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CHAPTER 2 FRACTIONS, DECIMALS AND PERCENTAGES

EXERCISE 5, Page 10

1. Change the improper fraction 15

7 into a mixed number.

15

7 =

12

7 as a mixed number

2. Change the mixed number 24

9 into an improper fraction.

24

9 = 2 +

4

9 and 2

18

9 hence, 2

4

9 =

18

9 +

4

9 =

22

9 as an improper fraction

3. A box contains 165 paper clips. 60 clips are removed from the box. Express this as a fraction in its simplest form.

60 12

165 33 by dividing numerator and denominator by 5

= 4

11 by dividing numerator and denominator by 5

4. Order the following fractions from the smallest to the largest: 4

9,

5

8,

3

7,

1

2,

3

5

4

9 = 0.44444

5

8 = 0.62500

3

7 = 0.42857

1

2 = 0.50000

3

5 = 0.60000

Ordering the fractions gives: 3

7,

4

9,

1

2,

3

5,

5

8

5. Evaluate, in fraction form: 1 2

3 5

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1 2 (5 1) (3 2)

3 5 15

=

11

15

6. Evaluate, in fraction form: 5 4

6 15

5 4 (5 5) (2 4)

6 15 30

=

17

30

7. Evaluate, in fraction form: 1 2

2 5

1 2 (5 1) (2 2)

2 5 10

=

9

10

8. Evaluate, in fraction form: 7 1

16 4

7 1 (1 7) (4 1)

16 4 16

=

3

16

9. Evaluate, in fraction form: 2 3

7 11

2 3 (11 2) (7 3) 22 21

7 11 77 77

=

43

77

10. Evaluate, in fraction form: 2 1 2

9 7 3

7 2 9 1 21 22 1 2 14 9 42

9 7 3 63 63

=

47

63

11. Evaluate, in fraction form: 2 1

3 25 3

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2 1 2 1 2 13 2 3 2 3 2

5 3 5 3 5 3

= 1 + 2 1

5 3

= 1 + 6 5

15

= 1 +

1

15 =

11

15

12. Evaluate, in fraction form: 7 2 5

27 3 9

7 2 5 7 18 15

27 3 9 27

=

4

27

13. Evaluate, in fraction form: 53 3

313 4

53 3

313 4

= 5 + 3 + 3 3 12 39 51

8 813 4 52 52

=

518

52

14. Evaluate, in fraction form: 45 2

38 5

45 2 5 2 25 16 9

3 (4 3) 1 18 5 8 5 40 40

=

91

40

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EXERCISE 6, Page 12

1. Evaluate: 2 4

5 7

2 4 2 4

5 7 5 7

=

8

35

2. Evaluate: 3 8

4 11

3 8 3 2

4 11 1 11 by cancelling

= 3 2

1 11

= 6

11

3. Evaluate: 3 5

4 9

3 5 1 5

4 9 4 3 by cancelling

= 1 5

4 3

=5

12

4. Evaluate: 17 15

35 68

17 15 1 3

35 68 7 4 by cancelling

= 1 3

7 4

= 3

28

5. Evaluate: 3 7 2

15 9 7

3 7 2 3 7 9

15 9 7 5 9 7 =

3

5 by cancelling

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6. Evaluate: 1 3 5

14 11 39

1 3 5 1 3 44 1 1 4 1 1 1

14 11 39 4 11 39 4 1 13 1 1 13 =

1

13 by cancelling

7. Evaluate: 2 4

9 27

2 4 2 27 1 3

9 27 9 4 1 2 =

3

2 or 1

1

2 by cancelling

8. Evaluate: 3 45

8 64

3 45 3 64 1 8

8 64 8 45 1 15 =

8

15 by cancelling

9. Evaluate: 3 5

8 32

3 5 3 32 3 4 12

8 32 8 5 1 5 5 = 2

2

5 by cancelling

10. Evaluate: 1 2

2 14 3

1 2 9 5 3 5 15

2 14 3 4 3 4 1 4 = 3

3

4 by cancelling

11. Evaluate: 11 5

23 9

1 5 4 23 4 9 4 3

1 23 9 3 9 3 23 1 23 =

12

23 by cancelling

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12. Evaluate: 3 2

2 34 3

3 2 11 11 11 3 1 3

2 34 3 4 3 4 11 4 1 =

3

4 by cancelling

13. Evaluate: 1 3 1

19 4 3

1 3 1 1 3 4 1 1 1

19 4 3 9 4 3 9 1 1 =

1

9

14. Evaluate: 1 3 2

3 14 5 5

1 3 2 13 8 2 13 8 5 13 2 1 13 1 1

3 14 5 5 4 5 5 4 5 2 1 1 2 1 1 1 = 13

15. If a storage tank is holding 450 litres when it is three-quarters full, how much will it contain

when it is two-thirds full?

