ 3

5= 3

10÷ 3

5= 3

10× 5

3= 1

2

(b) 11

10÷

(1

3

4+ 2

1

4

)= 11

10÷

(7

4+ 9

4

)= 11

10÷ 16

4= 11

10× 1

4= 11

40

(c) 21

7÷ 2

3

16+ 3

4= 15

7÷ 35

16+ 3

4= 15

7× 16

35+ 3

4= 48

49+ 3

4= 192 + 147

196= 339

196

Page 3: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 3

(d)

(3

4× 1

3

4

)× 5 =

(3

4× 7

4

)× 5 = 21

16× 5 = 105

16

(e)

(2

3

4+ 1

1

4

(1

1

5− 2

10

)=

(11

4+ 5

4

(6

5− 2

10

)= 16

(12

10− 2

10

)= 4

(f)1

3

4

23

4− 1

1

2

=7

411

4− 3

2

=7

411

4− 6

4

= 7

4× 4

5= 7

5.

5. (a) 23

4

(1

1

4+ 2

3

4

)= 11

4

(5

4+ 11

4

)= 11

4

(16

4

)= 11

(b) 31

5× 5

8= 16

5× 5

8= 2

(c) 52 − 32 = 25 − 9 = 16

(d)4

1

53

5+ 5

2

5

=21

53

5+ 27

5

=21

530

5

= 21

30= 7

10

(e) 17

8× 4

5− 1

3

4= 15

3

82× 4

5− 1

3

4= 3

2− 7

4= −1

4

(f)2

1

2− 3

4

41

8

=5

2− 3

433

8

=10

4− 3

433

8

=7

433

8

= 14

33.

Page 4: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

4 Pass Publications GCSE Mathematics Higher

EXERCISES

Topic 2 ∼ Percentage

1. Find the values of the following:

(a) 5% of 100 kg (b) 7% of 28 m (c) 10% of £125

(d) 22% of 120 tonnes (e) 25% of 50 N (f) 35% of 5 A.

2. Find 33% of 250 kg.

3. What is 150% of £1 000?

4. Write 250 kg as a percentage of 25 kg.

5. Write 52 p as a percentage of £3.

6. The bus fares in outer London have increased from 70 p to £1. What is the percentageincrease?

7. The following items cost:

(a) £70 (b) £125

(c) £3.75 before a sale discount of 37.5%. How much is now the cost?

8. A T.V. costs £575 before V.A.T. of 17.5% is applied. How much do you pay withV.A.T.?

9. A car costs £13750 includingV.A.T. of 17.5%. How much is the cost of the car withoutV.A.T.?

10. If Pythagoras who lived 2600 years ago invested one drachma at 1% p.a., calculate:

(a) the simple interest and

(b) the compound interest earned today.

11. The formula Cn = C(1 + r

100

)n is used to work out the capital accumulated after n

years, Cn, at r% and C is the capital invested. If C = £1500, r = 5 and n = 10,calculate Cn.

12. If Cn = £14500, r = 7, n = 2 find C.

13. If C5 = C1.055, what is the ratio C5C

?

14. £5000 is invested at 2.65% p.a. for 5 years. How much is invested at the end ofyear 5?

15. A car is bought for £14995. The depreciation of the car is 15% per annum, what isthe value of the car after 3 years?

Page 5: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 5

16. A house was bought for £72500 on the 15th November 1985, and it costs £525000 onthe 14th November 2004. What is the average value of increase p.a.? If inflation onaverage is 5% during this period of time, what is the net gain or loss of this investment?

17. Increase the following amounts by 5%:

(a) £10 (b) £125 (c) £500 (d) £1250.

18. Decrease the following amounts by 8%:

(a) 50 kg (b) 25 m (c) 425 tonnes (d) £525.

19. Write down the percentages of the following fractions:

(a) 34 (b) 7

8 (c) 2 (d) 113 (e) 4

5 (f) 110 (g) 3

7 (h) 915.

20. Write down the fractions of the following percentages:

(a) 2% (b) 35% (c) 85% (d) 125%.

21. Change these fractions into decimals:

(a) 12 (b) 1

4 (c) 38 (d) 9

16 (e) 34 (f) 11

40 (g) 532

(h) 1920 (i) 7

16 (j) 339 (k) 34

5 (l) 425 (m) 85

9 (n) 725.

22. Change these fractions into recurring decimals:

(a) 13 (b) 1

18 (c) 411 (d) 4

3 (e) 416 (f) 43

7 (g) 323 (h) 2 4

13.

23. Change these decimals into fractions:

(a) 0.1 (b) 0.35 (c) 0.25 (d) 0.15

(e) 0.75 (f) 0.03 (g) 0.111 (h) 0.775.

24. Express the following as fractions:

(a) 5.05 (b) 0.375 (c) 0.125 (d) 6.9

(e) 8.75 (f) 5.125 (g) 3.68 (h) 0.003.

25. Write the following as percentages:

(a)3

5(b)

3

8(c)

27

40(d)

14

15(e) 0.98 (f) 0.001 (g) 2.17 (h) 0.33

26. Change the following percentages to fractions:

(a) 25% (b) 35% (c) 33% (d) 335%

(e) 5% (f) 4112% (g) 871

2% (h) 3912%.

Page 6: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

6 Pass Publications GCSE Mathematics Higher

SolutionsTopic 2

1. (a)5

100× 100 kg = 5 kg (b)

7

100× 28 m = 1.96 m

(c)10

100× £125 = £12.50 (d)

22

100× 120 tonnes = 26.4 tonnes

(e)25

100× 50 N = 12.5 N (f)

35

100× 5 A = 1.75 A.

2.33

100× 250 kg = 82.5 kg . 3.

150

100× £1000 = £1500.

4.250 kg

25 kg× 100 = 1000%. 5.

52 p

300 p× 100 = 17.3% to 3 s.f..

6.30 p

70 p× 100 = 300

7= 42.9% to 3 s.f..

7. (a) £70 − 37.5

100£70 = £70 × 0.62.5 = £43.75

(b) £125 × 0.625 = £78.13 (c) £3.75 × 0.625 = £2.34.

8. £575 + 17.5

100× £575 = £575 × 1.175 = £675.63.

9.£13750

1.175= £11702.13.

10. (a) 0.01×2600 = 26 drachmae (b) 1×1.012600 = 1.7201718×1011 drachmae.

11. Cn = C(

1 + r

100

)n

C10 = £1500

(1 + 5

100

)10

= £2443.34.

12. £14500 = £C

(1 + 7

100

)2

£C = 14500

1.072 = £12664.86.

13.C5

C= 1.055 = 1.276 to 3 d.p. = 1.28 to 3 s.f..

14. C5 = £5000 × 1.02655 = £5698.56.

15. £14995 − 15

100£14995 = £14995 × 0.85

after one year £14995 × 0.853 = £9208.80 after 3 years.

Page 7: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 7

16. 525000 = 72500

(1 + 5

100

)19

525000

72500=

(1 + 5

100

)19

7.241379311

19 = 1 + 5

100(1.109823081 − 1)100 ⇒ r = 10.98% to 2 d.p.

72500 × 1.0519 = 183203.89 due to inflation

£525000 − £183203.89 due to inflation

£525000 − £183203.89 = 341796.1

= £342000 to 3 s.f. net gain.

17. (a) £10 × 1.05 = £10.50 (b) £125 × 1.05 = £131.25

(c) £500 × 1.05 = £525 (d) £1250 × 1.05 = £1312.50.

18. (a) 50 kg × 0.92 = 46 kg (b) 25 m × 0.92 = 23 m

(c) 425 tonnes × 0.92 = 391 tonnes (d) £525 × 0.92 = £483.

19. (a)3

4× 25

25= 75

100= 0.75% = 75% (b)

7

8× 12.5

12.5= 87.5

100= 87.5%

(c)2

1× 100

100= 200% (d) 1

1

3× 100

100= 400

3= 133.3̇%

(e)4

5× 20

20= 80

100= 80% (f)

1

10× 10

10= 10

100= 10%

(g)3

7= 0.428571428 = 42.9% to 3 s.f. (h)

9

15= 0.6 = 60%.

