Question Bank | Unit 1D - Arithmetic - Estimation, Significant Figures, Decimal Places and Standard Form

1. (a) Express 9 × 10³ + 3 × 10 + 5 as a single number.

(b) Find the value of $\frac{1.26 \times 10^{3}}{7 \times 10^{- 1}}$ expressing your answer in standard form.

2. (a) Express in standard form (i) 480, (ii) 0.016.

(b) Evaluate 480 ÷ 0.016, giving your answer in standard form.

3. (a) Express ⁷⁄₄₀ as a decimal.

(b) Express ⁹⁄₂₅ as a percentage.

(c) Express 3.086 correct to 2 significant figures.

4. Given that p = 2.4 × 10³ and that q = 6 × 10^-2, calculate, expressing each answer in standard form,

(a) pq,

(b) ^p⁄_q.

5. (a) Express 1.48247 correct to three decimal places.

(b) Taking the value of π to be 3.14159, find the value of (7π - 21.9), giving your answer correct to three significant figures.

6. Given that a = 5 × 10⁴ and b = 3 × 10² find, expressing each of your answers in standard form, the value of

(a) ab,

(b) a + b,

(c) ^b⁄_a.

7. (a) Estimate, correct to 1 significant figure, the value of $\frac{\sqrt{8.01 \times 25.01}}{1.999}$ .

(b) Evaluate 0.57 × 0.3, giving your answer in standard form.

8. (a) Write down the value of 16.048 correct to

(i) one significant figure,

(ii) one decimal place.

(b) Express 0.00345 in standard form.

9. (a) Express 0.003186 correct to 3 significant figures.

(b) Express 52 300 in standard form.

(c) Express ³³⁄₈₀ as a decimal.

10. Express

(a) 2.149 correct to one decimal place.

(b) 408 correct to two significant figures.

(c) 0.0054 in standard form.

11. (a) Express 486 mm in cm, giving your answer correct to the nearest cm.

(b) Express 5.126 correct to two significant figures.

(c) Write down, correct to the nearest whole number, the value of $\sqrt{898}$ .

12. Given that y = 1.7 × 10², evaluate

(a) $\sqrt{y - 1}$ ,

(b) 12y, expressing your answer in standard form.

13. Express $\frac{2}{7}$ as a decimal, giving your answer correct to 2 decimal places.

14. Estimate, correct to 2 significant figures, the value of 30.02 × 19.99 − 78.23.

15. Given that m = 5 × 10⁴ and that n = 4 × 10³ calculate, giving each answer in standard form,

(a) mn,

(b) m²,

(c) $\frac{1}{m}$ + n.

16. Given that t = 4 × 10^-3, express 3t in standard form.

17. (a) Measured correct to the nearest centimetre, the sides of a rectangle are 9 cm and 6 cm. Find the smallest possible perimeter of the rectangle.

(b) Estimate, correct to one significant figure, the value of $\frac{84.3}{0.0418}$ .

18. Evaluate 6 × 0.00475, giving your answer in standard form.

19. It is given that x = 3 × 10⁶. Find the value of each of the following expressing your answers in standard form:

(a) 4x,

(b) x²,

(c) $\frac{6}{x}$ .

20. Find the approximate value of the expression $\frac{5.02 \times \sqrt{15.98}}{9.97}$ giving your answer correct to the nearest whole number.

21. Express 0.0002 in standard form.

22. Estimate, correct to 1 significant figure, the value of 43.95812 − 28.34075.

23. Express

(a) $235.80 correct to the nearest $10,

(b) 0.0354 correct to two significant figures.

24. (a) Express $\frac{3}{7}$ as a decimal, correct to 2 decimal places,

(b) Express 576 000 in standard form.

25. (a) Estimate the value of $\frac{75.21}{1.53}$ , giving your answer correct to one significant figure.