If 450 litres is3

4full then

1

4full would be 450 3 = 150 litres.

Thus, a full tank would have 4 150 = 600 litres.

2

3 of the tank will contain

2600

3 = 400 litres

16. A tank contains 24,000 litres of oil. Initially, 7

10 of the contents are removed, then

3

5 of the

remainder is removed. How much oil is left in the tank?

If 7

10 is removed then

324,000

10 litres remains, i.e. 7200 litres.

If 3

5 of this is then removed, then

27200

5 litres remains, i.e. 2880 litres

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EXERCISE 7, Page 13

1. Evaluate: 1 3 20

22 5 27

1 3 20 5 3 20 5 1 4

22 5 27 2 5 27 2 1 9 by cancelling

= 5 4

2 9 (M)

= 45 8 37

18 18

=

12

18 (S)

2. Evaluate: 1 3 16

3 4 27

1 3 16 1 1 4

3 4 27 3 1 9 by cancelling

= 1 4

3 9 (M)

= 3 4

9

=

1

9 (S)

3. Evaluate: 1 3 9 1

2 5 15 3

1 3 9 1 1 3 15 1 1 1 1 3 6 2 7

2 5 15 3 2 5 9 3 2 1 3 6 6

= 1

16

4. Evaluate: 1 2 5 1

25 3 9 4

1 2 5 1 1 8 9 1

25 3 9 4 5 3 5 4 (D)

= 1 8 3 1

5 1 5 4 by cancelling

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= 1 24 1

5 5 4 (M)

= 4 96 5 95 15

420 20 20

=

34

4 (A/S)

5. Evaluate: 4 1 1 2 2

5 2 6 5 3

4 1 1 2 2 4 1 1 5 2

5 2 6 5 3 5 2 6 2 3 (D)

= 2 1 1 5 2

5 1 6 2 3 by cancelling

= 2 5 2

5 12 3 (M)

= 24 25 40 39

60 60

=

13

20 (A/S)

6. Evaluate: 3 2 1 5 3

5 3 2 6 2

3 2 1 5 3 3 4 3 15

5 3 2 6 2 5 6 12

(B)

= 3 1 12

5 6 15 (D)

= 3 1 2

5 1 15 by cancelling

= 3 2

5 15 (M)

= 9 2

15

=

7

15 (S)

7. Evaluate: 1 2 7 1 2 2

of 4 3 32 5 10 3 3 5

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1 2 7 1 2 2 1 22 37 10 2 2of 4 3 3

2 5 10 3 3 5 2 5 10 3 3 5

(O)

= 1 44 37 10 3 2

2 10 3 2 5

(B/D)

= 1 7 5 1 2

2 10 1 1 5 by cancelling

= 7 5 2

20 1 5 (M)

= 7 100 8 99

20 20

=

194

20 (A/S)

8. Evaluate:

2 2 16 1

3 5 33 1

6 14 2

2 2 1 20 7 1 4 7 1

6 13 5 3 3 5 3 3 1 3

3 1 27 3 27 26 1

4 2 4 2 4 3

by cancelling

=

4 7 13 1 3

9 12 1

by cancelling

=

28 13 3

92

=

28 1 279 23 3 9

9 9 9 92 2 2

= 2

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EXERCISE 8, Page 14

1. In a box of 333 paper clips, 9 are defective. Express the non-defective paper clips as a ratio of the defective paper clips, in its simplest form. Non-defective paper clips = 333 – 9 = 324 Non-defective paper clips as a ratio of the defective paper clips = 324:9 = 36:1 by dividing by 9 2. A gear wheel having 84 teeth is in mesh with a 24 tooth gear. Determine the gear ratio in its simplest form. Gear ratio = 84:24 = 42:12 = 21:6 = 7:2 or 3.5:1 in its simplest form 3. In a box of 2000 nails, 120 are defective. Express the non-defective nails as a ratio of the defective ones, in its simplest form. Non-defective nails = 2000 – 120 = 1880 Non-defective nails as a ratio of the defective nails = 1880:120 = 188:12 = 94:6 = 47:3 in its simplest form 4. A metal pipe 3.36 m long is to be cut into two in the ratio 6 to 15. Calculate the length of each piece. Number of parts = 6 + 15 = 21

