Unit 1A

1. Evaluate

(a) 156 ÷ 1.2,

(b) 1 ²⁄₃ × 1 ²⁄₇.

2.Write down the positive square roots of each of the following

(a) 6 400,

(b) 6 ¹⁄₄,

(c) 9a¹⁶.

3. Evaluate ⁶⁄₇ × ²⁄₃ ÷ ⁹⁄₁₄, giving your answer as a fraction in its simplest form.

4. Using as much of the information below as necessary, write down the value of $\sqrt{\mathrm{0.013}}$.

[ $\sqrt{\mathrm{13}}$ = 3.61, $\sqrt{\mathrm{1.3}}=\; 1.14$ ]

5. Calculate the exact value of

(a) 0.6 × 2.3

(b) 1.71 ÷ 0.3

6. Giving each answer as a fraction in its lowest terms, find the exact value of

(a) ¹ ⁄ ₅ ÷ 4,

(b) 1 ¹ ⁄ ₃ × (4¹ ⁄ ₂ - 1 ³ ⁄ ₄)

7. (a) Evaluate $\sqrt{1\frac{9}{16}}$

(b) Using as much of the information below as necessary, evaluate $\sqrt{\mathrm{3270}}$.

[ $\sqrt{\mathrm{3.27}}$ = 1.808, $\sqrt{\mathrm{32.7}}$ = 5.718 ]

8. (a) Express 0.64 as a fraction in its simplest form.

(b) Express ³ ⁄ ₈ as a decimal.

9. Evaluate giving your answer as a single fraction in its simplest form,

(a) 3 ¹ ⁄ ₄ - 2 ⁵ ⁄ ₉,

(b) ³ ⁄ ₄ ÷ (1 ¹ ⁄ ₂)².

10. Express as a single fraction in its simplest form (² ⁄ ₅ + ¹ ⁄ ₃) × 1¹ ⁄ ₄.

53. Given that 333 × 37 = 12 321, write down the value of

(a) 3330 × 0.37,

(b) 666 × 37,

(c) 123.21 ÷ 37.

54. Evaluate, giving each of your answers as a fraction in its lowest terms,

(a) ³⁄₄ + ¹⁄₆,

(b) 1 ²⁄₃ ÷ 5,

(c) 1 ¹⁄₃ × (4 - 3 ¹⁄₂).

55. Given that 253 × 44 = 11 132, write down the exact value of

(a) 2.53 × 440,

(b) 253 × 88,

(c) 11.132 ÷ 253.

56. Calculate the exact value of

(a) ²⁄₃ + ¹⁄₇,

(b) 0.86 ÷ 0.2,

(c) $\sqrt{0.0009}$.

57. Calculate the exact value of

(a) 4.2 - 0.67,

(b) 3.2 × 0.15,

(c) 0.8 ÷ 16.

58. Calculate the exact value of

(a) 2.4 × 0.35,

(b) 0.7 ÷ 140,

(c) 5.3 - 0.76.

59. Calculate the exact value of

(a) 0.86 × 0.03,

(b) ¹⁄_0.04,

(c) $\sqrt{\mathrm{36\; 000\; 000}}$.

60. (a) Express 0.45 as a fraction in its lowest terms.

(b) Express ¹⁄₇ as a decimal, giving your answer correct to 3 decimal places.

(c) Express 31.6824 correct to 3 significant figures.

61. Find the value of the following, expressing each answer as simply as possible,

(a) 3 ¹⁄₄ - 1 ²⁄₃,

(b) ³⁄₈ of 3 ¹⁄₅,

(c) ³⁄₁₀ ÷ ²⁄₃.

62. Calculate the exact value of

(a) 6 - 2.55,

(b) 35 ÷ 0.07,

(c) (0.3)³.

63. (a) Write down the next term in the sequence 0, 3, 8, 15, 24, 35, ...

(b) Express 28% as a fraction in its lowest terms.

64. (a) Find the value of

(i) 1 ¹⁄₂ - ⁵⁄₈,

(ii) 174 ÷ 0.03.

(b) Three numbers are given below. Write L opposite the largest, M opposite the next largest, and S opposite the smallest.

0.3 million, 87 555, four hundred and two thousand

65. Find the value of

(a) ¹⁄₄ + ²⁄₃,

(b) 3 ¹⁄₃ ÷ ⁵⁄₆.

66. Find the value of

(a) 1 ¹⁄₅ - ³⁄₇,

(b) ⁸⁄₉ × ³⁄₄.

67. Calculate the exact value of

(a) 69.2 ÷ 0.04,

(b) $\sqrt[3]{8000}$.

68. Use the fact that 372 × 52 = 19 344 to write down the exact value of

(a) 3.72 × 520,

(b) 193.44 ÷ 0.52.

69. A sum of money was divided between three people. Angela received ¹⁄₂ of the money. Ben received ³⁄₁₀ of the money. Charles received the remainder.

(a) Ben received $10. How much did Angela receive?

(b) What fraction of the money did Charles receive?

70. 100 g of beans contain 4.7 g of protein. Calculate the mass of protein in 3.3 kg of beans.

71. Arrange the following three numbers in order, starting with the smallest.

¹⁄₅,   0.22,   0.033

72. Evaluate

(a) 16 - 2 × 3,

(b) ¹⁄₄ × 2 ¹⁄₂,

(c) ⁷⁄₈ ÷ ¹⁄₁₆.