(b) A box contains three cans. Each can has a mass of 230 g, correct to the nearest gram. The box has a mass of 40 g, correct to the nearest gram. Find the smallest possible value of

(i) the mass of 1 can,

(ii) the total mass of the 3 cans and the box.

26. Given that m = 8.4 × 10⁵ and n = 2 × 10². Expressing your answers in standard form, find the value of

(a) 2m,

(b) $\frac{m}{n}$ .

27. It is given that t = 4 × 10⁴. Find the value of each of the following, giving your answers in standard form,

(a) 5t,

(b) $\sqrt{t}$ ,

(c) t².

28. Express

(a) 0.00052 in standard form,

(b) 35.2609 correct to 3 significant figures.

29. Estimate, correct to 1 significant figure, the value of 0.0123 × 5040.

30. Express

(a) 0.0062 in standard form,

(b) 45% as a fraction in its lowest terms,

(c) $\frac{4}{7}$ as a decimal, giving your answer correct to 2 decimal places.

31. (a) Express, correct to 2 significant figures,

(i) 249.526,

(ii) 0.040375.

(b) Hence estimate, correct to 1 significant figure, the value of 249.526 × 0.040375.

32. Express

(a) 0.07 in standard form,

(b) 0.375 as a fraction in its lowest terms.

33. (a) Express, correct to 2 significant figures,

(i) 819.345,

(ii) 5.032.

(b) Hence estimate, correct to 1 significant figure, the value of 819.345 × 5.032.

34. Find the value of each of the following, expressing each answer in standard form.

(a) 3 × (6.1 × 10⁵)

(b) $\frac{4}{8 \times 10^{6}}$

35. (a) Estimate the value of $\frac{34.123}{79.7}$ , giving your answer correct to one significant figure.

(b) Each side of a square has length 130 mm correct to the nearest millimetre. Find the smallest possible value of the perimeter of the square.

36. Estimate the value of $\frac{30}{0.049}$ , giving your answer correct to 1 significant figure.

37. (a) The Sun is approximately 150 000 000 kilometres from the Earth. Write this distance in standard form.

(b) Beta and Gamma are stars. Beta is 3 × 10¹⁴ kilometres from the Earth and Gamma is 6 × 10¹¹ kilometres from the Earth. How many times further from the Earth is Beta than Gamma?

61. (a) Calculate the exact value of $\sqrt[3]{8 \times 10^{6}}$ .

(b) The population of Nigeria was recently estimated to be 103 940 000. Express 103 940 000 in standard form, correct to 3 significant figures.

62. The length of a rectangle is 12 cm, correct to the nearest centimetre.

(a) What is the upper bound for the length of the rectangle?

(b) The width of the rectangle is 7 cm, correct to the nearest centimetre. Calculate the smallest possible perimeter of the rectangle.

63. Given that p = 8 × 10⁹ and q = 4 × 10⁷, find the value of each of the following, expressing your answers in standard form.

(a) p × q,

(b) q ÷ p,

(c) p + q.

64. The inside radius of a cylindrical pot is 10 cm. It contains 2100 cm³ of water. Estimate, correct to the nearest centimetre, the depth of water in this pot.

65. It is given that s = 4 × 10⁶ and t = 3 × 10⁴. Find the value of each of the following, expressing your answers in standard form.

(a) s²,

(b) t ÷ s,

(c) s − t.

66. The area of Pakistan is 803 944 km². Express this area in standard form, correct to 3 significant figures,

(a) in square kilometres,

(b) in square metres.

67. The distance from the Earth to the sun is 1.5 × 10⁸ km. The distance from Pluto to the sun is 6.0 × 10⁹ km.

(a) Find the ratio of the shorter distance to the longer. Give your answer in its simplest form.

(b) How much further is Pluto from the sun than the Earth from the sun? Give your answer in standard form.

68. The number of listeners to a radio station in 1995 was 350 000.

(a) In 1996, there were 392 000 listeners. Calculate the percentage increase in the number of listeners from 1995 to 1996.

