✏️ Paper 1 non-calculator warm-up — Foundation

Answer key

1. (a) £102 — Method: find the value of one part of the ratio, then work out each group's share before comparing them. Working: the ratio 5:2 has 5 + 2 = 7 parts, so one part is £238 ÷ 7 = £34. Adults donate 5 × £34 = £170 and children donate 2 × £34 = £68, so adults donate £170 − £68 = £102 more than children. So the difference is £102. Distractor £68 is only the children's donation, without finding the difference. Distractor £170 is only the adults' donation, without finding the difference. Distractor £136 comes from doubling the children's donation instead of subtracting it from the adults' donation.
2. (d) 3/5 — Method: a decimal with one digit after the point is a number of tenths, so it is written over 10 and then cancelled. Working: 0.6 is six tenths, so 0.6 = 6/10; the highest common factor of 6 and 10 is 2, and 6 ÷ 2 = 3 with 10 ÷ 2 = 5. Answer: 3/5. The distractors: 2/3 comes from confusing 0.6 with the recurring decimal 0.666..., which is the one that equals 2/3; 1/6 comes from putting 1 over the single digit after the point; 3/50 comes from using hundredths for a one-place decimal, giving 6/100, which then cancels by 2 to 3/50.
3. (a) 5 — 16 + 9 = 25, then √25 = 5. Splitting the root over the addition instead gives √16 = 4 and √9 = 3, then 4 + 3 = 7 — but a root does not split over a sum like this. Multiplying those two roots instead of adding them gives 4 × 3 = 12. Taking the negative square root instead of the positive one gives −5.
4. (d) 27/80 — Method: write the total as a fraction of a litre, then divide by the number of glasses. Working: 1.35 = 27/20, so each glass holds 27/20 ÷ 4 = 27/80 of a litre. Answer: 27/80. 27/20 comes from converting the total correctly to a fraction but forgetting to divide by the number of glasses. 27/5 comes from multiplying the total by 4 instead of dividing. 17/50 comes from rounding 1.35 ÷ 4 to 0.34 before converting to a fraction.
5. (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.
6. (a) 91 is not prime, because 91 = 7 × 13. — Check 91 for prime factors up to its square root, which is just under 10: 91 ÷ 7 = 13, and both 7 and 13 are prime, so 91 = 7 × 13 and 91 is not a prime number. Checking only 2, 3 and 5 misses that 7 also needs to be tried — 91 is odd, its digits do not sum to a multiple of 3 (9 + 1 = 10), and it does not end in 0 or 5, so those three checks alone wrongly suggest it is prime. Assuming any odd number ending in 1 must be prime ignores that 91 = 7 × 13 is a counterexample. Misapplying the digit-sum test for 3 by miscounting 9 + 1 as a multiple of 3 wrongly concludes 91 is divisible by 3, when the correct digit sum, 10, is not a multiple of 3. So 91 is not prime, because 91 = 7 × 13.
7. (d) 3 : 8 — Multiply both parts by 4 to clear the decimal: 0.75 × 4 = 3 and 2 × 4 = 8, giving 3 : 8, which has no common factor other than 1. Giving 75 : 200 multiplies by 100 instead of 4, and has not then been simplified down to 3 : 8. Giving 0.75 : 2 has not been converted into whole numbers at all. Giving 3 : 2 converts the first part correctly but leaves the second part unscaled.
8. (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.
9. (b) £60 — £48 represents 100% − 20% = 80% of the original price. 1% = £48 ÷ 80 = £0.60, so 100% = £0.60 × 100 = £60.
10. (a) 20% — Method: percentage increase = increase ÷ original amount × 100. Working: the increase is 84 − 70 = 14 marks, and 14 ÷ 70 = 0.2, so 0.2 × 100 = 20. Answer: an increase of 20%. The distractors: 14% comes from quoting the 14 mark increase as though marks and per cent were the same thing; 17% comes from dividing the 14 by the new mean 84 instead of by the original 70, which gives 17% to the nearest per cent; 120% is the new mean written as a percentage of the old one, which is the whole of the new mean rather than the increase.