20. (a) 2% = 2

100= 1

50(b) 0.35 = 7

20

(c) 0.85 = 85

100= 17

20(d) 1.25 = 5

4.

Page 8: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

8 Pass Publications GCSE Mathematics Higher

21. (a) 0.5 (b) 0.25 (c) 0.375 (d) 0.5625 (e) 0.75 (f) 0.275 (g) 0.15625

(h) 0.95 (i) 0.4375 (j) 3.6̇ (k) 3.8 (l) 4.4 (m) 8.5̇ (n) 7.4.

22. (a) 0.3̇3̇ (b) 0.05̇ (c) 0.3̇6̇ (d) 1.3̇

(e) 4.16̇ (f) 4.4̇2̇8̇5̇7̇1̇ (g) 3.6̇ (h) 2.3̇0̇7̇6̇9̇2̇.

23. (a)1

10(b) 0.35 =

35

100=

7

20(c) 0.25 =

1

4

(d) 0.15 =15

100= 3

20(e)

3

4(f)

3

100

(g)111

1000(h)

775

1000=

155

200=

31

40.

24. (a) 5.05 = 5 + 5

100= 5

1

20(b) 0.375 = 3

8(c)

1

8(d) 6 9

10 (e) 834 (f) 51

8

(g) 3 68100 = 334

50 = 31725 (h) 3

1000.

25. (a)3

5× 20

20= 60

100= 60%

(b)3

8× 12.5

12.5= 37.5

100= 37.5%

(c)27 × 2.5

40 × 2.5= 67.5

100= 67.5%

(d)14

15= 0.93̇ = 93.3̇% to 3 s.f .

(e) 98% (f) 0.1% (g) 217% (h) 33%.

26. (a) 25% = 25

100= 1

4(b)

35

100= 7

20

(c)33

100(d)

335

100= 3

35

100= 3

7

20

(e)5

100= 1

20(f)

41.5

100= 415

1000= 83

200.

(g)87.5

100= 875

1000= 175

200= 35

40= 7

8(h)

39.5

100= 395

1000= 79

200.

Page 9: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 9

EXERCISES

Topic 3 ∼ Decimals

1. Find the sum of the following decimal numbers:

3.56, 2.09, 0.065, 15.39, 1.70 without the use of a calculator.

2. Evaluate:

3.79 + 1.99 + 0.97 + 0.005 without the use of a calculator.

3. Determine the difference between 37.99 and 22.01, without the use of a calculator.

4. Evaluate:

25.09 − 16.98 without the use of a calculator.

5. Find the product 1.95 and 0.67, without the use of a calculator.

6. Evaluate:

15.69 × 0.87 without the use of a calculator.

7. Evaluate the quotient of 0.96 divided by 0.08, without the use of a calculator.

8. Evaluate:1.210.11 without the use of a calculator.

9. Write the following recurring decimals as fractions:

(a) 0.15̇ (b) 0.33̇ (c) 1.3̇5̇

(d) 6.87̇ (e) 0.9̇3̇ (f) 0.6̇6̇.

Show clearly all the steps.

10. Convert the following decimals to fractions:

(a) 0.607 (b) 0.65 (c) 1.35

(d) 2.75 (e) 0.00125 (f) 6.25.

11. Without the use of a calculator, add the following:

(a) 1.011 + 0.0003 + 2.046

(b) 2.003 + 0.003 + 1.003.

12. Subtract 1.999 from 2.000.

13. Multiply: (a) 0.97 × 0.369 (b) 3.767 × 1.001.

Page 10: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

10 Pass Publications GCSE Mathematics Higher

14. Divide 4.769 by 0.0023.

15. 4.765×23.9 = 113.8835. Find the following answers without the use of a calculator:

(a) 476.5 × 0.239 (b) 0.4765 × 239

(c)1138835

239(d)

11388.35

4765.

16. Find the following recurring decimals as fractions:

(a) 0.6̇7̇ (b) 0.75̇ (c) 1.1̇1̇3̇.

17. Express 0.2̇3̇5̇ as ND

where N and D are integers.

18. Change the following decimals to fractions:

(a) 0.825 (b) 0.125 (c) 1.35 (d) 3.725.

*Difficult

Page 11: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 11

SolutionsTopic 3 ∼ Decimals1. 3.56 2. 3.79 3. 37.99 −

2.09 1.99 22.010.065 0.97 15.98

15.39 0.0051.70 6.755

22.805

4. 25.09 − 5. 1.95 × 6. 15.69 ×16.98 0.67 0.878.11 1365 10983

1170 125521.3065 13.6503

7.0.96

0.08= 96

8= 12. 8.

1.21

0.11= 121

11= 11.

9. (a) 0.15̇ = 0.15555 . . . = x

10x = 1.55555 . . .

x = 0.15555 . . .

9x = 1.4

x = 1.4

9= 14

90= 7

45(b) 0.3̇3̇ = 0.3333 . . . = x

10x = 3.3333 . . .−

x = 0.3333 . . .

9x = 3

x = 1

3(c) 1.3̇5̇ = 1.353535 . . . = x

100x = 135.3535 . . .−

x = 1.3535 . . .

99x = 134

x = 134

99

Page 12: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

12 Pass Publications GCSE Mathematics Higher

(d) 6.87̇ = 6.8777 . . . = x

100x = 687.777 . . .−

10x = 68.777 . . .

90x = 619

x = 619

90

(e) 0.9̇3̇ = 0.939393 . . . = x

100x = 93.9393 . . .−

x = 0.9393 . . .

99x = 93

x = 93

99= 31

33

(f) 0.6̇6̇ = 0.666666 . . . = x

100x = 66.6666 . . .−

x = 0.6666 . . .

99x = 66

x = 66

99= 6

9= 2

3.

10. (a) 0.607 = 607

1000(b) 0.65 = 65

100= 13

20

(c) 1.35 = 135

100= 27

20(d) 2.75 = 275

100= 55

20= 2

3

4

(e) 0.00125 = 125

100000= 1

800(f) 6.25 = 625

100= 6

1

4.

11. (a) 1.011 (b) 2.0030.0003 + 0.003 +2.046 1.0033.0573 3.009

12. 2.000 −1.9990.001

Page 13: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 13

13. (a) 0.9700.369 ×8730

58202910

0.357930

(b) 3.7671.001 ×3767

000000003767

3.770767

14.4.769

0.0023= 47690

23= 2073.478 to 3 d.p.

2073.47826123

∣∣4769046 −1690161 −

8069 −110

92 −180161 −190184 −

6046 −14

15. 4.765 × 23.9 = 113.8835

(a) 476.5 × 0.239 = 4.765 × 100 × 23.9

100= 113.8835

(b) 0.476.5 × 239 = 4.765

10× 23.9 × 10 = 113.8835

(c)1138835

239= 113.8835 × 10000

23.9 × 10= 4.765 × 1000 = 4765

(d)11388.35

4765= 113.8835 × 100

4.765 × 1000= 23.9

10= 2.39.

Page 14: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

14 Pass Publications GCSE Mathematics Higher

16. (a) 0.6̇7̇ = 0.676767 . . . = x

100x − x = 99x

67.6767 . . . − 0.6767 . . . = 67

99x = 67

x = 67

99

(b) 0.75̇ = 0.7555 . . . = x

100x = 75.55 . . .

100x − 10x = 90x = 75.55 . . . − 7.55 . . .

90x = 68

x = 68

90= 34

45

(c) 1.1̇1̇3̇ = 1.113113113 . . . = x

1000x = 1113.113113 . . .

1000x − x = 999x = 1112

x = 1112

999.

17. 0.2̇3̇5̇ = 0.235235235 . . . = x

1000x = 235.235235 . . .

x = 0.235235 . . .

999x = 235

x = 235

999.

18. (a) 0.825 = 825

1000= 165

200= 33

40

(b) 0.125 = 125

1000= 1

8

(c) 1.35 = 135

100= 27

20= 1

7

20

(d) 3.725 = 3725

1000= 149

40= 3

29

40.

Page 15: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 15

EXERCISES

Topic 4 ∼ Irrational numbers

1. The square roots of the prime numbers are irrational numbers. Are the squares ofirrational numbers always rational? Explain by giving an example.

2. Sketch five right angle triangles whose vertical sides are unequal.

(a) The hypotenuse is rational and so are the vertical sides.