Each year the radio station invites its listeners to vote for their favourite pieces of music.

(b) In 1995, 18% of the listeners voted. Calculate the number who voted.

(c) In 1997, the number of listeners was 460 000 and 91 000 of these voted.

(i) Calculate the percentage of listeners who voted in 1997, giving your answer correct to the nearest whole number.

(ii) The number who voted in 1997 was 40% more than in 1996. Calculate the number who voted in 1996.

69. (a) A cube has side 6 cm. Find, correct to the nearest 10 square centimetres, the total surface area of the cube.

(b) Express 456 minutes in hours and minutes, giving your answer correct to the nearest 5 minutes.

70. The distance of the planet Pluto from the sun is 6.0 × 10⁹ km. The distance of the Earth from the sun is 1.5 × 10⁸ km.

(a) How many million kilometers is Pluto from the sun.

(b) Find, in its simplest form, the ratio,
distance of Pluto from the sun : distance of the Earth from the sun.

(c) Find the difference between the distance of Pluto from the sun and the distance of the Earth from the sun. Give your answer in standard form.

71. (a) Express 6700 in standard form.

(b) 3 × 10³ + 2 × 10¹ + 4 × 10^x + 5 × 10^y = 3024.05, where x and y are integers.
Write down the value of x and the value of y.

72. The length and breadth of a rectangle, both measured correct to the nearest centimetre, are 11 cm and 9 cm respectively.

Calculate the lower bound of the area of the rectangle.

73. (a) Write down, correct to the nearest integer, the value of $\sqrt{896}$ .

(b) In 1998 there were 4.9 × 10⁷ people living in England and of these 9.3 × 10⁶ were under 16 years of age. Find the number of people who were 16 or over. Give your answer in standard form.

74. (a) Write down, correct to the nearest integer, the value of $\sqrt[]{397}$ .

(b) Find the value of 3.6 × 10⁷ − 6.4 × 10⁶. Give your answer in standard form.

75. (a) Express Rs 15471 correct to

(i) the nearest thousand rupees,

(ii) 3 significant figures.

(b) The length of each side of an equilateral triangle is 8 cm, correct to the nearest centimetre. Complete the sentence in the answer space to show the smallest and largest possible values of the perimeter of the triangle.

cm ≤ perimeter < cm

76. It is given that m = 8 × 10⁶ and n = 4 × 10^-3. Expressing your answers in standard form, find

(a) m × n,

(b) $\frac{m}{n}$ .

77. A Football Club paid $15 million for a player. A newspaper reported that the Club would "have to sell 75174 tickets to cover the cost."

(a) Write down 75174 correct to two significant figures.

(b) Use your answer to (a) and the information above to estimate the cost, in dollars, of a ticket.

78. A man travelled by car from home to the library, and later from the library to the dentist. The kilometre readings in the car were

(a) The man passed a cinema between home and the dentist. At the cinema, the man had travelled half the distance between home and the dentist. Calculate the distance travelled between the cinema and the library.

(b) The given readings are accurate to the nearest tenth of a kilometre. Calculate the upper bound for the distance travelled between home and the dentist.

79. (a) Express, correct to two significant figures,

(i) 386.71,

(ii) 0.02049.

(b) Hence estimate, correct to one significant figure, the value of 386.71 × 0.02049.

80. (a) A pile of 1500 sheets of paper is 6 cm high. Find the thickness of one sheet of paper

(i) in centimetres, giving your answer as a decimal,

(ii) in metres, giving your answer in standard form.

(b) Each sheet of paper is 30 cm long and 20 cm wide. Find the total surface area of both sides of one sheet of paper, giving your answer in square metres.

81. (a) Express, correct to two significant figures,

(i) 288.54,

(ii) 0.03026.

(b) Hence estimate, correct to one significant figure, the value of 288.54 × 0.03026.