Printable · GCSE Foundation · ages 14-16
Estimation and checking worksheet — GCSE Foundation
Fifteen questions on "estimation and checking" — DfE statement N14. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
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Estimation and checking worksheet — GCSE Foundation
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- 1.A country's population is 8,340,000, and it is estimated that 1,950,000 of them live in the capital city. By rounding each number to 1 significant figure, work out an estimate for the number of people who do not live in the capital city.
- 2.A rectangular field measures 19.6 m by 48.3 m. Work out an estimate for the area of the field, by rounding each length to 1 significant figure.
- 3.Work out an estimate for 113 + 491, by rounding each number to the nearest 100.
- 4.Work out an estimate for 588 ÷ 31, by rounding each number to 1 significant figure.
- 5.Work out an estimate for 6.8 × 41, by rounding each number to 1 significant figure.
- 6.Work out an estimate for 397 ÷ 21, by rounding each number to 1 significant figure.
- 7.A cinema has 21 rows of seats with 29 seats in each row. Work out an estimate for the number of people the cinema can hold, by rounding each number to 1 significant figure.
- 8.Work out an estimate for 89 + 52, by rounding each number to the nearest 10.
- 9.Estimate the value of √45, giving your answer to the nearest whole number.
- 10.Work out an estimate for 6.4 × 3.9, by rounding each number to the nearest whole number.
- 11.Oliver uses his calculator to work out 25% of 80 and writes down 320. Without using a calculator, work out the correct value of 25% of 80.
- 12.Work out an estimate for 37 × 84, by rounding each number to 1 significant figure.
- 13.Amelia estimates 48 × 21 by working out 50 × 20 = 1,000. Work out whether her estimate is an under-estimate or an over-estimate, and by how much.
- 14.Freya uses her calculator to work out 7² and writes down 14. Work out the correct value of 7².
- 15.Work out an estimate for 79.3 − 24.6, by rounding each number to the nearest whole number.
Answer key
- (d) 6,000,000 — Method: round each number to 1 significant figure, then subtract. Working: 8,340,000 rounds to 8,000,000 (1 s.f.); 1,950,000 rounds to 2,000,000 (1 s.f.); 8,000,000 − 2,000,000 = 6,000,000. Answer: 6,000,000. 6,390,000 is the exact difference, found without rounding the numbers first. 8,000,000 comes from rounding the population correctly but forgetting to subtract the capital's population at all. 6,300,000 comes from rounding 8,340,000 to the nearest hundred thousand, 8,300,000, instead of to 1 significant figure, then subtracting the correctly rounded 2,000,000.
- (c) 1,000 m² — Method: round each length to 1 significant figure, then use area of a rectangle = length × width on the rounded lengths. Working: 19.6 m rounds to 20 m and 48.3 m rounds to 50 m, so the estimate is 20 × 50 = 1,000 and the area is about 1,000 m². Answer: 1,000 m². The distractors: 800 m² comes from rounding 48.3 down to 40 when the digit after its first significant figure is 8 and sends it up to 50, giving 20 × 40 = 800; 140 m² is the perimeter of the rounded rectangle, 2 × 20 + 2 × 50 = 140, not its area; 70 m² comes from adding the rounded lengths, 20 + 50 = 70, instead of multiplying them.
- (c) 600 — Method: round each number to the nearest 100, then add the rounded values. Working: 113 is nearer to 100 than to 200, so it rounds to 100; 491 is nearer to 500 than to 400, so it rounds to 500; adding those gives the estimate. Answer: 600. The distractors: 500 comes from rounding each number down to the hundred below instead of to the nearest hundred, giving 100 + 400; 700 comes from rounding each number up to the hundred above, giving 200 + 500; 604 is the exact total, worked out in full when the question asks for an estimate.
- (a) 20 — Method: round each number to 1 significant figure, then divide the rounded values. Working: 588 rounds to 600 (1 s.f.) and 31 rounds to 30 (1 s.f.). 600 ÷ 30 = 20. Answer: 20. 17 comes from cutting 588 down to 500, keeping the leading digit as it stands instead of rounding it up to 1 significant figure, 600, then dividing by the correctly rounded 30. 200 comes from misreading the rounded divisor 30 as 3, giving 600 ÷ 3 instead of 600 ÷ 30. 19 is the exact value of 588 ÷ 31 rounded to the nearest whole number, found without rounding the numbers first.
- (c) 280 — Method: round each number to 1 significant figure, then multiply the rounded numbers. Working: 6.8 rounds to 7 because the next digit is 8, and 41 rounds to 40 because its next digit is 1, so the estimate is 7 × 40 = 280. Answer: 280. The distractors: 240 comes from cutting 6.8 down to 6 instead of rounding it up to 7, giving 6 × 40 = 240; 350 comes from rounding 41 up to 50 when the digit after its first significant figure is less than 5, giving 7 × 50 = 350; 28 comes from multiplying the leading digits only and losing the place value of the 40, which makes the estimate ten times too small.
- (a) 20 — Method: round each number to 1 significant figure, then divide. Working: 397 rounds to 400 and 21 rounds to 20, so the estimate is 400 ÷ 20 = 20. Answer: 20. The distractors: 19 comes from not estimating at all, since 397 ÷ 21 = 18.9 to 1 decimal place, which rounds to 19, while the question asks for an estimate from rounded numbers; 200 comes from dividing 400 by 2 instead of by 20, losing the place value of the rounded divisor; 2 comes from dividing the leading digits only and losing the place value of both numbers.
