1
Working with Decimals
EXERCISE 1
WORKED EXAMPLES:
a. (i) 234 = 2 x 100 + 3 x 10 + 4 x 1 b. (i) 9 x 1000 + 8 x 100 + 7 x 10 + 6 x 1 = 9876
(ii) 23.4 = 2 x 10 + 3 x 1 + 4 x (ii) 9 x 100 + 8 x 10 + 7 x 1 + 6 x = 987.6
(iii) 2.34 = 2 x 1 + 3 x + 4 x (iii) 9 x 10 + 8 x 1 + 7 x + 6 x = 98.76
(iv) 0.234 = 2 x + 3 x + 4 x (iv) 9 x 1 + 8 x + 7 x + 6 x = 9.876
WRITTEN EXAMPLES: Write the following as expanded numbers.
1. 7650 2. 765 3. 76.5 4. 7.65
5. 0.765 6. 980 7. 98 8. 9.8
9. 0.98 10. 0.098 11. 604 12. 60.4
13. 6.04 14. 0.604 15. 3075 16. 307.5
17. 30.75 18. 3.075 19. 0.28 20. 0.028
21. 0.208 22. 0.9 23. 0.09 24. 0.009
25. 3.142 26. 20.6 27. 7.03 28. 0.905
29. 0.084 30. 0.45 31. 0.067 32. 0.809
Write the following as ordinary numbers.
33. 3 x 1000 + 2 x 100 + 5 x 10 34. 3 x 100 + 2 x 10 + 5 x 1
35. 3 x 10 + 2 x 1 + 5 x 36. 3 x 1 + 2 x + 5 x
37. 3 x + 2 x + 5 x 38. 6 x 1000 + 4 x 10 + 8 x 1
39. 6 x 100 + 4 x 1 + 8 x 40. 6 x 10 + 4 x + 8 x
41. 6 x 1 + 4 x + 8 x 42. 7 x 100 + 3 x 10 + 5 x
43. 7 x 10 + 3 x 1 + 5 x 44. 7 x 1 + 3 x + 5 x
45. 2 x + 9 x 46. 2 x + 9 x
47. 2 x + 9 x 48. 4 x
49. 4 x 50. 4 x
51. 6 x 10 + 2 x 52. 6 x 1 + 2 x
53. 6 x + 2 x 54. 3 x + 7 x
55. 8 x + 4 x 56. 5 x + 9 x
Write down the value of the 5 in the following numbers.
57. 6453 58. 5190 59. 7593 60. 3.504
61. 4.605 62. 7.053 63. 5.31 64. 7.05
65. 2.50 66. 0.675 67. 0.519 68. 0.258
69. 573 70. 3.75 71. 58.2 72. 2.58
73. 5940 74. 0.495 75. 54 268 76. 9.7135
EXERCISE 2
Note: 1. 0.375 = 3 x + 7 x + 5 x = + + =
2. 0.64 = 6 x + 4 x = + =
3. 0.5 = , 0.50 = , 0.500 =
i.e. if one decimal place the denominator is 10
if two decimal places the denominator is 100
if three decimal places the denominator is 1000
ORAL EXAMPLES:
State these decimals as fractions with denominators of 10, 100, 1000. (Do not simplify the fractions).
1. 0.2 2. 0.20 3. 0.200 4. 0.08
5. 0.080 6. 0.008 7. 0.19 8. 0.190
Note: Before comparing decimals write them to the same number of decimal places first.
WORKED EXAMPLES:
a. Compare 0.7 and 0.68 b. Compare 0.077 and 0.7
0.7 = 0.70 = whereas 0.68 = 0.7 = 0.700 = whereas 0.077 =
Hence 0.7 > 0.68 Hence 0.077 < 0.7
WRITTEN EXAMPLES:
Copy down and complete the following statements by putting <, > or = in the circles.
1. 0.2 0.20 2. 0.4 0.35 3. 0.79 0.8 4. 0.06 0.6
5. 0.01 0.010 6. 0.025 0.03 7. 0.06 0.059 8. 0.04 0.004
9. 0.7 0.700 10. 0.099 0.909 11. 0.4 0.325 12. 0.875 0.9
13. 0.080 0.08 14. 0.505 0.55 15. 0.22 0.202 16. 0.8 0.088
17. 0.600 0.6 18. 0.403 0.43 19. 0.97 0.907 20. 0.101 0.11
Write down the following decimals in order of size - smallest first and largest last.
