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Cambridge IGCSE0580

Estimation

Mathematics 0580 Chapter Notes

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Estimation
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Estimation notes

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In this chapter

  1. 1Nearest ten, hundred, thousand and decimal places
  2. 2Significant figures
  3. 3Estimating a calculation
  4. 4Estimates with a sum in the calculation
  5. 5Accuracy of a final answer

1. Nearest ten, hundred, thousand and decimal places

To round a value, look at the digit just to the right of the place you are rounding to. If it is 5 or more, round up. If it is 4 or less, leave the digit.

To round to the nearest hundred or nearest thousand, keep zeros as place holders. To round to a number of decimal places, count the digits after the decimal point.

5764 ≈ 6000 (nearest thousand)

0.04857 ≈ 0.049 (3 decimal places)

Examiner insight

“Answers that were rounded to the wrong level of accuracy were sometimes seen, for example 10 000.”

Worked example 12 marks

(a) Write 65 432 correct to the nearest hundred.

(b) Write 0.04857 correct to 3 decimal places.

  1. 1

    (a) the tens digit is 3, so round down: 65 400 [1]

  2. 2

    (b) the fourth decimal place is 5, so round up: 0.049 [1]

2. Significant figures

Significant figures are counted from the first non-zero digit. Zeros at the start of a decimal are not significant. Zeros after the first non-zero digit in a decimal are significant.

In a whole number, zeros that only hold the place are not significant. Write them in after rounding.

0.06173 ≈ 0.062 (2 s.f.)

46 280 ≈ 46 000 (2 s.f.)

Examiner insight

“Just over half of the candidates answered this correctly with the most common incorrect answer being 0.05 suggesting that many confused significant figures with decimal places.” “Candidates are advised that trailing zeroes such as in 0.04600 are considered significant unlike in whole numbers where the zeros are just place holders.”

Worked example 12 marks

Do not use a calculator. Write 0.06173 correct to 2 significant figures.

  1. 1

    the zeros at the start are not significant, so the first significant figure is 6 [1]

  2. 2

    the third significant figure is 7, so round up: 0.062 [1]

3. Estimating a calculation

To make an estimate, write each number in the calculation correct to 1 significant figure. Then work out the calculation with the rounded numbers.

Do not work out the original calculation and round the answer.

(0.43 × 22.8)/13.7 ≈ (0.4 × 20)/10

41.3/(9.79 × 0.765) ≈ 40/(10 × 0.8) = 5

Examiner insight

“Some candidates did no rounding as instructed leading to lengthy calculations.” “Unfortunately, many rounded 17.8 to 18 when the question asked for one significant figure, not to two or to the nearest whole number.”

Worked example 12 marks

Do not use a calculator. By writing each number in the calculation correct to 1 significant figure, work out an estimate for the value of (0.43 × 22.8)/13.7.

  1. 1

    0.43 ≈ 0.4, 22.8 ≈ 20 and 13.7 ≈ 10, so (0.4 × 20)/10 [1]

  2. 2

    8/10 = 0.8 [1]

4. Estimates with a sum in the calculation

When the calculation has a sum or a difference, round each number first. Then add or subtract the rounded numbers.

Work top and bottom separately, then divide.

(9.57 × 3.84)/(6.72 + 1.13) ≈ (10 × 4)/(7 + 1)

Examiner insight

“The instruction to round each number correct to 1 significant figure was frequently ignored, as candidates tried addition and long multiplication of the original expression.”

Worked example 12 marks

Do not use a calculator. By writing each number in the calculation correct to 1 significant figure, work out an estimate for the value of (9.57 × 3.84)/(6.72 + 1.13).

  1. 1

    9.57 ≈ 10, 3.84 ≈ 4, 6.72 ≈ 7 and 1.13 ≈ 1, so (10 × 4)/(7 + 1) [1]

  2. 2

    40/8 = 5 [1]

5. Accuracy of a final answer

Round only the final answer. Keep the unrounded values in your working until then.

Give an answer to at least 3 significant figures unless the question asks for another degree of accuracy. If the answer is exact, do not round it.

Examiner insight

“Rounding calculations at intermediate stages often produced inaccurate answers.” “Candidates should be reminded that when an answer is exact it should not be rounded.”

Worked example 12 marks

A car travels 250 km on 18 litres of fuel. Work out the distance travelled for each litre of fuel. Give your answer to 3 significant figures.

  1. 1

    250 ÷ 18 = 13.888… [1]

  2. 2

    13.9 km per litre to 3 significant figures [1]

Recap

  • Look at the digit to the right of the place you round to. 5 or more rounds up.
  • Significant figures start at the first non-zero digit. Leading zeros do not count.
  • To estimate, round each number to 1 significant figure, then calculate.
  • Round every number first, including those in a sum.
  • Round only the final answer. Give at least 3 significant figures unless told otherwise.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write 7849 correct to the nearest hundred.1 mark
  2. Do not use a calculator. Write 0.00736 correct to 2 decimal places.1 mark
  3. Do not use a calculator. Write 0.04058 correct to 2 significant figures.1 mark
  4. Do not use a calculator. By writing each number correct to 1 significant figure, estimate (58.3 × 0.19)/3.9.2 marks
  5. Do not use a calculator. By writing each number correct to 1 significant figure, estimate (4.8 + 5.2)/(0.51 × 3.9).2 marks
  6. Use a calculator. Work out 8.7/(1.9 × 2.3). Give your answer correct to 3 significant figures.2 marks
  7. Do not use a calculator. Write 0.079962 correct to 3 significant figures.1 mark
  8. Do not use a calculator. By writing each number correct to 1 significant figure, estimate ((6.9 - 2.1) × 31)/0.098.3 marks
  9. Do not use a calculator. A machine uses 0.0478 kg of ink for each poster. It prints 385 posters each hour for 28 hours. The ink comes in drums that each hold 19.6 kg. By writing each number correct to 1 significant figure, estimate the number of drums of ink the machine uses.4 marks
  10. Use a calculator. A rectangular wall measures 13.8 m by 2.75 m. One litre of paint covers 11.6 m². The wall needs two coats of paint. (a) Work out the area of the wall. (b) Work out the number of litres of paint needed for two coats. Give your answer correct to 3 significant figures. (c) Paint is sold in tins of 2.5 litres. Work out the number of tins needed.5 marks

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