- (d) 600 — Method: the number of seats is the number of rows multiplied by the number of seats in each row, so round each number to 1 significant figure and then multiply the rounded values, which is quick because a product of two multiples of ten is found by multiplying the non-zero digits and attaching the zeros. Working: 21 rounds to 20 and 29 rounds to 30; 2 × 3 = 6, and 20 and 30 carry one zero each, so two zeros follow the 6. Answer: about 600 seats. The distractors: 50 comes from adding the two rounded numbers instead of multiplying them, 20 + 30; 60 comes from multiplying 20 by the 3 of 30 and forgetting the zero in 30; 6,000 comes from attaching three zeros to 2 × 3 when 20 and 30 provide only two between them.
- (d) 140 — Method: round each number to the nearest 10, then add the rounded values. Working: 89 rounds to 90 (nearest 10) and 52 rounds to 50 (nearest 10). 90 + 50 = 140. Answer: 140. 141 is the exact value of 89 + 52, found without rounding first, so it is not an estimate. 130 comes from rounding 89 down to 80 instead of up to the nearest 10, 90. 150 comes from rounding 52 up to 60 instead of down to the nearest 10, 50.
- (b) 7 — Method: trap the number between the two square numbers on either side of it, then decide which of them it is nearer to. Working: 6² = 36 and 7² = 49, so √45 lies between 6 and 7; 49 − 45 = 4 while 45 − 36 = 9, so 45 is nearer to 49. Answer: 7. The distractors: 6 comes from taking the square number below 45 and stopping there, without checking which of 36 and 49 is nearer; 22.5 comes from halving 45 instead of looking for the number that multiplies by itself to give 45; 2,025 comes from squaring 45 instead of taking its square root.
- (c) 24 — Method: round each number to the nearest whole number, then multiply the rounded values. Working: 6.4 rounds to 6 (nearest whole number) and 3.9 rounds to 4 (nearest whole number). 6 × 4 = 24. Answer: 24. 18 comes from rounding 3.9 down to 3 instead of up to the nearest whole number, 4, giving 6 × 3. 28 comes from rounding 6.4 up to 7 instead of down to the nearest whole number, 6, giving 7 × 4. 25 is the exact value of 6.4 × 3.9, which is 24.96, rounded to the nearest whole number after multiplying, rather than estimated by rounding first.
- (b) 20 — Method: check a calculator answer by replacing the percentage with a simple fraction — 25% is one quarter, so the calculation becomes a division by 4. Working: 80 ÷ 4 = 20, and the calculator answer of 320 is 80 × 4, which is what happens when the amount is multiplied by 4 instead of divided by it; a quarter of an amount must be smaller than the amount. Answer: 20. The distractors: 25 comes from writing the percentage itself down as the answer; 16 comes from dividing by 5 instead of by 4, which finds 20% rather than 25%; 3.2 comes from working out 80 ÷ 25 instead of a quarter of 80.
- (a) 3200 — Method: round each number to 1 significant figure, then multiply the rounded numbers. Working: 37 rounds to 40 (the digit after the first, 7, rounds the 3 up to 4), and 84 rounds to 80, so the estimate is 40 × 80 = 3200. 2400 comes from rounding 37 down to 30, keeping the first digit as it is instead of letting the 7 round it up, giving 30 × 80 = 2400. 3108 comes from multiplying the exact numbers, 37 × 84, without rounding either of them first. 120 comes from adding the rounded numbers, 40 + 80 = 120, instead of multiplying them. Answer: 3200.
- (c) An under-estimate, by 8 — Method: work out the exact product, then compare it with the estimate; an estimate that is smaller than the exact value is an under-estimate, and the difference between them is the size of the error. Working: 48 × 21 = 48 × 20 + 48 = 960 + 48 = 1,008, and 1,008 − 1,000 = 8, so the estimate falls short. Answer: an under-estimate, by 8. The distractors: an over-estimate by 8 has the size of the error right but the direction wrong, and comes from assuming that rounding 48 up to 50 must push the estimate above the exact value, without allowing for 21 being rounded down; an over-estimate by 19 comes from working out 48 × 21 as 48 × 20 + 21 = 981, adding a 21 where another 48 belongs; the claim that the estimate is exactly right comes from arguing that one number was rounded up and the other down, so the two changes must cancel.
- (a) 49 — Method: 7² means 7 multiplied by itself. Working: 7 × 7 = 49. Answer: 49. 14 comes from working out 7 × 2, treating the power 2 as a number to multiply by rather than an instruction to multiply 7 by itself. 77 comes from writing the digit 7 twice side by side, treating the power as an instruction to repeat the digit rather than to multiply. 9 comes from working out 7 + 2, adding the base and the power instead of multiplying the base by itself.
- (b) 54 — Method: round each number to the nearest whole number, then subtract the rounded values. Working: 79.3 rounds to 79 (nearest whole number) and 24.6 rounds to 25 (nearest whole number). 79 − 25 = 54. Answer: 54. 54.7 is the exact value of 79.3 − 24.6, found without rounding first, so it is not an estimate. 55 comes from rounding 24.6 down to 24 instead of up to the nearest whole number, 25, giving 79 − 24. 59 comes from rounding 24.6 to the nearest 10, 20, instead of to the nearest whole number, 25, giving 79 − 20.
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