(b) The hypotenuse is irrational and the vertical sides are rational.

(c) All three sides are irrational.

(d) The hypotenuse is rational and the vertical sides are irrational.

(e) The hypotenuse is irrational, one vertical side is rational and the other irrational.Insert the numbers on the diagrams that you have chosen.

3. Simplify the following surds:

(a)√

24 (b)√

12 (c)√

8 (d)√

18

(e)√

500 (f)√

45 (g)√

48 (h)√

1000

(i)√

125 (j)√

800 (k)√

300.

The term surd is derived from the word absurd.

4. Simplify the following surds:

(a) 3√

12 × 4√

3 (b)√

125 × 5√

5

(c)√

243 × √27 (d) 2

√60 × √

15.

5. Write down two irrational numbers between 3 and 4.

6. Write down two irrational numbers between 1 and 2.

7. Prove that√

3 is irrational.

8. Is a recurring number irrational?

*9. For GCSE π = 227 to 3 s.f. Can you find a better and closer to π fraction? Clue – use

the numbers 113355.*

10. Find two irrational numbers x and y which represent the sides of a rectangle such thatthe perimeter is rational.

Page 16: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

16 Pass Publications GCSE Mathematics Higher

11. The sides of a rectangle are irrational and the area is rational. Find these sides if thearea of the rectangle is 50 m2.

*12. Find the value of(√

15 − 2√

3)2

.

*13. Find the value of(√

5 − √3) (√

5 + √3)

.

*14. 2 � 1

2 � 1

The above rectangle has irrational sides as shown.Determine (a) the perimeter (b) the area.

*15. If x = 2√

2 − 1, y = 2√

2 + 1 determine the value of xyx−y

.

16. Simplify

√125

20.

17. Simplify√

26(

2√

13 − 2√

2)

.

18. Simplify the quotient

(7 − √

2) (

7 + √2)

47.

19. Simplify the quotient

(6 − √

5) (

6 + √5)

√31

20. a and b are two positive irrational numbers. The sum and the product are rational.Express 1

a+ 1

bas a single fraction, explain why a+b

ab or aba+b

are always rational.

21. Find the value√

7 × √63.

22. Find the value of k if√

5 × √500 = k

√5.

23. Find the value of

√243 − √

3√12

.

Page 17: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 17

SolutionsTopic 4

1. π2 is an irrational number

π = 3.141592654 to 9 d.p.

π2 = 9.869604401 to 9 d.p..

2.

5 4

3

(a)

32 + 42 = 52

9 + 16 = 25

5 1

2

(b)

22 + 12 = (√

5)2

4 + 1 = 5

52

3

2(c) (√

3)2 +

(√5)2 =

(2√

2)2

3 + 5 = 8

7

(d)

2

3(√

2)2 +

(√7)2 = 32

2 + 7 = 9

3

(e)

2

7(√

3)2 + 22 =

(√7)2

3 + 4 = 7.

3. (a)√

24 = √2 × 2 × 2 × 3 = √

4√

2√

3 = 2√

6

(b)√

12 = √2 × 2 × 3 = √

4√

3 = 2√

3

(c)√

8 = √2 × 2 × 2 = √

2 × 2√

2 = 2√

2

(d)√

18 = √2 × 3 × 3 = √

2√

9 = 3√

2

(e)√

500 = √10 × 10 × 5 = √

100√

5 = 10√

5

(f)√

45 = √3 × 9 = √

9√

3 = 3√

3

Page 18: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

18 Pass Publications GCSE Mathematics Higher

(g)√

48 = √3 × 16 = √

16√

3 = 4√

3

(h)√

1000 = √100

√10 = 10

√10

(i)√

125 = √25

√5 = 5

√5

(j)√

800 = √100

√2 × 2 × 2 = 10 × 2

√2 = 20

√2

(k)√

300 = √100

√3 = 10

√3.

4. (a) 3√

12 × 4√

3 = 12√

12 × 3 = 12√

36 = 12 × 6 = 72

(b)√

125 × 5√

5 = 5√

5 5√

5 = 125

(c)√

243 × √27 = √

3 × 81 × √27 = √

3 × 27 × 81 = √81 × 81 = 81

(d) 2√

60 × √15 = 2 × √

4 × 15 × √15 = 2 × 2 × 15 = 60.

5. 3,√

11,√

13, 4. 6. 1,√

2,√

3, 2.

7. If√

3 = a

bwhere a and b are integers and have common factor, squaring both sides

3 = a2

b2 a2 = 3b2

3b2 is a multiple of 3 and therefore odd. So a2 is an odd as well and so a is odd. Thismeans a = 3c, ∴ a2 = 9c2 9c2 − 3b2 hence 3c2 = b2.

8. 0.1̇2̇5̇ = 0.125125.125 . . . = x

1000x = 125.125125 . . .

x = 0.125125 . . .

999x = 125

x = 125

999= 0.1̇2̇5̇

Which is a rational number therefore a recurring number is not an irrational number.

9. π = 3.141592654 = 3.14 to 3 s.f.22

7= 3.142857143 = 3.14 to 3 s.f.

355

113= 3.14159292 = 3.141593 to 7 s.f.

π = 3.141592654 = 3.141593 to 7 s.f.

therefore355

113is equal to π to 7 s.f..

Page 19: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 19

10.

5� 3

9� 3

9� 3

5� 3

perimeter = 9 + √3 + 9 + √

3 + 5 − √3 + 5 − √

3 = 28 units.

11. (8� 14 ) m

(8� 14 ) m

(8 + √

14) (

8 − √14

)= 64 − 14 = 50 m2.

12.(√

15 − 2√

3)2

=(√

15)2 + 2

(√15

) (−2

√3)

+(−2

√3)2

= 15 − 4√

45 + 4 × 3 = 27 − 4√

45 = 27 − 12√

5.

13.(√

5 − √3) (√

5 + √3)

=(√

5)2 −

(√3)2 = 5 − 3 = 2.

14.(√

2 + 1)

2 + 2(√

2 − 1)

= 2√

2 + 2 + 2√

2 − 2

perimeter = 4√

2

area =(√

2 + 1) (√

2 − 1)

=(√

2)2 − 12 = 2 − 1 = 1.

15.xy

x − y=

(2√

2 − 1) (

2√

2 + 1)

(2√

2 − 1)

−(

2√

2 + 1)

=(

2√

2)2 − 12

−2= −7

2.

16.

√125

20=

√5 × 5 × 5

4 × 5= 5

2.

17.√

26(

2√

13 − 2√

2)

= √2√

13(

2√

13 − 2√

2)

= 26√

2 − 4√

13.

Page 20: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

20 Pass Publications GCSE Mathematics Higher

18.

(7 − √

2) (

7 + √2)

47=

(7)2 −(√

2)2

47= 49 − 2

47= 47

47= 1.

19.

(6 − √

5) (

6 + √5)

√31

= 36 − 5√31

= 31√31

×√

31√31

= √31.

20.1

a+ 1

b= a + b

ab

a = 2 − √2, b = 2 + √

2 positive irrational numbers

a + b

ab=

(2 − √

2)

+(

2 + √2)

(2 − √

2) (

2 + √2)

= 4

4 − 2= 4

2= 2 ⇒ ab

a + b= 2

4= 1

2.

21.√

7 × √63 = √

7 × √7√

9 = 7 × 3 = 21.

22.√

5 × √500 = k

√5

√5 × √

5√

100 = 50

k = 50√5

×√

5√5

= 10√

5.

23.

√243 − √

3√12

=√

3√

81 − √3√

3√

4

=√

3(9 − 1)√3(2)

= 8

2= 4.

Page 21: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 21

EXERCISES

Topic 5 ∼ Significant figures and decimal places

1. Consider the number 3.141592654 which is correct to 10 significant figures. Writedown the number correct to:

(a) nine significant figures (b) seven significant figures

(c) five significant figures (d) three significant figures and

(e) one significant figure.

2. Consider the number 0.00125. Write down this number to 2 and 1 significant figures.

3. Write the following numbers correct to the approximation given in brackets:

(a) 0.12345 (3 s.f.) (b) 476.7 (3 s.f.)