Length of 1 part = 3.36 m 21 = 336 cm 21 = 336 112

21 7 = 16 cm

Hence, 6 parts = 6 16 = 96 cm and 15 parts = 15 16 = 240 cm

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5. On the instructions for cooking a turkey it says that it needs to be cooked 45 minutes for every kilogram. How long will it take to cook a 7 kg turkey? If 1 kg takes 45 minutes, then 7 kg takes 7 45 = 315 minutes

= 5 hours 15 minutes or 1

54

hours

6. In a will, £6440 is to be divided between three beneficiaries in the ratio 4:2:1. Calculate the amount each receives. Number of parts = 4 + 2 + 1 = 7

Amount for each part = £6440

7 = £920

Hence, 4 parts = 4 £920 = £3680, 4 parts = 2 £920 = £1840 and 1 part = 1 £920 = £920 7. A local map has a scale of 1:22,500. The distance between two motorways is 2.7 km. How far are they apart on the map?

Distance apart on map = 2.7 km 2700m 2700 100cm

22500 22500 22500

= 12 cm

8. A machine produces 320 bolts in a day. Calculate the number of bolts produced by 4 machines in 7 days. The machine produces 320 bolts in 1 day

If there were 4 machines, then 4 × 320 bolts would be produced daily, i.e. 1280 bolts.

In 7 days, number of bolts produced = 7 × 1280 = 8960 bolts

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EXERCISE 9, Page 15

1. Express 14.1794 correct to 2 decimal places

14.1794 = 14.18, correct to 2 decimal places 2. Express 2.7846 correct to 4 significant figures

2.7846 = 2.785, correct to 4 significant figures 3. Express 65.3792 correct to 2 decimal places

65.3792 = 65.38, correct to 2 decimal places 4. Express 43.2746 correct to 4 significant figures

43.2746 = 43.27, correct to 4 significant figures 5. Express 1.2973 correct to 3 decimal places

1.2973 = 1.297, correct to 3 decimal places 6. Express 0.0005279 correct to 3 significant figures.

0.0005279 = 0.000528, correct to 3 significant figures.

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EXERCISE 10, Page 15

1. Evaluate 37.69 + 42.6, correct to 3 significant figures. 37.69 + 42.60 80.29 1 1

Hence, 37.69 + 42.6 = 80.29 = 80.3, correct to 3 significant figures

2. Evaluate 378.1 – 48.85, correct to 1 decimal place. 378.10 - 48.85 329.25 Hence, 378.1 – 48.85 = 329.25 = 329.3, correct to 1 decimal place 3. Evaluate 68.92 + 34.84 – 31.223, correct to 4 significant figures. 68.92 103.760 + 34.84 - 31.223 103.76 72.337 Hence, 68.92 + 34.84 – 31.223 = 72.537 = 72.54, correct to 4 significant figures 4. Evaluate 67.841 – 249.55 + 56.883, correct to 2 decimal places. 67.841 249.550 + 56.883 - 124.724 124.724 – 249.550 = - (249.55 – 124.724) 124.724 124.826 Hence, 67.841 – 249.55 + 56.883 = - 124.826 = - 124.83, correct to 2 decimal places 5. Evaluate 483.24 – 120.44 – 67.49, correct to 4 significant figures. 120.44 483.24 + 67.49 - 187.93 187.93 295.31

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Hence, 483.24 – 120.44 – 67.49 = 295.31 = 295.3, correct to 4 significant figures

© John Bird Published by Taylor and Francis

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EXERCISE 11, Page 17

1. Evaluate without using a calculator: 3.57 1.4 357 × 14 1428 3570 4998 357 14 = 4998, hence 3.57 1.4 = 4.998 2. Evaluate without using a calculator: 67.92 0.7 6792 × 7 47544 6792 7 = 47544, hence 67.92 0.7 = 47.544 3. Evaluate without using a calculator: 548.28 1.2

548.28 1.2 = 548.28

1.2 The denominator is multiplied by 10 to change it into an integer. The

numerator is also multiplied by 10 to keep the fraction the same.