21 0.005, 0.05, 0.5 22. 0.777, 0.077, 0.007
23. 0.3, 0.33, 0.333 24. 0.6, 0.66, 0.06
25. 0.004, 0.044, 0.04 26. 0.808, 0.88, 0.088
27. 0.022, 0.22, 0.202 28. 0.99, 0.099, 0.909, 0.999
29. 0.1, 0.01, 0.11, 1 30. 0.001, 0.101, 0.011, 0.111
EXERCISE 3
0.02 = and so 0.02 x 10 = x = = 0.2
i.e. when multiplying by 10 the decimal point moves 1 place to the right, when multiplying by 100 the decimal point moves 2 place to the right and when multiplying by 1000 the decimal point moves 3 place to the right.
0.02 = and so 0.02 ÷ 10 = x = = 0.002
i.e. when dividing by 10 the decimal point moves 1 place to the left, when dividing by 100 (=10 x 10) the decimal point moves 2 place to the left and when dividing by 1000 (=10 x 10 x 10) the decimal point moves 3 place to the left.
WORKED EXAMPLES:
a. (i) 0.324 x 10 = 3.24 b. (i) 798 ÷ 10 = = 79.8
(ii) 0.324 x 100 = 32.4 (ii) 798 ÷ 100 = = 7.98
(iii) 0.324 x 1000 = 324 (iii) 798 ÷ 1000 = = 0.798
1. 6.875 x 10 2. 6.875 x 100 3. 6.875 x 1000 4. 4650 ÷ 10
5. 4650 ÷ 100 6. 4650 ÷ 1000 7. 0.98 x 10 8. 0.98 x 100
9. 0.98 x 1000 10. 37 ÷ 10 11. 37 ÷ 100 12. 37 ÷ 1000
13. 0.04 x 10 14. 0.04 x 100 15. 0.04 x 1000 16.
17. 18. 19. 0.9 x 10 20. 0.9 x 100
21. 0.9 x 1000 22. 23. 24.
25. 1 litre is equal to 0.22 gallons. 26. 100 gallons is equal to 450 litres.
Find how many gallons in a. 10 litres. Find how many litres in a. 10 gallons.
b. 100 litres. b. 1 gallon.
27. 1 acre is equal to 0.405 hectares. 28. 100 hectares is equal to 247 acres.
Find how many hectares in a. 10 acres. Find how many acres in a. 10 hectares.
b. 100 acres. b. 1 hectare.
29. 10 wire staples cost $3.50. 30. A car uses 7.5 litres of petrol every 100 Km.
Find the cost of Find how many litres of petrol it would use for
a. one wire staple. a. 10 Km.
b. 100 wire staples. b. 1000 Km.
31. a. A nautical mile is equal to 1.852 Km.
Find (to the nearest whole number) how many kilometres there are in 100 nautical miles.
b. A knot is equal to 1.852 Km/hr.
Find (to the nearest whole number) how many Km/hr there are in 10 knots.
32. 1 millilitre of milk has a mass of 1.03 grams.
a. Find, in grams, the mass of 1 litre of milk. (1 litre = 1000 millilitres)
b. Find, in kilograms, the mass of 1 litre of milk. (1 kilogram = 1000 grams)
EXERCISE 4
ORAL EXAMPLES: Convert the decimals to fractions and fractions to decimals.
1. 0.3 2. 0.9 3. 0.03 4. 0.09
5. 0.33 6. 0.99 7. 0.003 8. 0.009
9. 0.033 10. 0.099 11. 0.333 12. 0.999
13. 14. 15. 16.
17. 18. 19. 20.
21. 22. 23. 24.
WORKED EXAMPLES:
a. Convert 0.24 and 0.024 to fractions (in their simplest form).
0.24 = =
0.024 = = =
b. Convert and to decimals.
= = 0.36
= = 0.036
WRITTEN EXAMPLES: Convert the decimals to fractions (in their simplest form) and fractions to decimals.
