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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Answer key: Number worksheet — GCSE Foundation
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- (a) 36 kg — The mix is 3 + 5 = 8 equal shares, so the cement is 3/8 of the mass. One share is 96 ÷ 8 = 12 kg, and the cement is 3 shares: 3 × 12 = 36 kg. Working out 3/5 of 96 gives 57.6 kg, which uses the sand as the denominator instead of the whole mix; 60 kg is the mass of the sand; 12 kg is one share only.
- (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) 3/7 — Total parts = 3 + 4 = 7. Boys are 3 of the 7 parts, so the fraction is 3/7. 4/7 comes from finding the fraction of girls instead of boys. 3/4 comes from writing the ratio itself as a fraction, without adding the parts to find the total. 7/3 comes from putting the total number of parts over the number of boys instead of the number of boys over the total.
- (d) 8 — Method: whatever is inside the brackets is worked out first, and the multiplication is carried out afterwards. Working: inside the brackets, −3 + 7 = 4, and then 2 × 4 = 8. Answer: 8. The distractors: 1 comes from ignoring the brackets and multiplying first, giving 2 × (−3) = −6 and then −6 + 7 = 1; −20 comes from reading −3 + 7 as −(3 + 7) = −10, so that 2 × (−10) = −20; 6 comes from working the brackets out correctly and then adding the 2 instead of multiplying by it, giving 2 + 4 = 6.
- (b) 364 — Divide in stages using multiples of 12. 12 × 300 = 3600, leaving a remainder of 4368 − 3600 = 768. Then 12 × 64 = 768, so 4368 ÷ 12 = 300 + 64 = 364. Placing the decimal point as though dividing 436.8 by 12 gives 36.4. Transposing the last two digits of 364 gives 346. Working out 768 ÷ 12 as 4 instead of 64, losing the tens digit, and adding 300 + 4 gives 304. So 4368 ÷ 12 = 364.
- (b) 4/9 — Method: double the butter part of the ratio, keeping flour the same, find the new total, then write butter's part over the new total. Working: the new ratio is flour : butter = 5 : 4, since butter doubles from 2 to 4. New total = 5 + 4 = 9. Fraction of butter = 4/9. Answer: 4/9. 2/9 comes from forgetting to double the butter part and using the original value 2 over the new total of 9. 4/7 comes from doubling the butter part correctly to 4 but keeping the old total of 7 instead of working out the new total. 2/5 comes from using the original ratio 5:2 directly as butter over flour without doubling anything.
- (b) 8π cm² — A diameter of 8 cm gives a radius of 4 cm. The area of a full circle would be π × r² = π × 4² = 16π cm², and a semicircle is exactly half of this, giving 16π ÷ 2 = 8π cm². Forgetting to halve the area for the semicircle gives 16π cm², the area of the whole circle. Halving the diameter twice, using a radius of 2 instead of 4, gives π × 2² = 4π cm². Using the diameter itself as the radius, so π × 8² = 64π, and then halving that for the semicircle gives 32π cm².
- (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.
- (a) 1/2 — The ratio 1:2:3 has 1 + 2 + 3 = 6 parts in total. Amir and Bo together receive 1 + 2 = 3 of those parts, so together they receive 3/6 of the £60, which simplifies to 1/2. Using only Amir's single part, 1/6, ignores Bo's share entirely. Adding Bo's and Chen's parts instead of Amir's and Bo's, 2 + 3 = 5, gives 5/6. Comparing Amir and Bo's 3 parts to Chen's 3 parts, rather than to the total of 6 parts, gives 3/3 = 1.
- (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.
- (a) 10π cm — The circumference of a circle is found from C = 2 × π × r. With a radius of 5 cm, this gives C = 2 × π × 5 = 10π cm. Using the radius directly in the formula without doubling it gives 5π cm, missing the factor of 2. Using the formula for area, π × r², instead of circumference gives 25π cm, which is also the wrong units for a length. Doubling the radius to get a diameter of 10 and then applying the circumference formula a second time gives 20π cm, doubling the answer that is already correct.
- (d) 64 — Method: each time the power increases by 1, the value doubles, since one more 2 is multiplied in. Working: 2⁶ = 2⁵ × 2 = 32 × 2 = 64. Answer: 64. (37 comes from adding the power to the value, 32 + 5, instead of multiplying by the base. 192 comes from multiplying 32 by the new power, 6, instead of by the base, 2. 34 comes from adding the base to the value, 32 + 2, instead of multiplying.)
- (b) −0.7 < −0.25 — Method: compare the two negative decimals by their distance from zero on a number line. Working: −0.7 is 0.7 away from zero and −0.25 is 0.25 away from zero, so −0.7 is further from zero in the negative direction, making it the smaller number. Answer: −0.7 < −0.25 is true. "−0.7 > −0.25" comes from comparing 0.7 and 0.25 as if both numbers were positive, ignoring the negative signs. "−0.7 = −0.25" comes from assuming the two numbers are equal because they are both negative decimals. "−0.7 ≥ −0.25" combines the false statement "−0.7 > −0.25" with the false statement "−0.7 = −0.25".
- (a) 1/16 — Method: terms can only be subtracted once they share a denominator, so write every term over the largest denominator, 16, and then subtract the numerators in order from left to right. Working: 1 = 16/16, 1/2 = 8/16, 1/4 = 4/16 and 1/8 = 2/16, so the numerators give 16 − 8 − 4 − 2 − 1 = 1, over a denominator of 16. Answer: 1/16. The distractors: 1/8 comes from stopping one term early, after 16 − 8 − 4 − 2 = 2; 3/16 comes from a sign slip on the last term, adding it instead of subtracting it, which gives 2 + 1 = 3; 15/16 comes from working from the right-hand end as though the last four terms were bracketed together, so that only a single sixteenth is taken away from 1.
- (b) 2 — Method: the cube root of a number is the value that multiplies by itself three times to give that number. Working: 2 × 2 × 2 = 8, so ∛8 = 2. 4 comes from working out 8 ÷ 2 = 4, halving the number instead of finding its cube root. 24 comes from working out 8 × 3 = 24, multiplying by 3 instead of cube-rooting it. 64 is 8², the square of 8, not its cube root. Answer: 2.
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