Thus, 548.28 548.28 10 5482.8

1.2 1.2 10 12

The long division is similar to the long division of integers. 456.9 12 5482.8

48 68 60 82 72 108 108 0 Hence, 548.28 1.2 = 456.9

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4. Evaluate without using a calculator: 478.3 1.1, correct to 5 significant figures

478.3 1.1= 478.3

1.1 The denominator is multiplied by 10 to change it into an integer. The

numerator is also multiplied by 10 to keep the fraction the same.

Thus, 478.3 478.3 10 4783

1.1 1.1 10 11

The long division is similar to the long division of integers. 434.818.. 11 4783.000

44 38 33 53 44 90 88 20 11 90 88 2 Hence, 478.3 1.1= 434.818.. = 434.82, correct to 5 significant figures 5. Evaluate without using a calculator: 563.48 0.9, correct to 4 significant figures

563.48 0.9 = 563.48

0.9 The denominator is multiplied by 10 to change it into an integer. The

numerator is also multiplied by 10 to keep the fraction the same.

Thus, 563.48 563.48 10 5634.8

0.9 0.9 10 9

The long division is similar to the long division of integers.

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626.088.. 9 5634.800

54 23 18 54 54 0 8 0 0 80 72 80 72 8 Hence, 563.48 0.9 = 626.088.. = 626.1, correct to 4 significant figures

6. Express 4

9 as a decimal fraction correct to 3 significant figures.

0.4444 9 4.000

36 40 36 40 36 40 36 4

Hence, 4

9= 0.4444 = 0.444, correct to 4 significant figures

7. Express 17

27 as a decimal fraction, correct to 5 decimal place

0.629629 27 17.000000

162 80 54 260 243 170 162 80 54 26

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Thus, 17

27 = 0.62963, correct to 5 decimal places.

8. Express 19

16 as a decimal fraction correct to 4 significant figures

0.5625 16 9.00000

Thus, 9

116

= 1.563, correct to 4 significant figures.

9. Express 1331

37 as a decimal fraction, correct to 2 decimal places

0.837

37 31.00

296 140 111 290 259 31

Thus, 1331

37 = 13.837 = 13.84, correct to 2 decimal places

10. Evaluate 421.8 17, (a) correct to 4 significant figures and (b) correct to 3 decimal places.

24.811764 17 421.800000

34 81 68 138 136 20 (a) 421.8 17 = 24.81, correct to 4 significant figures 17 30 17 (b) 421.8 17 = 24.812, correct to 3 decimal places 130 119 110 102 80 68 12

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11. Evaluate 0.0147

2.3, (a) correct to 5 decimal places and (b) correct to 2 significant figures.

0.0147 0.147

2.3 23

0.0063913 23 0.1470000

138 90 69 (a) 0.0147 2.3 = 0.00639, correct to 5 decimal places 210 207 30 (b) 0.0147 2.3 = 0.0064, correct to 2 significant figures 23 70 69 1

12. Evaluate (a) 12.6

1.5 (b) 5.2 12

(a) 12.6 12.6666 126.666

1.5 1.5 15

8.4444

15 126.666

120 66 60 66 60 66 60 66 60 6

Hence, 12.6

1.5 = 8.4444 = 8.4

(b) 5.2 12 = 5.2222 12

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5222222 × 12 6266664

Hence, 5.2 12 = 62.6666 = 62.6

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EXERCISE 12, Page 18

1. Express 0.0032 as a percentage 0.0032 = 0.0032 × 100% = 0.32% 2. Express 1.734 as a percentage

1.734 = 1.734 × 100% = 173.4% 3. Express 0.057 as a percentage

0.057 = 0.057 × 100% = 5.7% 4. Express 20% as a decimal number

20% = 20

100 = 0.20 as a decimal number

5. Express 1.25% as a decimal number

1.25% = 1.25

100 = 0.0125 as a decimal number

6. Express 11

16 as a percentage

11 11 1100

100%16 16 16

%

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68.75

16 1100.00

96 140 128 120 112 80 80 0

Hence, 11

16 = 68.75%

7. Express as percentages, correct to 3 significant figures:

(a) 7

33 (b)

19

24 (c) 1

11

16

(a) 7 7 700

100%33 33 33

= 21.21212…% = 21.2%, correct to 3 significant figures.

