✏️ Paper 1 non-calculator warm-up — Foundation

Answer key

1. (a) £1,000 — Method: find the increase, then add it to the original price; the multiplier 1.25 does both steps at once. Working: 25% is one quarter, so 25% of £800 = £800 ÷ 4 = £200, and £800 + £200 = £1,000. Answer: £1,000. The distractors: £200 is the increase on its own, not the new price; £825 comes from adding £25 to £800, treating the 25% as £25; £600 comes from taking the 25% off the price instead of adding it on.
2. (d) 24 km — Method: a fraction acts as an operator, so finding 8/12 of a distance means dividing by the denominator and multiplying by the numerator. Working: 36 ÷ 12 = 3, so one twelfth of the walk is 3 km, and eight twelfths is 3 × 8 = 24 km. Answer: 24 km. The distractors: 12 km comes from working out the part of the walk still left, the other four twelfths, instead of the part already completed; 288 km comes from multiplying by the numerator without dividing by the denominator, giving 36 × 8 = 288; 4.5 km comes from dividing by the numerator instead of multiplying by it, giving 36 ÷ 8 = 4.5.
3. (c) 18 — 45 ÷ 5 = 9, and 2 × 9 = 18. A candidate who stops after finding one fifth gets 9. A candidate who uses 3/5 instead of 2/5 gets 27. A candidate who uses 4/5 instead of 2/5 gets 36.
4. (b) £76.00 — One part of the ratio is £47.50 ÷ 5 = £9.50. The school receives 8 parts, so its share is 9.50 × 8 = £76.00. Dividing £47.50 by 8 instead of 5, treating the charity's amount as if it were 8 parts, gives 47.50 ÷ 8 = 5.9375, then × 5 = £29.69. Adding the charity's amount to the school's amount instead of stopping at the school's own share gives the total collected, 9.50 × 13 = £123.50. Adding one part to the charity's amount instead of multiplying one part by 8 gives 47.50 + 9.50 = £57.00.
5. (a) 25 — Reverse the operations in reverse order: undo the subtraction by adding 6, then undo the division by multiplying by 5. −1 + 6 = 5, so the number divided by 5 equals 5, and 5 × 5 = 25 — checking, 25 ÷ 5 − 6 = 5 − 6 = −1. A candidate who subtracted 6 again instead of adding worked out −1 − 6 = −7, then −7 × 5 = −35. A candidate who multiplied by 5 before undoing the subtraction, doing the inverse operations in the wrong order, worked out −1 × 5 = −5, then −5 + 6 = 1. A candidate who multiplied by 5 but forgot to undo the subtraction at all worked out −1 × 5 = −5 and stopped there.
6. (b) −18 — Method: the multiplication is carried out before the addition, and a negative multiplied by a positive is negative. Working: (−4) × 3 = −12, so the calculation becomes (−6) + (−12) = −18. Answer: −18. The distractors: −30 comes from adding first and multiplying afterwards, giving (−6 + −4) × 3 = −10 × 3 = −30; 6 comes from treating (−4) × 3 as +12 on the grounds that a minus sign makes a product positive, giving −6 + 12 = 6; 18 comes from ignoring both minus signs and working out 6 + 4 × 3 = 18.
7. (d) £70 — Method: the discount is a percentage of the original price only, so work it out, subtract it, and add the fixed delivery charge afterwards. Working: 25% of £80 is £80 ÷ 4 = £20, so the discounted price is £80 − £20 = £60, and the total is £60 + £10 = £70. Answer: £70. The distractors: £60 comes from working out the discounted price and stopping there, leaving the delivery charge out of the total; £65 comes from taking £25 off the price instead of 25% of it, giving £80 − £25 = £55 and then £55 + £10 = £65; £67.50 comes from adding the delivery charge before the discount and reducing the whole amount, giving 75% of £90 = £67.50.
8. (c) 0.7, 0.72, 3/4 — Method: numbers written in different forms cannot be compared as they stand, so every fraction is turned into a decimal by dividing the numerator by the denominator, and the decimals are then compared place by place from the left. Working: 3/4 means 3 ÷ 4 = 0.75, so the three values to compare are 0.7, 0.75 and 0.72; written to two decimal places they are 0.70, 0.75 and 0.72, and the hundredths digits 0, 5 and 2 put 0.70 first, 0.72 next and 0.75 last; written again in the forms the question used, the order from smallest is 0.7, then 0.72, then 3/4. Answer: 0.7, 0.72, 3/4. The distractors: 3/4, 0.7, 0.72 comes from turning 3/4 into 0.34 by writing the numerator and the denominator as the two digits after the point, which makes the fraction the smallest of the three; 0.72, 3/4, 0.7 comes from the belief that the more digits a decimal has the smaller it must be, which puts both 0.72 and 0.75 below 0.7 and 0.72 below 0.75; 3/4, 0.72, 0.7 comes from comparing the three values correctly but listing them largest first, against an instruction to start with the smallest.
9. (c) 600 — Method: round each number to the nearest 100, then add the rounded values. Working: 113 is nearer to 100 than to 200, so it rounds to 100; 491 is nearer to 500 than to 400, so it rounds to 500; adding those gives the estimate. Answer: 600. The distractors: 500 comes from rounding each number down to the hundred below instead of to the nearest hundred, giving 100 + 400; 700 comes from rounding each number up to the hundred above, giving 200 + 500; 604 is the exact total, worked out in full when the question asks for an estimate.