11. (d) 25% — Method: percentage decrease = decrease ÷ original amount × 100. Working: the reduction is £60 − £45 = £15, and 15 ÷ 60 = 0.25, so 0.25 × 100 = 25%. Answer: 25%. The distractors: 15% comes from quoting the £15 reduction as though pounds and per cent were the same thing; 33% comes from dividing the £15 by the new price £45 instead of by the original £60, which gives 33% to the nearest per cent; 75% is the new price written as a percentage of the old one, which is what is still paid rather than what has been taken off.
12. (b) £72 — One fifth of £60 = £12. New price = £60 + £12 = £72. A candidate who gives the increase instead of the new price gets £12. A candidate who subtracts the increase instead of adding it gets £60 − £12 = £48. A candidate who uses 1/4 instead of 1/5 gets £60 + £15 = £75.
13. (c) 200 students — Method: find the percentage who have a brother or a sister, taking care that the students with both are not counted twice, then take that percentage from 100% and apply the result to the 400 students. Working: 25% + 40% = 65%, but the 15% with both has been counted in each of those figures, so 65% − 15% = 50% have a brother or a sister; that leaves 100% − 50% = 50%, and 50% of 400 = 200. Answer: 200 students. The distractors: 260 students is 65% of 400, the number with a brother or a sister when the 15% overlap is counted twice; 140 students comes from taking that same uncorrected 65% away from the 400; 300 students comes from subtracting only the 25% with a brother and ignoring the sisters altogether.
14. (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.
15. (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.
16. (d) 4:3 — Boys = 28 − 16 = 12. Girls to boys is 16 : 12. The highest common factor of 16 and 12 is 4. Divide both parts by 4: 16 ÷ 4 = 4 and 12 ÷ 4 = 3, so the ratio is 4 : 3. Dividing by 2 instead of 4 gives 8 : 6, which still shares a common factor of 2, so it is not fully simplified. Swapping the order gives 3 : 4, boys to girls instead of girls to boys. Writing girls to the whole class, 16 : 28, simplifies to 4 : 7 — that compares girls to everyone, not girls to boys.
17. (a) 23 — Division undoes multiplication, so the missing number is 391 ÷ 17 = 23. Writing down 17 repeats the number already given instead of solving for the missing one. Subtracting instead of dividing gives 391 − 17 = 374. Multiplying instead of dividing gives 391 × 17 = 6647.
18. (c) 1:2 — Convert £1.50 into pence: £1.50 = 150p, so the ratio is 75 : 150. Dividing both parts by 75 gives 1 : 2. Getting 50 : 1 comes from not converting the units at all and simplifying 75 : 1.5. Getting 2 : 1 has the two parts the wrong way round. Getting 3 : 4 comes from reading 75p as 3/4 of a pound and then comparing it with £1 instead of £1.50.
19. (b) 3 : 5 — The highest common factor of 27 and 45 is 9. Divide both parts by 9: 27 ÷ 9 = 3 and 45 ÷ 9 = 5, giving 3 : 5. Giving 9 : 15 divides by 3, a common factor but not the highest one, so it is not yet in simplest form. Giving 5 : 3 swaps the order of the two parts. Giving 27 : 45 has not been simplified at all.
20. (a) 36 — Method: for the highest common factor, take the LOWER power of each prime that appears in both numbers. Working: for 2, the lower power is 2² (from 108); for 3, the lower power is 3² (from 72), so the highest common factor is 2² × 3² = 4 × 9 = 36. 216 comes from taking the higher power of each prime instead, 2³ × 3³ = 8 × 27 = 216, which gives the lowest common multiple, not the highest common factor. 108 is simply one of the two numbers, not their highest common factor. 6 comes from multiplying the primes without any powers at all, 2 × 3 = 6. Answer: 36.