73. Express as a single fraction ⁴⁄_2x-1 - ³⁄_5x+6.

74. Find the value of

(a) 2 ¹⁄₄ ÷ ³⁄₅,

(b) 1.3 ÷ 0.2.

75. (a) Find, in its simplest form, the fraction which is exactly half-way between ³⁄₄ and 1.

(b) Write down the value of this fraction as a decimal, giving your answer correct to one significant figure.

76. (a) Write down the value of $\sqrt{0.04}$.

(b) Write down the following in order of size, starting with the smallest.

¹⁄₃,   0.3²,   0.32

77. Find the value of 3 ¹⁄₅ - 1 ¹⁄₂.

78. (a) Find the value of $\sqrt{0.000025}$.

(b) Write down the following, in order of size, starting with the smallest.

³³⁄₁₀₀,   0.3,   ³⁄₁₀

79. Calculate the exact value of 21.08 ÷ 8.

80. Find the value of

(a) ²⁄₃ - ⁴⁄₇,

(b) ²⁄₅ ÷ ³⁄₄.

81. One milligram, 1 mg, is one thousandth of a gram.

One microgram, 1 μg, is one millionth of a gram.

(a) Express 3 mg in grams, as a decimal.

(b) How many micrograms are equivalent to 1 mg?

(c) Every 100 g of a certain cereal contains 300 μg of Folic Acid. What fraction of the cereal is Folic Acid?

82. Find the value of

(a) 21.6 - 2.47,

(b) ⁴⁄₅ ÷ 1 ¹⁄₂.

83. Find the value of

(a) 1.2 × 0.35,

(b) ³⁄₅ ÷ 1 ¹⁄₉.

84. Evaluate

(a) 6 + 4 ÷ 2,

(b) ³⁄₄ ÷ ¹⁄₁₀.

85. Consider the nine numbers 0.4, ²⁄₃, $\sqrt{2}$, π, 5, 9, 17, 40, 121. Write down

(a) the two prime numbers,

(b) the square numbers,

(c) the irrational numbers.

86. (a) Express 0.85 as a fraction in its lowest terms.

(b) Giving your answer as a fraction, find the value of 3 ¹⁄₂ - 2 ²⁄₃.

87. Find the value of

(a) 4.5 × 0.4,

(b) 12 ÷ 0.8.

88. (a) Write down the positive square root of 2 ¹⁄₄.

(b) Express 1.65 hours in minutes.

89. (a) Express each of the numbers 10, 15 and 25 as a product of prime factors.

(b) Three bus services operate from the same depot. The first service leaves at 10 minute intervals, the second at 15 minute intervals and the third at 25 minute intervals. All three services leave the depot together at 08 00. Find the time when the three services next leave the depot together.

90. Find the value of

(a) 0.02 × 0.7,

(b) 0.96 ÷ 1.2.

91. (a) Express ¹³⁄₂₅ as an exact decimal.

(b) Find the value of 1 ⁴⁄₅ + 2 ¹⁄₄, leaving your answer as a fraction.

92. Find

(a) the square of 3 ¹⁄₂,

(b) the positive square root of 1 ⁷⁄₉.

93. (a) Express each of the numbers 35 and 63 as a product of their prime factors.

(b) Find the largest number which will divide exactly into both 35 and 63.

(c) Find the smallest number which is exactly divisible by both 35 and 63.

94. Find the value of

(a) 28 ÷ 3 ¹⁄₂,

(b) 0.3 × 0.02.

95. Express ¹³⁄₄₀ as a decimal.

96. (a) Express 0.052 square metres in square centimetres.

(b) (i) Express 90 as the product of its prime factors.

(ii) The lowest common multiple of 6, 15 and x is 90. Find the two possible values of x which are odd numbers.

97. Find the value of 35 ÷ 2 ¹⁄₇.

98. Express ¹¹⁄₂₅ as a decimal.

99. Find the value of 0.06 ÷ 0.002.

100. (a) Evaluate 6 + 27 ÷ 3.

(b) Express 0.4 as a fraction in its lowest terms.

101. (a) Add together 2 kilograms and 2 grams. Give your answer in grams.

(b) On a particular day, the lowest temperature was -12 °C. The difference between the lowest and highest temperatures was 7 °C. Find the highest temperature on that day.

102. Giving your answer as simply as possible, evaluate

(a) ³⁄₇ + ¹⁄₂,

(b) 1 ²⁄₅ ÷ 2 ¹⁄₃.

103. Find the value of

(a) 0.2 × 0.45,

(b) 1 ³⁄₅ ÷ 2 ¹⁄₃.

104. Find the value of $\sqrt{0.0081}$.

105. The numbers 168 and 324, written as the products of their prime factors, are

168 = 2³ × 3 × 7,          324 = 2² × 3⁴.

Find

(a) $\sqrt{324}$,

(b) the largest integer which is a factor of both 168 and 324,

(c) the smallest positive integer value of n for which 168n is a multiple of 324.

106. Find the value of

(a) ^0.64⁄_0.008,

(b) 2 ²⁄₃ × 1 ¹⁄₅.

107. Find the value of $\sqrt{0.0025}$.

108. The numbers 480 and 576, written as the products of their prime factors, are

480 = 2⁵ × 3 × 5,          576 = 2⁶ × 3².

Find

(a) $\sqrt{576}$,

(b) the largest integer which is a factor of both 480 and 576,

(c) the smallest positive integer value of n for which 480n is a multiple of 576.