10. (a) 30 cm — The tile's side length must be a common factor of 90 and 120. The factors of 90 include 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; the factors of 120 include 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. The highest number common to both lists is 30, so the largest square tile has a side length of 30 cm. Picking 15 cm, a common factor but not the largest, gives tiles that are smaller than necessary. Picking 10 cm, also a common factor but smaller still, wastes even more of the possible tile size. Working out the lowest common multiple instead of the highest common factor gives 360 cm, a length far bigger than either side of the patio. So the largest square tile Ben can use has a side length of 30 cm.
11. (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.
12. (b) £300 — The increased price is 110% of the original, so the original price = £330 ÷ 1.1 = £300. A candidate who finds 10% of £330 and subtracts it, wrongly treating £330 as the original, gets £330 − £33 = £297. A candidate who adds 10% of £330 again instead of reversing the increase gets £330 + £33 = £363. A candidate who divides by 0.1 instead of 1.1 gets £3,300.
13. (a) 12 — Method: for any two numbers, their highest common factor multiplied by their lowest common multiple equals the product of the two numbers. This holds because the HCF collects every prime factor the two numbers share, and the LCM collects every prime factor that appears in either number, so between them they use each prime factor of the two numbers exactly once — the same primes as the product. Working: 4 × 60 = 240, and 240 ÷ 20 = 12. 15 comes from working out 60 ÷ 4 = 15, dividing the wrong pair of numbers. 16 comes from working out 20 − 4 = 16, subtracting the highest common factor instead of using the product rule. 240 is 4 × 60, the product of the highest common factor and the lowest common multiple, left un-divided by 20. Answer: 12.
14. (a) 20 cm — Method: match the measurement you are given to its own part of the ratio, use it to find the value of one part, then multiply by the parts belonging to the measurement asked for. Working: the length is the second measurement listed, so it matches 4 parts and one part = 16 ÷ 4 = 4 cm; the height is 5 parts, so 5 × 4 = 20. Answer: 20 cm. The distractors: 12 cm is the width, which is the 3-part measurement; 4 cm is the value of one part only; 80 cm comes from multiplying the 16 cm by 5 without first dividing by the 4 parts the length is worth.
15. (a) 5 — Method: the number of glasses is the amount in the jug divided by the amount one glass holds. Write the mixed number as an improper fraction, then divide by multiplying by the reciprocal. Working: 3 1/3 = (3 × 3 + 1)/3 = 10/3, and 10/3 ÷ 2/3 = 10/3 × 3/2 = 30/6 = 5. Answer: 5. The distractors: 2 comes from writing 3 1/3 as 4/3, adding the whole number to the numerator instead of multiplying it by the denominator first, and then dividing 4/3 by 2/3; 20/9 comes from multiplying by 2/3 instead of dividing by it; 5/3 comes from dividing by 2 rather than by 2/3, as though each glass held 2 litres.
16. (b) 88.8 — Multiply as whole numbers first, ignoring the decimal point: 37 × 24. Split it as 37 × 20 = 740 and 37 × 4 = 148, so 37 × 24 = 740 + 148 = 888. 3.7 has 1 decimal place and 24 has none, so the answer needs 1 decimal place: 88.8. Counting the 2 digits in "3.7" as though that were the number of decimal places gives 8.88 instead of 1 decimal place. Leaving the decimal point out altogether gives 888. Misreading 37 × 4 as 138 rather than 148 gives a running total of 878, placed with 1 decimal place as 87.8. So 3.7 × 24 = 88.8.
17. (b) −2 — Method: the multiplication is carried out before the addition, and a negative multiplied by a positive is negative. Working: (−3) × 4 = −12, so the calculation becomes −12 + 10 = −2. Answer: −2. The distractors: 22 comes from ignoring the minus sign and working out 3 × 4 + 10 = 22; −42 comes from adding before multiplying, giving (4 + 10) × (−3) = 14 × (−3) = −42; −22 comes from adding 12 and 10 and then writing a minus sign in front of the total, instead of moving 10 places up from −12.
18. (c) 25/36 — Method: square a fraction by squaring its numerator and its denominator separately, then add the two results over a common denominator. Working: (2/3)² = 4/9 and (1/2)² = 1/4; the lowest common denominator of 9 and 4 is 36, so 4/9 = 16/36 and 1/4 = 9/36, and 16 + 9 = 25. Answer: 25/36. The distractors: 49/36 comes from adding the two fractions first and squaring the total, giving (7/6)²; 5/13 comes from squaring correctly but then adding the numerators and the denominators, as (4 + 1)/(9 + 4); 7/3 comes from doubling each fraction instead of squaring it, giving 4/3 + 1.
19. (b) £45 — Since 2/3 of the amount is £30, one third is £30 ÷ 2 = £15, and the whole amount is three thirds: £15 × 3 = £45. Applying the fraction forwards to £30 instead of reversing it, £30 × 2/3 = £20, treats the given amount as the whole rather than as two thirds of it. Finding one third correctly as £15 but forgetting to multiply by 3 to get the whole amount leaves £15 as the final answer. Reading £30 as one third of the amount rather than as two thirds, and so multiplying straight by 3, gives £30 × 3 = £90.
20. (c) 40 cm — Method: split the total length into the number of parts shown by the ratio, then find the value of the shorter share. Working: the ratio 4:5 has 4 + 5 = 9 parts, so one part is 90 ÷ 9 = 10 cm, and the shorter piece is 4 × 10 = 40 cm. So the shorter piece is 40 cm. Distractor 50 cm is the length of the LONGER piece, not the shorter one. Distractor 45 cm comes from splitting the ribbon into two equal halves, ignoring the ratio. Distractor 10 cm is the value of one part, found correctly but never multiplied by 4.