Note: 100 = 2 x 50 = 4 x 25 = 5 x 20
1000 = 2 x 500 = 4 x 250 = 5 x 200 = 8 x 125 = 40 x 25
1. 0.4 2. 0.6 3. 0.04 4. 0.06
5. 0.004 6. 0.006 7. 2.5 8. 0.25
9. 0.025 10. 1.5 11. 0.15 12. 0.015
13. 3.5 14. 0.35 15. 0.035 16. 1.6
17. 0.16 18. 0.016 19. 1.4 20. 0.14
21. 0.014 22. 2.25 23. 0.225 24. 0.125
25. 26. 27. 28. 3
29. 30. 31. 32.
33. 34. 35. 1 36. 2
37. 38. 39. 40.
41. 42. 43. 44.
45. 46. 47. 48.
49. Find (in minutes) a. 0.1 of an hour. 50. Find (in degrees) a. 0.1 of a right angle.
b. 0.5 of an hour. b. 0.5 of a right angle.
c. 0.05 of an hour. c. 0.25 of a revolution.
d. 0.25 of an hour. d. 0.125 of a revolution.
e. 0.75 of an hour. e. 0.025 of a revolution.
EXERCISE 5
Find the following sums and differences.
Note: When adding or subtracting decimals keep the decimal points under each other.
1. 0.6 + 0.2 2. 0.7 + 0.3 3. 0.8 + 0.4 4. 0.9 + 0.5
5. 2.5 + 1.2 6. 3.6 + 2.4 7. 4.7 + 3.6 8. 5.8 + 4.8
9. 0.06 + 0.03 10. 0.08 + 0.02 11. 0.07 + 0.05 12. 0.09 + 0.04
13. 0.27 + 0.18 14. 0.45 + 0.36 15. 0.86 + 0.57 16. 0.64 + 0.79
17. 2.36 + 3.81 18. 4.74 + 5.63 19. 6.52 + 7.49 20. 8.98 + 9.27
21. 2.5 + 0.06 22. 3.7 + 1.28 23. 4.6 + 2.42 24. 5.8 + 3.64
25. 0.9 - 0.5 26. 0.8 - 0.6 27. 2.7 - 1.4 28. 5.6 - 3.5
29. 3.5 - 0.7 30. 6.4 - 0.9 31. 4.3 - 2.8 32. 7.2 - 4.6
33. 0.07 - 0.02 34. 0.06 - 0.04 35. 0.59 - 0.13 36. 0.48 - 0.25
37. 0.65 - 0.36 38. 0.84 - 0.48 39. 2.72 - 2.57 40. 4.93 - 1.69
41. 0.31 - 0.26 42. 0.54 - 0.47 43. 3.72 - 1.68 44. 5.93 - 2.85
45. 6.34 - 3.86 46. 8.28 - 4.79 47. 7.12 - 5.67 48. 9.46 - 6.58
49. 4 - 0.8 50. 2 - 0.6 51. 5 - 0.9 52. 3 - 0.7
53. 4 - 0.09 54. 2 - 0.07 55. 5 - 0.08 56. 3 - 0.06
57. 0.4 - 0.07 58. 0.2 - 0.09 59. 0.5 - 0.06 60. 0.3 - 0.08
61. 0.9 + 0.3 + 0.6 62. 0.9 + 0.3 - 0.6 63. 0.9 - 0.3 + 0.6 64. 0.9 - 0.3 - 0.6
65. 1.2 + 0.8 + 0.4 66. 1.2 + 0.8 - 0.4 67. 1.2 - 0.8 + 0.4 68. 1.2 - 0.8 - 0.4
69. 2.5 + 1.5 + 0.5 70. 2.5 + 1.5 - 0.5 71. 2.5 - 1.5 + 0.5 72. 2.5 - 1.5 - 0.5
73. 0.75 + 0.5 + 0.25 74. 0.75 + 0.5 - 0.25 75. 0.75 - 0.5 + 0.25 76. 0.75 - 0.5 - 0.25
77. 1 + 0.2 + 0.03 78. 1 + 0.2 - 0.03 79. 1 - 0.2 + 0.03 80. 1 - 0.2 - 0.03
EXERCISE 6
WORKED EXAMPLES:
a. (i) 2 x 0.3 = x = = 0.6 b. (i) 5 x 0.4 = x = = 2.0
(ii) 2 x 0.03 = x = = 0.06 (ii) 5 x 0.04 = x = = 0.20
(iii) 0.2 x 0.3 = x = = 0.06 (iii) 0.5 x 0.4 = x = = 0.20
(iv) 0.2 x 0.03 = x = = 0.006 (iv) 0.5 x 0.04 = x = = 0.020
WRITTEN EXAMPLES: Find the following products (as shown above).