(c) 46.9539 (4 s.f.) (d) 0.0098765 (5 s.f.)

(e) 0.0098765 (4 s.f.) (f) 0.0098765 (3 s.f.)

(g) 35 × 19 (3 s.f.) (h) 137 × 679 (4 s.f.)

(i) 37.5 × 139.65 (3 s.f.) (j) 479 × 0.012567 (3 s.f.)

(k) 29 × 39 × 767 (3 s.f.).

4. The mass of the earth is calculated to be 5.976 × 1024 kg to 4 significant figures. Writedown this mass to 3, 2 and 1 s.f.

5. The electronic mass is given as 9.109534 × 10−31 kg. Write down this to 3 s.f.

6. Round off to the nearest whole number the following:

(a) 9.3 (b) 7.56

(c) 7.499 (d) 999

(e) 4.75 × 3.76 (f) 0.37 × 6.99 × 37.

7. Round off the following to the nearest 10:

(a) £375 (b) 524 kg

(c) 35.95 mm (d) 10.99 m.

8. 125975 to:

(a) the nearest 10 (b) the nearest 100

(c) the nearest 1000 (d) the nearest 10000.

Page 22: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

22 Pass Publications GCSE Mathematics Higher

SolutionsTopic 5

1. 3.141592654

(a) 3.14159265 (b) 3.141593

(c) 3.1416 (d) 3.14 (e) 3.

2. 0.0013 to 2 s.f.

0.001.

3. (a) 0.123 (b) 477 (c) 46.95

(d) 0.0098765 (e) 0.009877 (f) 0.00988

(g) 35 × 19 = 665 (h) 137 × 679 = 93020

(i) 37.5 × 139.65 = 5236.875 = 5240 to 3 s.f.

(j) 479 × 0.012567 = 6.019593 = 6.02 to 3 s.f.

(k) 29 × 39 × 767 = 867477 = 867000 to 3 s.f.

4. 5.976 × 1024 kg

5.98 × 1024 kg to 3 s.f.

6.0 × 1024 kg to 2 s.f.

6 × 1024 kg to 1 s.f.

5. 9.109534 × 10−31 kg

9.11 × 10−31 kg to 3 s.f.

6. (a) 9 (b) 8 (c) 7 (d) 999

(e) 4.75 × 3.76 = 17.86 (18)

(f) 0.37 × 6.99 × 37 = 95.6931 (96).

7. (a) £375 = £380 to the nearest 10 (b) 524 kg = 520 kg to the nearest 10

(c) 35.95 mm = 40 mm to the nearest 10 (d) 10.99 m = 10 m to the nearest 10.

8. (a) 125980 (b) 126000 (c) 126000 (d) 130000.

Page 23: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 23

EXERCISES

Topic 6 ∼ Upper and lower bounds

1. The mass of a young girl is 45 kg correct to the nearest kg. What are the upper andlower bounds of the mass?

2. The height of a basket ball player is 2.05 m to the nearest centimetre. What are theupper and lower bounds of the height?

3. Asif weighs a bag of potatoes. He records the mass as 1.1 kg. The mass is recorded tothe nearest tenth of a kilogram. What are the upper and lower bounds of the possiblemass?

4. A straight road is measured to the nearest 5 m, the road is 1000 m long. Find the actuallength.

5. A rectangular garden is 10 m × 6 m, the measurements are given to the nearest metre.Determine the upper and lower bounds of the sides and of the area.

6. The area of a triangle is given as 12× base × height

h

b

If h = 20 cm to the nearest centimetre and b = 10 cm to the nearest centimetre, findthe maximum and minimum areas.

7. In the diagram there are three shapes, a square, a rectangle and an equilateral triangle

3.5 4.5 1.5

3.51.5 1.5

2.5

(i) (ii) (iii)

All the sides are given in mm and to the nearest tenth of a mm.

(a) Find the longest and the shortest sides.

(b) Find the maximum and minimum perimeters.

Page 24: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

24 Pass Publications GCSE Mathematics Higher

8. The sides are given to the nearest hundredth of a centimetre.

h

5.00 cm

5.00 cm5.00 cm

(a) Find the longest and the shortest height, h.

(b) Find the maximum and minimum area.

9. The volume of a metal block of copper is 50 cm3 to the nearest 1 cm3, the density ofcopper is 8.9 g/cm3 to the nearest tenth. Calculate:

(a) The upper bound of its mass.

(b) The lower bound of its mass.

Approximations

10. Write down the following numbers to the nearest thousand:

(a) 7399 (b) 90501 (c) 13099

(d) 656935 (e) 75555 (f) 2599.

11. Write down the following numbers to the nearest integer:

(a) 9.6 (b) 10.7 (c) 10.45 (d) 15.55

(e) 5.125 (f) 5.509 (g) 5.499 (h) 6.53

(i) 56.63 (j) 839.6 (k) 1023.5 (l) 15235.435.

12. Write down the following numbers to the nearest ten:

(a) 29 (b) 24 (c) 35 (d) 127 (e) 3,059

(f) 5,395 (g) 6,949 (h) 79,499 (i) 856.79 (j) 7,509.

13. Find the sum of 7,679, 9,699, 10,950, 15,999. Give your answer as an approximationto the nearest ten.

14. Find the quotients of the following numbers to the nearest unit:

(a) 769 by 29 (b) 76 by 23

(c) 623 by 24 (d) 87 by 4

(e) 425 by 23 (f) 576 by 26

(g) 975 by 22 (h) 699 by 28.

15. A book contains 21,976 words. Write down the number of words to the nearest

(i) ten (ii) hundred (iii) thousand.

Page 25: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 25

Estimations

16. Without the use of a calculator, workout the following estimations to the nearest ten:

(a)7.69 × 9.49

3.75(b)

(3.72 − 2.22

) × 103

10.49

(c)47.9 × 8.90

24.9.

17. Without the use of a calculator give the answer to 1 s.f. for the following:

(a)4.93 × 5.47

24.9(b)

5.79 × 7.79

4.75

(c)655 × 856

66 × 207.

18. Estimate the following quotients to 1 s.f.:

(a)7699 × 3333

8888(b)

279.5 ÷ 62.9

55.5 ÷ 14.9

(c)3.25 × 37.5 × 7.55

39.9.

19. (a) Use the calculator to work out the value of√

2.562+4.353.95−1.05 . Write down all the figures

on your calculator display.

(b) Give your answer to part (a) to an appropriate degree of accuracy.

20. Work out an estimate for the following quotients:

(a)48.0356

4.012 (b)27.975

7.035(c)

77.95

0.01.

21. Use the calculator to work out the value of 27.9×13.932.7−14.9

(a) Write down all the figures on your calculator display.

(b) Write your answer to part (a) to an appropriate degree of accuracy.

Page 26: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

26 Pass Publications GCSE Mathematics Higher

SolutionsTopic 6 ∼ Estimations and Errors

1. 45.5 kg is the upper bound of the mass, 44.5 kg is the lower bound of the mass.

If m is the mass of the young girl

44.5 < m < 45.5

m = 45.499 this is 45 to 2 s.f.

m = 44.501 this is 45 to 2 s.f.

2. 2.055 m the upper bound of the height

2.045 m the lower bound of the height.

If h is the height of the basket ball player

204.5 cm < h < 205.5 cm

h = 205.499 cm which to the nearest cm is 205

h = 204.501 cm which to the nearest cm is 205.

3. 1.15 kg is the upper bound of the mass. 1.05 kg is the lowest bound of the mass.

4. The length of the road is between 997.5 m and 1002.5 m.

1000 m ± 2.5 m recorded to the nearest x, M ± 1

2x.

5.6 m

10 m

9.5 m < length < 10.5 m

5.5 m < width < 6.5 m

9.5 × 5.5 m2 < Area < 10.5 × 6.5 m2

52.25 m2 < A < 68.25 m2.

6. Area max = 20.5 × 10.5

2= 107.625 cm2 108 cm2 to 3 s.f.

Area min = 19.5 × 9.5

2= 92.625 cm2 92.6 cm2 to 3 s.f.

