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.
Answer key: Number worksheet — GCSE Foundation
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- (c) 6.3 × 10⁷ — To add numbers in standard form, first write them with the same power of 10. 6 × 10⁷ = 60 × 10⁶, so the sum is 60 × 10⁶ + 3 × 10⁶ = 63 × 10⁶ = 6.3 × 10⁷. A candidate who added the A values without adjusting for the different powers worked out 6 + 3 = 9 and kept the larger power, writing 9 × 10⁷. A candidate who added the powers of 10 as if multiplying wrote 9 × 10¹³. A candidate who added the A values but used the smaller power wrote 9 × 10⁶.
- (c) 9π cm² — The area of a circle is π × r². With a radius of 3 cm this is π × 3² = 9π cm², and this is exact because π has not been replaced by any approximation. Writing 28.3 cm² replaces π with a rounded decimal value, 3.14, and then rounds the result again, so it is only an approximation. Writing 28.26 cm² uses π ≈ 3.14 without a final rounding step, but this is still only an approximation of 9π, not the exact value. Writing 27 cm² comes from replacing π with the rough approximation 3, which is even further from the true value.
- (c) −1 — Method: find each cube root separately, keeping its sign, and then add the two results. Working: (−3) × (−3) × (−3) = −27, so ∛(−27) = −3, and 2 × 2 × 2 = 8, so ∛8 = 2. Adding gives −3 + 2 = −1. Answer: −1. The distractors: 5 comes from taking the cube root of a negative number as positive, giving 3 + 2; −5 comes from reading the minus sign as applying to the whole sum and working out −(3 + 2); −6 comes from multiplying the two roots, −3 × 2, instead of adding them.
- (c) 87 litres — Work out how much water is drained: 8 × 6 = 48 litres. Subtract this from the starting amount: 120 − 48 = 72 litres. Then add the 15 litres from the hose: 72 + 15 = 87 litres. Subtracting the 15 litres instead of adding it, as though the hose also removed water, gives 120 − 48 − 15 = 57 litres. Stopping after the drain step, without adding the hose water back in, leaves the working at 72 litres. Adding the rate and the time instead of multiplying them, 8 + 6 = 14 litres drained, and then working from there gives 120 − 14 + 15 = 121 litres. So 87 litres of water is left in the tank.
- (c) 0.94 — Method: a fraction is written as a decimal by making the denominator a power of ten, because the decimal places record tenths, hundredths and thousandths. Working: 50 × 2 = 100, so the numerator must also be multiplied by 2, giving 47 × 2 = 94 and the equivalent fraction 94/100; 94 hundredths is written with two digits after the decimal point. Answer: 0.94. The distractors: 0.47 comes from treating the denominator as though it were already 100 and writing the digits of the numerator straight after the point; 4.7 comes from dividing 47 by 10 instead of by 50; 0.094 comes from multiplying the denominator by 20 to reach 1000 but the numerator by only 2, giving 94/1000.
- (a) 42.5 ≤ t < 47.5 — Rounding to the nearest 5 minutes means the actual time can be up to half of 5 minutes, 2.5 minutes, below or above 45 before it would round to a different multiple of 5. The lower bound is 45 − 2.5 = 42.5 and the upper bound is 45 + 2.5 = 47.5. A time of exactly 47.5 minutes would round up to 50, not 45, so 47.5 is excluded while 42.5 does still round to 45. Writing 42.5 ≤ t ≤ 47.5 wrongly includes 47.5. Writing 40 ≤ t < 50 uses a whole rounding unit, 5, either side instead of half of it. Writing 44.5 ≤ t < 45.5 treats the rounding unit as 1 minute instead of 5 minutes.
- (a) 40 — Multiply the number of choices for each item: 4 × 5 × 2 = 40. 11 comes from adding the three numbers instead of multiplying them. 20 comes from multiplying only the crisps and chocolate bars, 4 × 5, and forgetting the drink. 10 comes from multiplying only the chocolate bars and drinks, 5 × 2, and forgetting the crisps.
- (d) 8 — Method: the square root of a number is the value that multiplies by itself to give that number. Working: 8 × 8 = 64, so √64 = 8. 32 comes from working out 64 ÷ 2 = 32, halving the number instead of finding its square root. 4 is the cube root of 64, since 4 × 4 × 4 = 64, not the square root. 16 comes from working out 8 × 2 = 16, doubling the correct answer by mistake. Answer: 8.
- (c) 6 — 2 × 3 = 6, then 36 ÷ 6 = 6. Ignoring the brackets and working left to right gives 36 ÷ 2 = 18, then 18 × 3 = 54. Multiplying by the bracket instead of dividing by it gives 2 × 3 = 6, then 36 × 6 = 216. Dividing by only the 2 inside the bracket, and ignoring the × 3, gives 36 ÷ 2 = 18.
- (b) 5/27 — Method: multiply the numerators together and the denominators together, then simplify. Working: (5 × 2)/(6 × 9) = 10/54 = 5/27. Answer: 5/27. 7/15 comes from adding the fractions instead of multiplying: (5+2)/(6+9) = 7/15. 15/4 comes from flipping the second fraction, as if dividing: (5 × 9)/(6 × 2) = 45/12 = 15/4. 5/3 comes from cancelling the two denominators against each other, dividing both 6 and 9 by 3 to leave 5/2 × 2/3 = 10/6 = 5/3; cancelling is only valid between a numerator and a denominator, never between two denominators.
- (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.
- (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.
- (d) 5 — Method: the greatest number of identical bunches is the highest common factor of the two flower totals; then divide the red roses by that number of bunches. Working: 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7, so their highest common factor is 2² × 3 = 12. That means 12 bunches, and 60 ÷ 12 = 5 red roses in each. 7 is the number of white roses in each bunch, since 84 ÷ 12 = 7, not red roses. 12 is the number of bunches itself, not the number of red roses in one bunch. 20 comes from working out 60 ÷ 3 = 20, dividing by only part of the highest common factor. Answer: 5.
- (b) 1.45 ≤ m < 1.55 — The flour's mass is labelled 1.5 kg, correct to the nearest 0.1 kg, so half of 0.1 kg is added to and subtracted from 1.5 kg to find the interval: 1.5 − 0.05 = 1.45 and 1.5 + 0.05 = 1.55, giving 1.45 ≤ m < 1.55. '1.4 ≤ m < 1.6' comes from taking the whole 0.1 kg as the margin either side, instead of half of it. '1.45 < m ≤ 1.55' comes from writing the inequality signs the wrong way round — the lower bound should be included and the upper bound excluded, not the other way round. '1.45 ≤ m ≤ 1.55' comes from including the upper bound, when the convention is that the upper bound is never actually reached.
- (d) 60 mph — Average speed = distance ÷ time, with the time measured in hours. 45 minutes is 45/60 of an hour, which is 0.75 of an hour, so the journey takes 1.75 hours. Speed = 105 ÷ 1.75 = 60 mph. 52.5 mph rounds the time up to 2 hours, 72.4 mph writes 1 hour 45 minutes as 1.45 hours, and 183.75 mph multiplies the distance by the time instead of dividing.
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