1. 2 x 0.1 2. 2 x 0.01 3. 0.2 x 0.1 4. 0.2 x 0.01
5. 3 x 0.4 6. 3 x 0.04 7. 0.3 x 0.4 8. 0.3 x 0.04
9. 5 x 0.6 10. 5 x 0.06 11. 0.5 x 0.6 12. 0.5 x 0.06
Note: When multiplying decimals the number of decimal places in the answer is equal to the number of decimal places in the question.
EXERCISE 7
ORAL EXAMPLES: State the following products.
1. 3 x 0.1 2. 3 x 0.01 3. 0.3 x 0.1 4. 0.3 x 0.01
5. 4 x 0.2 6. 4 x 0.02 7. 0.4 x 0.2 8. 0.4 x 0.02
9. 3 x 0.5 10. 3 x 0.05 11. 0.3 x 0.5 12. 0.3 x 0.05
13. 5 x 0.2 14. 5 x 0.02 15. 0.5 x 0.2 16. 0.5 x 0.02
WRITTEN EXAMPLES: Find the following products.
1. 0.1 x 6 2. 0.1 x 0.7 3. 8 x 0.01 4. 0.09 x 0.1
5. 0.9 x 2 6. 0.2 x 0.6 7. 7 x 0.02 8. 0.02 x 0.8
9. 8 x 0.3 10. 0.3 x 0.9 11. 0.06 x 3 12. 0.3 x 0.07
13. 0.4 x 0.7 14. 8 x 0.04 15. 0.04 x 0.9 16. 0.6 x 4
17. 0.5 x 0.1 18. 0.07 x 5 19. 0.5 x 0.08 20. 9 x 0.5
21. 0.01 x 4 22. 0.6 x 0.09 23. 7 x 0.6 24. 0.6 x 0.8
25. 9 x 0.08 26. 0.08 x 0.7 27. 0.9 x 7 28. (0.5)2
29. 3 x 2 x 0.1 30. 3 x 0.2 x 0.1 31. 0.3 x 0.2 x 0.1 32. 2 x 0.3 x 4
33. 0.2 x 3 x 0.4 34. 0.2 x 0.3 x 0.4 35. 0.5 x 4 x 3 36. 0.5 x 0.4 x 3
37. 0.5 x 0.4 x 0.3 38. 4 x 0.5 x 6 39. 0.4 x 5 x 0.6 40. 0.4 x 0.5 x 0.6
EXERCISE 8
WORKED EXAMPLES:
a. (i) = 0.3 (Check that 0.3 x 2 = 0.6) b. (i) = 0.4 (Check that 0.4 x 3 = 1.2)
(ii) = 0.03 (Check that 0.03 x 2 = 0.06) (ii) = 0.04 (Check that 0.04 x 3 = 0.12)
c. (i) = 2.5 (Check that 2.5 x 5 = 12.5)
(ii) = 0.25 (Check that 0.25 x 5 = 1.25)
(iii) = 0.025 (Check that 0.025 x 5 = 0.125)
d. (i) = = 0.5 (Check that 0.5 x 4 = 2.0)
(ii) = = 0.05 (Check that 0.05 x 4 = 0.20)
(iii) = = 0.005 (Check that 0.005 x 4 = 0.020)
ORAL EXAMPLES: State the following quotients.
1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11. 12.
13. 14. 15. 16.
17. 18. 19. 20.
WRITTEN EXAMPLES: Find the following quotients.
1. 2. 3. 4.
5. 6. 7. 8.
9. 10. 11. 12.
13. 14. 15. 16.
17. 18. 19. 20.
21. 22. 23. 24.
25. 26. 27. 28.
29. 30. 31. 32.
33. 34. 35. 36.
37. 38. 39. 40.
41. 42. 43. 44.
45. 46. 47. 48.
EXERCISE 9
WORKED EXAMPLES:
a. = = = 30 (Check that 30 x 0.2 = 6.0 = 6)
b. = = = 3 (Check that 3 x 0.2 = 0.6)
c. = = = 0.3 (Check that 0.3 x 0.2 = 0.06)
d. = = = 0.03 (Check that 0.03 x 0.2 = 0.006)