Page 27: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 27

7. (a) (i) 3.55 mm longest; 3.45 mm shortest side

(ii) 4.55 mm, 4.45 mm, 2.25 mm, 2.45 mm

(iii) 1.55 mm, 1.45 mm

(b) (i) 4 × 3.55 = 14.2 mm maximum perimeter

4 × 3.45 = 13.8 mm minimum perimeter

(ii) 2 × 4.55 + 2 × 2.55 = 14.2 mm

2 × 4.45 + 2 × 2.45 = 13.8 mm

(iii) 3 × 1.55 = 4.65 mm

3 × 1.45 = 4.35 mm.

8.

h

2.50cm

2.50cm

5.00 cm5.00 cm

(a) hmax = √5.0052 − 2.4952

= 4.33877863 = 4.34 cm to 3 s.f.

hmin =√

4.9952 − 2.5052 = 4.321458087

= 4.32 cm to 3 s.f.

(b) A = 1

2hb �⇒ Amax = 1

2hmaxbmax

= 1

24.33877863 × 5.005

= 10.85779352 = 10.9 cm2 to 3 s.f.

Amin = 1

2hmin × bmin

= 1

2× 4.321458087 × 4.995

= 10.79284157 = 10.8 cm2 to 3 s.f.

Page 28: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

28 Pass Publications GCSE Mathematics Higher

9. Vcu = 50 cm3, ρcu = 8.9 g/cm3

(a) mmax = ρcuVcu = 50.5 × 8.95

= 451.975 g = 452 g to 3 s.f.

min = ρcuVcu = 49.5 × 8.85

= 438.075 = 438 g to 3 s.f.

Approximation.

10. (a) 7000 (b) 91000 (c) 13000

(d) 657000 (e) 76000 (f) 3000.

11. (a) 10 (b) 11 (c) 10 (d) 16 (e) 5 (f) 6

(g) 5 (h) 7 (i) 57 (j) 840 (k) 1024 (l) 15235

12. (a) 30 (b) 20 (c) 40 (d) 130 (e) 3060

(f) 5400 (g) 6950 (h) 79500 (i) 860 (j) 7510

13. 76799699

1095015999 +44327 44330 to the nearest 10

14. (a)769

29= 26.51724138 = 27 to the nearest unit

(b)79

23= 3.434782609 = 3 to the nearest unit

(c)623

24= 25.95833333 = 26 to the nearest unit

(d)87

4= 21.75 = 22 to the nearest unit

(e)425

23= 18.47876087 = 18 to the nearest unit

(f)576

26= 22.15384615 = 22 to the nearest unit

(g)975

22= 44.31818182 = 44 to the nearest unit

(h)699

28= 24.96428571 = 25 to the nearest unit

Page 29: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

Pass Publications GCSE Mathematics Higher 29

15. (i) 21980 (ii) 22000 (iii) 22000

Estimations.

16. (a)7.69 × 9.49

3.75= 8 × 9

4= 18

(b)

(3.72 − 2.22

) × 103

10.49= 1.5 × 5.9 × 103

10.49

= 1.5 × 6 × 100

10= 900

10= 90

(c)47.9 × 8.90

24.9= 50 × 9

25= 18.

17. (a)4.93 × 5.47

24.5= 5 × 5

25= 1 to 1 s.f.

(b)5.79 × 7.79

24.9= 6 × 8

25= 48

25= 2 to 1 s.f.

(c)655 × 856

66 × 207=70010 ×8004

70 ×200= 40.

18. (a)7699 × 3333

8888= 7700 × 3000

9000= 7700

3= 2900 = 3000 to 1 s.f.

(b)279.5 ÷ 62.9

55.5 ÷ 14.9=

300

6060

15

= 5

4= 1 to 1 s.f.

(c)3.25 × 37.5 × 7.55

39.9= 3 × 40 × 7

40= 20 to 1 s.f.

19. (a)

√2.562 + 4.35

3.95 − 1.05=

√10.9036

2.9= 3.302059963

2.9= 1.138641367 = 1.14 to 3 s.f.

20. (a)48.0356

4.012 = 48

16= 3 (b)

27.975

7.035= 28

7= 4 (c)

77.95

0.01= 7795 = 7800.

21. (a)27.9 × 13.9

32.7 − 14.9= 387.81

17.8= 21.78707865 (b) 21.8 to 3 s.f.

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30 Pass Publications GCSE Mathematics Higher

EXERCISES

Topic 7 ∼ Fractions

1.8

3− 5

3÷ 20

212. 1 −

(1

4+ 3

4× 4

5

)

3.1

2+ 1

3+ 1

44. 1

1

6+ 3

7+ 1

42

5. 1 −(

2

3+ 1

4

)6. 2

2

3+ 1

3

6+ 1

1

9

7.

(3 − 5

2

(1 + 1

5

)8. 3

2

3+ 2

3

5

9.7

4− 3

4× 8

910. 1

3

8+ 2

7

8

11.1

5− 6

5÷ 10

3512. 12

2

3+ 6

3

7

13.1

5− 3

5÷ 12

714.

1

8+ 2

8+ 3

8

15.

(2 + 3

4

(3

4− 5

)16.

1

2+ 2

3+ 3

4+ 4

5

17.26

7− 22

7× 10

3318. 2

1

4+ 3

1

8+ 4

1

16

19. 13

4÷ 7

420. 4

3

4− 1

3

8

21.16

5− 6

5÷ 10

2522. 2

15

16− 1

5

8

23. 23

8÷ 1

3

424. 2

3

9− 1

3

25. 33

4÷ 1

1

426.

43

64− 7

16

27. 23

8+ 1

3

428. 5

4

5− 2

3

5

29. 33

4− 1

1

430. 1

3

4+ 5

6− 1

3

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Pass Publications GCSE Mathematics Higher 31

31.33

4 × 45

178 ÷ 3

8

32.3

11× 5

2

5

33.3

7− 6

7÷ 30

734.

1 − 35

310 − 1

5

35. 21

2+ 3

7

8÷ 3

1

436.

(3

4− 1

2

(3

1

2+ 2

1

4

)2

37. Find the sum of 234, 33

5 and 412.

38. Subtract 178 from the sum of 23

8 + 349.

39. Find the difference between 735 and 32

3.

40. From the sum of 338 and 21

4 subtract 3 116.

41. Subtract 238 from the difference between 4 3

16 and 1 432.

42. Find the product of 7 316 and 13

4.

43. 32

5÷ 2

1

1044. 2

3

5÷ 1

1

20

45. Simplify51

4 × 23

114 × 11

8

46.43

5

3 310 ÷ 1

10

47.41

4 × 23

14 × 23

5

48.

(1

4+ 1

3

)× 3

549.

145 ÷ 310

1523 × 13

5

50. Multiply the sum of 335 and 23

7 by 237.

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32 Pass Publications GCSE Mathematics Higher

SolutionsTopic 7 ∼ Fractions

1.8

3− 5

3÷ 20

21= 8

3− 5

3× 21

20= 8

3− 7

4= 32

12− 21

12= 11

12.

2. 1 −(

1

4+ 3

4× 4

5

)= 1 −

(1

4+ 3

5

)= 1 − 5 + 12

20= 1 − 17

20= 3

20.

3.1

2+ 1

3+ 1

4= 6

12+ 4

12+ 3

12= 13

12= 1

1

12.

4. 11

6+ 3

7+ 1

42= 7

6+ 3

7+ 1

42= 7 × 7

6 × 7+ 3 × 6

7 × 6+ 1

42

= 49

42+ 18

42+ 1

42= 68

42= 1

26

42= 1

13

21.

5. 1 −(

2

3+ 1

4

)= 1 −

(2 × 4

3 × 4+ 3 × 1

4 × 3

)

= 1 −(

8

12+ 3

12

)= 1 − 11

12= 1

12.

6. 22

3+ 1

3

6+ 1

1

9= 8

3+ 9

6+ 10

9

= 8 × 6

3 × 6+ 9 × 3

6 × 3+ 10 × 2

9 × 2

= 48

18+ 27

18+ 20

18= 95

18+ 5

5

18.

7.

(3 − 5

2

(1 + 1

5

)=

(6

2− 5

2

(5

5+ 1

5

)

= 1

2÷ 6

5= 1

2× 5

6= 5

12.