(b) 19 19 1900

100% 79.1666...%24 24 24

= 79.2%, correct to 3 significant figures.

(c) 11

1 1.6875 1.6875 100%16

= 168.75% = 169%, correct to 3 significant figures.

8. Place the following in order of size, the smallest first, expressing each as percentages, correct to

1 decimal place: (a) 12

21 (b)

9

17 (c)

5

9 (d)

6

11

(a) 12

21 = 0.5714 = 57.1% (b)

9

170.5294 = 52.9% (c)

5

9 = 0.5555 = 55.6%

(d) 6

11 = 0.5454 = 54.5%

Hence, the order is: (b), (d), (c) and (a) 9. Express 31.25% as a fraction in its simplest form

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31.25% = 31.25 3125 125

100 10000 400 =

5

16

10. Express 56.25% as a fraction in its simplest form.

56.25% = 56.25 5625

100 10000 =

9

16

11. Calculate 43.6% of 50 kg

43.6% of 50 kg = 43.6

50100

kg = 21.8 kg

12. Determine 36% of 27 m

36% of 27 m = 36

27100

m = 9.72 m

13. Calculate correct to 4 significant figures:

(a) 18% of 2758 tonnes (b) 47% of 18.42 grams (c) 147% of 14.1 seconds

(a) 18% of 2758 = 18

2758 18 27.58100

= 496.44 t = 496.4 t, correct to 4 significant figures.

(b) 47% of 18.42 = 47

18.42 47 0.1842100

= 8.6574 g = 8.657 g, correct to 4 significant figures.

(c) 147% of 14.1 = 147

14.1 147 0.141100

= 20.727 s, = 20.73 s, correct to 4 significant figures.

14. Express: (a) 140 kg as a percentage of 1 t (b) 47 s as a percentage of 5 min

(c) 13.4 cm as a percentage of 2.5 m

(a) 140 kg as a percentage of 1 t 140 140

100%1000 10

= 14%

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(b) 47 s as a percentage of 5 min = 47 47 47

100% 100%5 60 300 3

= 15. 6 %

(c) 13.4 cm as a percentage of 2.5 m =13.4 13.4 134

100%250 2.5 25

= 5.36%

15. Express 325 mm as a percentage of 867 mm, correct to 2 decimal places.

325 mm as a percentage of 867 mm = 325

100%867

= 37.49%

16. Express 408 g as a percentage of 2.40 kg. 408 g as a percentage of 2.40 kg = 408 g as a percentage of 2400 g

= 408

100%2400

= 17%

17. When signing a new contract, a Premiership footballer’s pay increases from £15,500 to £21,500 per week. Calculate the percentage pay increase, correct 3 significant figures.

Percentage pay increase = 21500 15500 6000

100% 100%15500 15500

= 38.7%

18. A metal rod 1.80 m long is heated and its length expands by 48.6 mm. Calculate the percentage increase in length.

Percentage increase in length = 3

48.6mm100%

1.80 10 mm

= 2.7%

19. A machine part has a length of 36 mm. The length is incorrectly measured as 36.9 mm. Determine the percentage error in the measurement.

Percentage error in the measurement = 36.9 36 0.9

100% 100%36 36

= 2.5% too high

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20. A resistor has a value of 820 Ω ± 5%. Determine the range of resistance values expected.

5% of 820 = 5

820100

= 41

The lowest value expected is 820 – 5% of 820 i.e. 820 – 41 = 779 Ω

The highest value expected is 820 + 5% of 820 i.e. 820 + 41 = 861 Ω

Hence, range of values expected is: 779 Ω to 861 Ω

21. For each of the following resistors, determine the (i) minimum value, (ii) maximum value: (a) 680 Ω ± 20% (b) 47 kΩ ± 5%

(a) 20% of 680 Ω = 20

680100

= 136 Ω

Hence, (i) minimum value = 680 – 136 = 544 Ω (ii) maximum value = 680 + 136 = 816 Ω

(b) 5% of 47 kΩ = 5

47100

= 2.35 kΩ

Hence, (i) minimum value = 47 – 2.35 = 44.65 kΩ (ii) maximum value = 47 + 2.35 = 49.35 kΩ 22. An engine speed is 2400 rev/min. The speed is increased by 8%. Calculate the new speed.

8% of 2400 rev/min = 8

2400100

= 192 rev/min

New speed = 2400 + 192 = 2592 rev/min