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.Tickets to a theme park cost £38.50 for an adult and £19.75 for a child. A family estimates the total cost for 4 adults and 3 children, by rounding each ticket price to the nearest £5. Work out their estimate for the total cost.
- 2.Work out an estimate for 79.3 − 24.6, by rounding each number to the nearest whole number.
- 3.A shop assistant says that 7.2 × 3.9 = 56.16. Work out an estimate for 7.2 × 3.9, by rounding each number to the nearest whole number, to show that the assistant’s answer cannot be correct.
- 4.Hannah works out 3.1 × 19.6 on her calculator and writes down 6.076. Work out an estimate for 3.1 × 19.6, by rounding each number to 1 significant figure.
- 5.Oliver drives 95 km at an average speed of 50 km/h. Work out an estimate for the time the journey takes, by rounding the distance to the nearest 100 km.
- 6.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.
- 7.A charity bake sale sells 187 cakes at £2.95 each. By rounding each number to 1 significant figure, work out an estimate for the total amount raised.
- 8.A minibus can carry 16 passengers. A school is taking 179 pupils on a trip. By rounding 179 to the nearest 10, work out an estimate for the number of minibuses needed, given that the school cannot hire part of a minibus.
- 9.Work out an estimate for 312 × 19, by rounding each number to 1 significant figure.
- 10.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.
- 11.Freya uses her calculator to work out 7² and writes down 14. Work out the correct value of 7².
- 12.A school orders 187 packed lunches for a trip. Each packed lunch costs £4.85. The school has £900 to spend. By rounding each number to 1 significant figure, work out an estimate for the total cost and decide whether £900 is enough.
- 13.Estimate the value of √45, giving your answer to the nearest whole number.
- 14.Work out an estimate for 397 ÷ 21, by rounding each number to 1 significant figure.
- 15.Work out an estimate for 588 ÷ 31, by rounding each number to 1 significant figure.
Answer key
- (b) £220 — Method: round each ticket price to the nearest £5, multiply each rounded price by the number of tickets, then add the two totals. Working: the adult price £38.50 rounds to £40, and 4 × £40 = £160; the child price £19.75 rounds to £20, and 3 × £20 = £60; £160 + £60 = £220. Answer: £220. £213.25 is the exact total cost, found without rounding the prices first, so it is not an estimate. £160 comes from including the cost of the adult tickets only and forgetting the three children's tickets. £200 comes from swapping the two ticket quantities, using 3 adults and 4 children instead of 4 adults and 3 children.
- (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.
- (d) 28 — Method: round each number to the nearest whole number, then multiply the rounded numbers to get an estimate that can be compared with the assistant's answer. Working: 7.2 rounds to 7, and 3.9 rounds to 4, so the estimate is 7 × 4 = 28. Since 28 is much smaller than 56.16, the assistant's answer cannot be correct. 56 comes from rounding the assistant's answer to the nearest whole number, instead of rounding the two numbers being multiplied and then multiplying them. 35 comes from rounding both numbers correctly but then slipping in the seven times table, writing 7 × 5 = 35 in place of 7 × 4 = 28. 21 comes from rounding 3.9 down to 3 instead of 4, giving 7 × 3 = 21. Answer: 28.
- (d) 60 — Method: round each number to 1 significant figure and multiply; the estimate then shows whether the calculator answer is sensible. Working: 3.1 rounds to 3 and 19.6 rounds to 20, so the estimate is 3 × 20 = 60. Answer: 60. Hannah's 6.076 is about ten times too small, which is what happens when 19.6 is keyed in as 1.96. The distractors: 62 comes from rounding 19.6 only and leaving 3.1 as it stands, giving 3.1 × 20 = 62; 6 comes from trusting the calculator display rather than checking it against an estimate; 600 comes from rounding 19.6 to 200 instead of to 20, a place-value slip, giving 3 × 200 = 600.
- (d) 2 hours — Method: the time for a journey is the distance divided by the speed, so round the distance first and then divide by the speed. Working: 95 km rounds to 100 km, and 100 ÷ 50 = 2; the speed is in kilometres per hour, so the answer is a number of hours. Answer: 2 hours. The distractors: 1 hour comes from rounding the distance down to 50 km to match the speed, so that the journey looks like a single hour of driving; 30 minutes comes from dividing the speed by the distance, 50 ÷ 100, instead of the distance by the speed; 1 hour 54 minutes is the exact time, 95 ÷ 50 = 1.9 hours, worked out in full when the question asks for an estimate.
- (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.
- (a) £600 — Method: round the number of cakes and the price of each cake to 1 significant figure, then multiply the rounded values. Working: 187 rounds to 200, and £2.95 rounds to £3 (the digit after the first, 9, rounds the 2 up to 3), so the estimate is 200 × £3 = £600. £400 comes from rounding £2.95 down to £2 instead of up to £3, giving 200 × £2 = £400. £561 comes from rounding only the price and using the exact number of cakes, 187 × £3 = £561. £570 comes from rounding 187 to the nearest 10 as 190 instead of to 1 significant figure as 200, giving 190 × £3 = £570. Answer: £600.
- (b) 12 — Method: round the number of pupils to the nearest 10, divide by the number of passengers each minibus can carry, then round up because a part-full minibus still needs a whole vehicle. Working: 179 rounds to 180 (nearest 10); 180 ÷ 16 = 11.25; 11 minibuses only carry 176 passengers, so a 12th minibus is needed for the rest. Answer: 12. 11 comes from rounding 11.25 to the nearest whole number in the usual way, without checking that the leftover pupils still need transporting. 10 comes from rounding 179 down to 170 instead of to the nearest 10, 180. 180 comes from stopping after rounding the number of pupils, without dividing by the number of passengers each minibus carries at all.
- (c) 6000 — Method: round each number to 1 significant figure, then multiply the rounded values. Working: 312 rounds to 300 (1 s.f.) and 19 rounds to 20 (1 s.f.). 300 × 20 = 6000. Answer: 6000. 5928 is the exact value of 312 × 19, found by multiplying without rounding first, which is not an estimate. 600 comes from rounding 19 down to 2 instead of to 20, losing a zero from its place value. 6200 comes from rounding 312 to the nearest 10, 310, instead of to 1 significant figure, 300, then multiplying by the correctly rounded 20.
- (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) 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) £1,000, so £900 is not enough — Method: round each number to 1 significant figure, multiply to estimate the total cost, then compare the estimate with the money available. Working: 187 rounds to 200 and £4.85 rounds to £5, so the estimate is 200 × 5 = 1,000, and £1,000 is more than the £900 the school has. Answer: £1,000, so £900 is not enough. The distractors: £800 comes from cutting £4.85 down to £4 instead of rounding it up to £5, giving 200 × 4 = 800, and that estimate wrongly suggests the money stretches; £935 comes from rounding the price only and keeping 187 lunches, giving 187 × 5 = 935; £950 comes from rounding 187 to the nearest 10 rather than to 1 significant figure, giving 190 × 5 = 950.
- (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.
- (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.
- (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.
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