8. 32

3+ 2

3

5= 11

3+ 13

5= 11 × 5

15+ 13 × 3

5 × 3

= 55

15+ 39

15= 94

15= 6

4

15.

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Pass Publications GCSE Mathematics Higher 33

9.7

4− 3

4× 8

9= 7

4− 1 × 2

1 × 3= 7

4− 2

3= 7 × 3

4 × 3− 2 × 4

3 × 4= 21

12− 8

12= 13

12= 1

1

12.

10. 13

8+ 2

7

8= 11

8+ 23

8= 34

8= 4

2

8= 4

1

4.

11.1

5− 6

5÷ 10

35= 1

5− 6

5× 35

10= 1

5− 3 × 7

1 × 5= 1

5− 21

5= −20

5= −4.

12. 122

3+ 6

3

7= 38

3+ 45

7= 38 × 7

3 × 7+ 45 × 3

7 × 3= 266

21+ 135

21= 401

21= 19

2

21.

13.1

5− 3

5÷ 12

7= 1

5− 3

5× 7

12= 1

5− 1 × 7

5 × 4= 4 × 1

5 × 4− 7

20= 4 − 7

20= − 3

20.

14.1

8+ 2

8+ 3

8= 6

8= 3

4.

15.

(2 + 3

4

(3

4− 5

)=

(2 × 4

4+ 3

4

(3

4− 5 × 4

1 × 4

)

=(

8

4+ 3

4

(3

4− 20

4

)= 11

4÷ −17

4= 11

4× − 4

17= −11

17.

16.1

2+ 2

3+ 3

4+ 4

5

= 30

2 × 30+ 2 × 20

3 × 20+ 3 × 15

4 × 15+ 4 × 12

5 × 12

= 30

60+ 40

60+ 45

60+ 48

60= 163

60= 2

43

60.

17.26

7− 22

7× 10

33= 26

7− 2

7× 10

3= 26 × 3

7 × 3− 20

21= 78 − 20

21= 58

21= 2

16

21.

18. 21

4+ 3

1

8+ 4

1

16= 9

4+ 25

8+ 65

16

= 9 × 4

4 × 4+ 25 × 2

8 × 2+ 65

16= 36

16+ 50

16+ 65

16= 151

16= 9

7

16.

19. 13

4÷ 7

4= 7

4÷ 7

4= 7

4× 4

7= 1.

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34 Pass Publications GCSE Mathematics Higher

20. 43

4− 1

3

8= 19

4− 11

8= 19 × 2

4 × 2− 11

8= 38

8− 11

8= 27

8= 3

3

8.

21.16

5− 6

5÷ 10

25= 16

5− 6

5× 25

10= 16

5− 3 × 5

1 × 5= 16

5− 15

5= 1

5.

22. 215

16− 1

5

8= 2

15

16− 1

10

16= 1

5

16.

23. 23

8÷ 1

3

4= 2

3

8÷ 1

6

8= 19

8÷ 14

8= 19

8× 8

14= 19

14= 1

5

14.

24. 23

9− 1

3= 21

9− 3

9= 18

9= 2.

25. 33

4÷ 1

1

4= 15

4÷ 5

4= 15

4× 4

5= 3.

26.43

64− 7

16= 43

64− 7 × 4

16 × 4= 43 − 28

64= 15

64.

27. 23

8+ 1

3

4= 2

3

8+ 1

6

8= 3

9

8= 4

1

8.

28. 54

5− 2

3

5= 29

5− 13

5= 16

5= 3

1

5.

29. 33

4− 1

1

4= 2

2

4= 2

1

2.

30. 13

4+ 5

6− 1

3= 7

4+ 5

6− 1

3= 7

4+ 5

6− 2

6

= 7

4+ 3

6= 7 × 3

4 × 3+ 3 × 2

6 × 2= 21 + 6

12= 27

12= 2

3

12= 2

1

4.

31.3

3

4× 4

5

17

8÷ 3

8

=15

4× 4

515

8× 8

3

= 3

5.

32.3

11× 5

2

5= 3

11× 27

5= 81

55= 1

26

55.

33.3

7− 6

7÷ 30

7= 3

7− 6

7× 7

30= 3

7− 1

5= 3 × 5

7 × 5− 1 × 7

5 × 7= 15 − 7

35= 8

35.

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Pass Publications GCSE Mathematics Higher 35

34.1 − 3

53

10− 1

5

=5

5− 3

53

10− 2

10

=2

51

10

= 2

5× 10

1= 4.

35. 21

2+ 3

7

8÷ 3

1

4= 5

2+ 31

8÷ 13

4= 5

2+ 31

8× 4

13= 5

2+ 31 × 1

26

= 5 × 13

2 × 13+ 31

26= 65 + 31

26= 96

26= 3

18

26= 3

9

13.

36.

(3

4− 1

2

(3

1

2+ 2

1

4

)2

=(

3

4− 2

4

(7

2+ 9

4

)2

= 1

(14

4+ 9

4

)2

= 1

(23

4

)2

= 1

4× 16

232 = 4

529.

37. 23

4+ 3

3

5+ 4

1

2= 2 + 3 + 4 + 3

4+ 3

5+ 1

2= 9 + 3

4+ 3

5+ 2

4

= 9 + 5

4+ 3

5= 10 + 1

4+ 3

5= 10 + 5

20+ 12

20= 1017

20.

38.

(2

3

8+ 3

4

9

)− 1

7

8= 19

8+ 31

9− 15

8

= 31

9+ 4

8= 31

9+ 1

2= 62 + 9

18= 71

18= 3

17

18.

39. 73

5− 3

2

3= 38

5− 11

3= 38 × 3

5 × 3− 11 × 5

3 × 5= 114

15− 55

15= 59

15= 3

14

15.

40.

(3

3

8+ 2

1

4

)− 3

1

16=

(27

8+ 9

4

)− 49

16= 27

8+ 18

8− 49

16= 45

8− 49

16

= 45 × 2

8 × 2− 49

16= 90

16− 49

16= 41

16= 2

9

16.

41.

(4

3

16− 1

4

32

)− 2

3

8= 67

16− 36

32− 19

8

= 67 × 2

16 × 2− 36

32− 76

32= 134 − 36 − 76

32= 22

32= 11

16.

Page 36: Number - static.premiersite.co.ukstatic.premiersite.co.uk/32733/docs/1197857_1.pdf · Number EXERCISES Topic 1 ∼ Bodmas 1. ... Topic 1 Bodmas 1. (a) ... 4 +2 1 4 = 11 10 ÷ 7 4

36 Pass Publications GCSE Mathematics Higher

42. 73

16× 1

3

4= 115

16× 7

4= 805

64= 12

37

64.

43. 32

5÷ 2

1

20= 17

5÷ 21

10= 17

5× 10

21= 34

21= 1

13

21.

44. 23

5÷ 1

1

20= 13

5÷ 21

20= 13

5× 20

21= 52

21= 2

10

21.

45.5

1

4× 2

3

11

4× 1

1

8

=21

4× 2

35

4× 9

8

=7

245

32

= 7

2× 32

45= 7 × 16

45= 112

45= 2

22

45.

46.4

3

5

33

10÷ 1

10

=23

533

10× 10

1

= 23

5 × 33= 23

165.

47.4

1

4× 2

31

4× 2

3

5

=17

4× 2

31

4× 13

5

=17

613

20

= 17

6× 20

13= 17 × 10

39= 170

39= 4

14

39.

48.

(1

4+ 1

3

)× 3

5=

(3

12+ 4

12

)× 3

5= 7

12× 3

5= 7

20.

49.1

4

5÷ 3

10

152

3× 1

3

5

=9

5÷ 55

152

3× 8

5

=9

5× 15

5516

15

= 27

55÷ 16

15

= 27

55× 15

16= 27 × 3

11 × 16= 81

176.

50.

(3

3

5+ 2

3

7

)× 2

3

7=

(18

5+ 17

7

)× 17

7= 18 × 7 + 17 × 5

35× 17

7

= 126 + 85

35× 17

7= 211 × 17

35 × 7= 3587

245= 14

157

245.

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Pass Publications GCSE Mathematics Higher 37

EXERCISES

Topic 8 ∼ Sequences

1. Write down the 6th and 7th terms of the sequence 12, 22, 32, 42, 52, . . . , . . . .

2. Write down the 5th and 6th terms of the sequence, 3, 12, 27, 48, . . . , . . . .

3. Write down the 9th and 10th terms of the sequence 11,

12,

13, . . . .

