Printable · GCSE Foundation · ages 14-16
Number worksheet — GCSE Foundation
Fifteen questions across the number statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
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Number worksheet — GCSE Foundation
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- 1.In a science experiment, the temperature of a liquid is recorded as 18.6 °C, correct to the nearest 0.2 °C. Write down the error interval for the actual temperature, T °C.
- 2.Work out the total cost of 3 pens at 45p each and one notebook at £2.35. Give your answer in pounds.
- 3.Work out an estimate for 8,900 ÷ 29, by rounding each number to 1 significant figure.
- 4.On one night in Manchester the temperature at 3 am was 8 °C below zero and at 9 am it was 3 °C below zero. Write down the warmer of the two readings.
- 5.Work out 2/3 × 3/4 exactly, giving your answer in its simplest form.
- 6.Work out √25 + 4² − 12 ÷ 3
- 7.Work out 8% of £250.
- 8.Write 60 as a product of its prime factors, using index notation.
- 9.Find the missing number: ▢ ÷ 15 = 24
- 10.A recipe uses 160 g of flour. Sam wants to increase the amount by 1/4. Work out the new amount of flour.
- 11.A space probe is 7.5 × 10⁸ km from Earth. Write this distance as an ordinary number.
- 12.Round 6.283 to 1 significant figure.
- 13.Work out the value of .
- 14.Grace buys three items costing £9.95, £19.90 and £4.99. Work out an estimate for the total cost, by rounding each price to the nearest pound.
- 15.Work out 20 − 8 ÷ 2 + 1
Answer key
- (b) 18.5 ≤ T < 18.7 — Method: the error interval reaches half the rounding unit either side of the recorded value. Working: half of 0.2 is 0.1, so the interval runs from 18.6 − 0.1 to 18.6 + 0.1. Answer: 18.5 ≤ T < 18.7. (18.4 ≤ T < 18.8 comes from using the full rounding unit, 0.2, either side instead of half of it. 18.5 ≤ T ≤ 18.7 comes from including the upper bound with ≤ instead of excluding it with <. 18.6 ≤ T < 18.8 comes from treating the recorded value as the start of the interval and adding the whole rounding unit, 0.2, above it.)
- (b) £3.70 — Work in one unit throughout. 45p is £0.45, so three pens cost 3 × £0.45 = £1.35. Adding the notebook gives £1.35 + £2.35 = £3.70. £2.80 is the cost of one pen and the notebook, so the multiplication by 3 has been left out; £8.40 multiplies the notebook by 3 as well as the pens; £15.85 comes from reading 45p as £4.50.
- (a) 300 — Method: round each number to 1 significant figure, then divide the rounded values. Working: 8,900 rounds to 9,000 and 29 rounds to 30; cancelling a zero from each gives 900 ÷ 3. Answer: 300. The distractors: 450 comes from rounding 29 down to 20 instead of to the nearest ten, giving 9,000 ÷ 20; 3,000 comes from rounding 29 to 3 rather than to 30, a place-value slip that divides by a number ten times too small; 307 is the exact quotient rounded to the nearest whole number, worked out in full when the question asks for an estimate.
- (a) −3 °C — Method: write each reading as a signed temperature, then choose the one further to the right on a number line. Working: 8 °C below zero is −8 °C and 3 °C below zero is −3 °C. On a number line −3 lies to the right of −8, so it is the warmer reading. Answer: −3 °C. The distractors: −8 °C comes from ordering negatives by the size of their digits, treating −8 as the larger number; 3 °C has the right size but the sign dropped, and a reading of 3 °C is above zero rather than below it; 5 °C comes from working out the difference between the two readings instead of choosing one of them.
- (b) 1/2 — To multiply fractions, multiply the numerators together and multiply the denominators together: 2 × 3 = 6 and 3 × 4 = 12, giving 6/12, which simplifies to 1/2. Adding the fractions instead of multiplying them, using a common denominator of 12, gives 8/12 + 9/12 = 17/12. Dividing by 3/4 instead of multiplying by it, so multiplying by its reciprocal 4/3, gives 2/3 × 4/3 = 8/9. Multiplying only the numerators, 2 × 3 = 6, and keeping the first denominator, 3, unchanged gives 6/3 = 2.
- (b) 17 — Roots and powers are worked out first: √25 = 5 and 4² = 16. Division comes next: 12 ÷ 3 = 4. Then addition and subtraction, left to right: 5 + 16 − 4 = 17. A candidate who treated 4² as 4 × 2 = 8, multiplying the base by the exponent instead of squaring it, worked out 5 + 8 − 4 = 9. A candidate who did not evaluate the root and used 25 itself worked out 25 + 16 − 4 = 37. A candidate who ignored the priority of division and worked through 5 + 16 − 12 ÷ 3 strictly left to right got 5 + 16 = 21, then 21 − 12 = 9, then 9 ÷ 3 = 3.