4. Calculate the nth and 50th terms:

(a) 1, 4, 7, 10, 13, 16, . . .

(b) 1, 2, 4, 8, 16, 32, . . .

(c) 10, 8, 6, 4, 2, 0, −2, . . . .

5. 2 × 3, 3 × 4, 4 × 5, . . . , . . . . Find the next two terms.

6. Write the next three terms of the sequence 1, 1, 2, 3, 5, 8, . . . , . . . . . . . .

7. Write the next three terms of the sequence a, a, 2a, 3a, 5a, 8a, . . . .

8. 1, 12,

34,

58,

1116,

2132, . . . , . . . . Write the next term.

9. 10, 20, 15, 1712, 161

4, 1678, . . . . Write the next three terms.

10. Write down three terms of an arithmetic sequence.

11. Write down five terms of a geometric series.

12. Write down the geometric mean of the sequence 3, 9, 27.

13. What is the common ratio of the sequence 1, 13,

19,

127,

181, . . .

14. Write down two harmonic sequences.

15. Here are the first five terms of a sequence 1, 3, 6, 10, 15, . . . , . . . . Find the followingfour terms.

16. Find the 4th and 5th terms:

(a) 0.01, 0.0001, 0.000001, . . . , . . .

(b) 12,

34,

58, . . . , . . .

(c) 15, 11, 7, . . . , . . .

17. 1, 23,

35,

47,

59, . . . , . . . . Find the missing terms.

18. 2, 212, 22

3, 234, 24

5, . . . . What is the convergent limit?

19. Write down a divergent sequence.

20. Write down an oscillating sequence.

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38 Pass Publications GCSE Mathematics Higher

SolutionsTopic 8 ∼ Sequences

1. 62, 72.

2. 3, 12, 27, 48, . . .

12 × 3, 22 × 3, 32 × 3, 42 × 3, 52 × 3, 62 × 3

∴ 75, 108.

3.1

9,

1

10since

1

nis the nth term

substitute n = 9 and n = 10.

4. (a) 1, 4, 7, 10, 13, 16, . . .

the differences are constant, 3, therefore the nth term is linear an + b

If n = 1, a + b = 1 . . . (1)

n = 2, 2a + b = 4 . . . (2)

(2) − (1) = a = 3, substitute in (1)

3 + b = 1 ⇒ b = −2

therefore 3n − 2 is the nth term.

If n = 50, 3 × 50 − 2 = 150 − 2 = 148

the nth and 50th terms are 3n − 2 and 148.(b) 1, 1×2, 1×2×2, 1×2×2×2, . . .

1, 2, 4, 8, . . . 2n−1

If n = 1, 21−1 = 20 = 1;

If n = 2, 22−1 = 2 etc

If n = 50, 249 is the 50th term.

(c) −2n + 12, −2(50) + 12 = −100 + 12 = −88.

5. 2 × 3, 3 × 4, 4 × 5, 5 × 6, 6 × 7.

6. 1, 1, 2, 3, 5, 8, 13, 21, 34 Fibonacci series

1 + 1 = 2, 1 + 2 = 3, 3 + 5 = 8, 5 + 8 = 13,

8 + 13 = 21, 13 + 21 = 34.

7. a, a, 2a, 3a, 5a, 8a, 13a, 21a, 34a Fibonacci series.

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Pass Publications GCSE Mathematics Higher 39

8. 1,1

2,

3

4,

5

8,

11

16,

21

32

1 + 1

22

=3

22

= 3

41

2+ 3

42

=2

4+ 3

42

=5

42

= 5

83

4+ 5

82

=6

8+ 5

82

=11

82

= 11

165

8+ 11

162

=10

16+ 11

162

=21

162

= 21

3211

16+ 21

322

=22

32+ 21

322

= 43

64.

9. 10, 20, 15, 171

2, 16

1

2, 16

7

8, 16

9

16, 16

23

32, 16

41

6410 + 20

2= 30

2= 15

20 + 15

2= 35

2= 17

1

2

15 + 171

22

=32

1

22

= 161

4

171

2+ 16

1

42

=33

3

42

= 161

2+ 3

8= 16

7

8

161

4+ 16

7

82

=32 + 9

82

= 169

16

167

8+ 16

9

162

=32 + 23

162

= 16 + 23

32= 16

23

32

169

16+ 16

23

322

= 16 + 41

64= 16

41

64.

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40 Pass Publications GCSE Mathematics Higher

10. 2, 4, 6 we add 2 to the first term, makes 4

we add 2 to the second term, makes 6.

11. 1, 1 × 1

3, 1 × 1

3× 1

3, 1 × 1

3× 1

3× 1

3, 1 × 1

3× 1

3× 1

3× 1

3, 1,

1

3,

1

9,

1

27,

1

81.

12. 3, 9, 27; a, b, c is a geometric series

b = √ac is the geometric mean

therefore√

3 × 27 = √81 = 9 is the geometric mean.

13.

1

31

=1

91

3

= 1

3= common ratio.

14.1

5,

1

8,

1

11; 5, 8, 11 form an arithmetic sequence

1

6,

1

10,

1

146, 10, 14 form an arithmetic sequence.

15. 1, 3, 6, 10, 15, 21, 28, 36, 45

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

10 + 5 = 15

15 + 6 = 21

21 + 7 = 28

28 + 8 = 36

36 + 9 = 45

The rule is, we addone more every timeafter the first one.

16. (a) 0.01, 00001, 0.000001, 0.00000001, 0.0000000001

we multiply each time by 0.01

(b)1

2,

3

4,

5

8,

11

16,

21

321

2+ 3

42

=2

4+ 3

42

= 5

85

8+ 3

42

= 11

16

5

8+ 11

162

= 21

32

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Pass Publications GCSE Mathematics Higher 41

(c) 15 − 4 = 11, 11 − 4 = 7,

7 − 4 = 3, 3 − 4 = −1.

17.1

1,

2

3,

3

5,

4

7,

5

9,

6

11,

7

13

The rule is: we add one to the numerator and the denominator is an odd numberincreasing by 2.

18. 2, 21

2, 2

2

3, 2

3

4, 2

4

5, . . . 3

5

6,

6

7,

7

8,

8

9,

9

10, ....

n

n + 1→ 1.

19. 3, 9, 27, 81, ....3 × 3n−1

we multiply each term by 3 indefinitely.

20. 2, −2, 2, −2.

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42 Pass Publications GCSE Mathematics Higher

EXERCISES

Topic 9 ∼ Standard form

For the following, calculators are not allowed:

1. Simplify the following indicial arithmetic expressions:

(a) 25 (b) 2−4 (c) 4−12 (d) 25−1

2

(e) 813 (f) 8−1

3 (g) 16−14 (h) 125

13 .

2. Find the values of the following root expressions:

(a)√

25 (b) 3√

125 (c) 3√

64 (d)√

256 (e) 5√

32

(f) 4√

256 (g) 8√

256 (h) 4√

81 (i) 5√

243.

3. Evaluate the following:

(a)(2−1

)−2(b)

(3−1

)2(c)

(52

)−2(d)

(102

)−3(e)

(10−2

)−1

(f)1(

3−1)−3 (g)

1

10−5(h) 10−3 (i)

(10−2

)3(j)

((10−1

)−2)−3

.

4. Evaluate the following:

(a)

√9

4(b) 3

√64

27(c)

√1

32(d)

√36 × 4 (e) 3

√125

243.

5. Simplify:

(a) 3√

3 − √3 + 5

√3 (b) 5

√5 + √

5 (c)√

5 × √5 × √

5 × √5

(d)√

8 (e)√

27 (f)√

125 (g)√

75

(h)√

45 (i)

√1

20(j)

√25

8(k)

√64

9.

6. Express√

27(√

3 − 1)

in terms of prime numbers.

7. Express√

8(

3 − √32

)in terms of prime numbers.

8. Write in standard form and to 3 s.f.:

(a) 0.0396 (b) 0.569 (c) 5.89 (d) 0.0000349

(e) 11000 (f) 23.695 (g) 6495000 (h) 73990

(i)10

0.0002(j)

100

0.00001(k)

1

0.005.

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Pass Publications GCSE Mathematics Higher 43

9. Write in full the following standard index forms:

(a) 7.35 × 10−6 (b) 3.65 × 109 (c) 1.23 × 105

(d) 3.5 × 10−6 (e) 4.756 × 101 (f) 1.234 × 10−5

(g) 3.75 × 10−3 + 2.75 × 10−6 (h) 7.95 × 105 − 2.95 × 104

(i) 2.5 × 106 × 5 × 10−5

(j)1.25 × 107

6.25 × 10−6 (k)2.187 × 10−10

2.43 × 10−12 .

10. Write one billion and one trillion in standard form.

11. Find the smallest number of the following:

3.5 × 103, 3.55 × 103, 3.55 × 10, 3.5 × 10−3.

12. In physics, the electronic charge is given as 1.6021892 × 10−19 C. Write this numberin full and to 3 s.f.

13. In physics, the Avogadro number in mol−1 is given as 6.022045 × 1023. Write thisnumber in full and to 3 s.f.

14. In electronics, Boltzman’s constant in Jk−1 is given as 1.38062 × 10−23. Write thisnumber in full and to 3 s.f.

15. In electronics, the electronic mass in kg is given as 9.109534 × 10−31. Write thisnumber in full and to 3 s.f.

16. In physics, Planck’s constant in Js is given as 6.626176 × 10−34. Write this numberin full and to 3 s.f.

17. In physics, the gravitational constant in Nm2kg−2 is given as 0.00000000006672.Write this number in standard form to 3 s.f.

18. Work out:

(a)(3 × 105

) + (7 × 104

)(b) 4 × 10−6 − 3 × 10−7

(c)(5 × 104

) × (6 × 105

)(d)

8 × 106

4 × 10−12 .

19. The wealth of the UK is estimated to be 5 trillion pounds. Write this number in full.

20. The wealth of the world is estimated to be £1022. Write this out in full and also writeit in words.

21. The mass of the earth is 5.976 × 1024 kg and the volume is given as 43π (6800 km)3.

What is the average density of the earth in standard form to 3 s.f.

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44 Pass Publications GCSE Mathematics Higher

SolutionsTopic 9 ∼ standard form

1. (a) 25 = 2 × 2 × 2 × 2 × 2 = 32 (b) 2−4 = 1

24 = 1

2 × 2 × 2 × 2= 1

16

(c) 4−12 = 1

412

= 1√4

= 1

2(d) 25−1

2 = 1

2512

= 1√25

= 1

5

(e) 813 = 3

√8 = 2 (f) 8−1

3 = 1

813

= 13√

8= 1

2

(g) 16−14 = 1

1614

= 14√

16= 1

2(h) 125

13 = 3

√125 = 5.

2. (a)√

25 = 5 (b) 3√

125 = 5 (c) 3√

64 = 4

(d)√

256 = 16 (e) 5√

32 = 2 (f) 4√

256 = 4

(g) 8√

256 = 8√

16 × 16

= 8√

2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2 = 8√

28 = 21 = 2

(h) 4√

81 = 3 (i) 5√

243 = 5√35 = 3

55 = 31 = 3.

3. (a)(2−1

)−2 = 22 = 4 (b)(3−1

)−2 = 3−2 = 1

32 = 1

9

(c)(52

)−2 = 5−4 = 1

54 = 1

625(d)

(102

)−3 = 10−6 = 0.000001

(e)(10−2

)−1 = 102 = 100 (f)1

(3−1)−3 = 1

33 = 1

27

(g)1

10−5= 105 = 100000 (h) 10−3 = 1

103 = 1

1000

(i)(10−2

)3 = 10−6 = 1

1000000(j)

((10−1

)−2)−3 = 10−6 = 1

1000000.

4. (a)

√9

4= 3

2(b) 3

√64

27= 4

3(c)

√1

32= 1

4√

2

(d)√

36 × 4 = √36

√4 = 6 × 2 = 12

(e) 3

√125

243= 3

√5 × 5 × 5

3 × 3 × 3 × 3 × 3= 5

3 3√

9.

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Pass Publications GCSE Mathematics Higher 45

5. (a) 3√

3 − √3 + 5

√3 = 7

√3 (b) 5

√5 + √

5 = 6√

5

(c)√

5 × √5 × √

5 × √5 = 5 × 5 = 25 (d)

√8 = √

2 × 2 × 2 = 2√

2

(e)√

27 = √3 × 3 × 3 = 3

√3 (f)

√125 = √

5 × 5 × 5 = 5√

5

(g)√

75 = √5 × 5 × 3 = 5

√3 (h)

√45 = √

3 × 3 × 5 = 3√

5

(i)1√20

= 1√2 × 2 × 5

= 1

2√

5(j)

√25

8=

√5 × 5

2 × 2 × 2= 5

2√

2

(k)

√64

9= 8

3.

6.√

27(√

3 − 1)

= √3 × 3 × 3

(√3 − 1

)= 3

√3

(√3 − 1

)= 3

(3 − √

3)

.

7.√

8(

3 − √32

)= 3

√8 − √

8√

8√

4 = 3 × 2√

2 − 8 × 2 = 2(

3√

2 − 23)

.

8. (a) 0.0396 = 3.96 × 10−2 (b) 0.569 = 5.69 × 10−1

(c) 5.89 = 5.89 × 100 (d) 0.0000349 = 3.49 × 10−5

(e) 11000 = 1.10 × 104 (f) 23.695 = 2.37 × 101

(g) 6495000 = 6.50 × 106 (h) 73990 = 7.40 × 104

(i)10

0.0002= 100000

2= 5.00 × 104 (j)

100

0.00001= 10000000 = 1.00 × 107

(k)1

0.005= 1000

5= 200 = 2.00 × 102.

9. (a) 7.35 × 10−6 = 0.00000735 (b) 3.65 × 109 = 3650000000

(c) 1.23 × 105 = 123000 (d) 3.5 × 10−6 = 0.0000035

(e) 4.756 × 101 = 47.56 (f) 1.234 × 10−5 = 0.00001234

(g) 3.75 × 10−3 + 2.75 × 0−6 = 0.00375 + 0.00000275 = 0.00375275

(h) 7.95 × 105 − 2.95 × 104 = 795000 − 29500 = 765500

(i) 2.5 × 106 × 5 × 10−5 = 12.5 × 101 = 125

(j)1.25 × 107

6.25 × 10−6 = 1

5× 1013 = 0.2 × 1013

(k)2.187 × 10−10

2.43 × 10−12 = 2187 × 10−13

243 × 10−14 = 9 × 10−13+14 = 9 × 10 = 90.

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46 Pass Publications GCSE Mathematics Higher

10. One billion = 1000000000 = 1.00 × 109

One trillion = 1000000000000 = 1.00 × 1012.

11. 3500, 3550, 35.5, 0.0035

3.5 × 10−3 is the smallest number.

12. 1.6021892 × 10−19

= 0.00000000000000000016021892 C

= 0.000000000000000000160 C to 3 s.f.

13. 6.022045 × 1023

= 602204500000000000000000 mol−1

= 602000000000000000000000 mol−1.

14. 1.38062 × 10−23

= 0.0000000000000000000000138 Jk−1.

15. 9.109534 × 10−31 kg

= 0.000000000000000000000000000000911kg.

16. 0.000000000000000000000000000000000663 Js.

17. 0.00000000006672 = 6.67 × 10−11 Nm2 Kg−2.

18. (a)(3 × 105

) + (7 × 104

) = 300000 + 70000 = 370000 = 3.7 × 105

(b) 4 × 10−6 − 3 × 10−7 = 0.000004 − 0.0000003 = 0.0000397 = 3.97 × 10−5

(c)(5 × 104

) × (6 × 105

) = 30 × 109 = 3.00 × 1010

(d)8 × 106

4 × 10−12 = 2 × 1018 = 2.00 × 1018.

19. £5000000000000000.

20. £10000000000000000000000

Ten million trillion.

21. ρ = m

V= 5.976 × 1024

4

3π(6800000)3

= 4537.276454 = 4.54 × 103 kg/m3.