- (d) £20 — 1% of £250 = £2.50, so 8% = 8 × £2.50 = £20. A candidate who misplaces the decimal point and finds 0.8% instead gets £2. A candidate who confuses 8% with 80% gets £200. A candidate who rounds 8% up to the nearby 10% gets £25.
- (b) 2² × 3 × 5 — Repeatedly divide 60 by prime numbers: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, and 5 is itself prime. So 60 is 2 × 2 × 3 × 5, which in index notation is 2² × 3 × 5. Stopping the factor tree after only three divisions and writing 2 × 3 × 5 misses that the 2 divides in twice, and gives only 30, not 60. Squaring the 3 as well as the 2 gives 2² × 3² × 5, which comes to 180, far too big. Squaring the 5 instead of the 2 gives 2 × 3 × 5², which comes to 150, also too big. So 60 = 2² × 3 × 5.
- (a) 360 — The inverse of ÷ 15 is × 15, so the missing number is 24 × 15 = 360. Subtracting instead of multiplying gives 24 − 15 = 9. Dividing by 15 again instead of multiplying gives 24 ÷ 15 = 1.6. Adding instead of multiplying gives 24 + 15 = 39.
- (a) 200 g — One quarter of 160 g is 40 g. Increasing the amount means adding this on: 160 + 40 = 200 g. Finding the increase, 1/4 of 160 = 40 g, but stopping there without adding it to the original amount leaves just 40 g. Using 4/5 instead of 5/4 as the scaling fraction, 160 × 4/5 = 128 g, actually decreases the amount rather than increasing it. Increasing by a half instead of a quarter, 160 + 80 = 240 g, uses the wrong fraction of 160.
- (b) 750,000,000 — Method: multiplying by 10⁸ moves the decimal point eight places to the right, and every empty place is filled with a zero. Working: 10⁸ = 100,000,000, and moving the decimal point in 7.5 eight places to the right gives 7.5 × 100,000,000. Answer: 750,000,000. The distractors: 7,500,000,000 comes from removing the decimal point first to make 75 and then writing eight zeros after it, which carries the digits one place too far; 600 comes from reading 10⁸ as 10 × 8 = 80 and working out 7.5 × 80; 0.000000075 comes from moving the decimal point eight places to the left, as though the index were negative.
- (c) 6 — Method: the first significant figure is the first non-zero digit; round using the digit after it to decide whether to round up or down. Working: the first significant figure of 6.283 is the 6; the next digit is 2, which rounds down, so 6.283 rounds to 6. 6.3 comes from rounding to 2 significant figures instead of 1. 10 comes from rounding up to the nearest 10 instead of finding 1 significant figure of the number itself. 0.6 comes from misplacing the decimal point after rounding. Answer: 6.
- (c) 0.001 — Method: a negative index means 'one over' the positive power, so $10^{-3}$ means one over $10^{3}$. Working: ten cubed is 1000, and one over 1000 is 0.001. 1000 comes from ignoring the negative sign and working out ten cubed instead of its reciprocal. −1000 comes from ignoring what the negative index does to the size, while still writing a negative sign on the large value. −0.001 comes from correctly finding the size, 0.001, but wrongly keeping a negative sign because the index was negative. Answer: 0.001.
- (b) £35 — Method: round each price to the nearest pound, then add the rounded prices. Working: £9.95 rounds to £10, £19.90 rounds to £20 and £4.99 rounds to £5, and £10 + £20 + £5 gives the estimate. Answer: £35. The distractors: £40 comes from rounding each price up to the nearest £10 rather than to the nearest pound, giving £10 + £20 + £10; £32 comes from cutting the pence off each price instead of rounding it, giving £9 + £19 + £4; £34.84 is the exact total, worked out in full when the question asks for an estimate.
- (d) 17 — 8 ÷ 2 = 4, then 20 − 4 = 16, then 16 + 1 = 17. Stopping after the subtraction and forgetting to add the final 1 leaves 16. Adding the 4 and the 1 together before subtracting gives 4 + 1 = 5, then 20 − 5 = 15 — the subtraction should use the 4 from the division, not a combined total. Working strictly left to right without giving division priority gives 20 − 8 = 12, then 12 ÷ 2 = 6, then 6 + 1 = 7.
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