✏️ Paper 1 non-calculator warm-up — Foundation

Answer key

1. (d) 11/12 — Convert both mixed numbers to improper fractions with a common denominator. 2 3/4 = 11/4, which is 33/12, and 1 5/6 = 11/6, which is 22/12. Subtracting, 33/12 − 22/12 gives 11/12, already in its simplest form. Forgetting to borrow, and instead subtracting the fraction parts the other way round to avoid a negative, 10/12 − 9/12 gives 1/12; adding that to the whole-number difference of 1 gives 13/12. Subtracting only the fraction parts, 9/12 − 10/12, and reporting just the size of that difference gives 1/12, which ignores the whole numbers altogether. Adding the two improper fractions instead of subtracting them, 33/12 + 22/12, gives 55/12. So 2 3/4 − 1 5/6 = 11/12.
2. (d) £50.00 — The sale price is 85% of the original, so the original price = £42.50 ÷ 0.85 = £50.00. 15% of £42.50 is £6.375. A candidate who finds 15% of £42.50 and subtracts it from the sale price gets £42.50 − £6.375 = £36.125, which is £36.13 to the nearest penny. A candidate who adds 15% of £42.50 instead of reversing the decrease gets £42.50 + £6.375 = £48.875, which is £48.88 to the nearest penny. A candidate who divides by 0.15 instead of 0.85 gets £283.33.
3. (c) a decrease of 25% — Method: use multipliers. An increase of 50% is × 1.5 and a decrease of 50% is × 0.5. Working: 1.5 × 0.5 = 0.75, so the final price is 75% of the original. Answer: a decrease of 25%. The distractors: no change comes from assuming +50% and −50% cancel; a decrease of 50% comes from applying only the second change; an increase of 25% has the direction wrong.
4. (a) 5 — Method: 10% is 10/100, which cancels to 1/10, so finding 10% of an amount means dividing it by 10. Working: 10% = 10/100 = 1/10, and 50 ÷ 10 = 5. Answer: 5 questions. The distractors: 10 comes from reading the 10% as a count of 10 questions rather than as an operator acting on the 50; 45 comes from working out how many questions Freya answers correctly, the other 90% of the quiz, instead of how many she gets wrong; 500 comes from multiplying by 10 instead of dividing by 10.
5. (a) 9 °C — Method: the fall is the difference between the two readings, so subtract the lower reading from the higher one; subtracting a negative number is the same as adding its positive. Working: 3 − (−6) = 3 + 6 = 9. Counting it out, the temperature drops 3 degrees to reach zero and a further 6 degrees below zero. Answer: 9 °C. The distractors: −9 °C comes from subtracting the readings the wrong way round, as −6 − 3, and reporting a fall as a negative amount; 3 °C comes from ignoring the minus sign and working out 6 − 3; 6 °C comes from counting only the part of the fall that happens below zero and forgetting the 3 degrees above it.
6. (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.
7. (a) 4 : 15 — Convert to the same unit first: 1.5 kg = 1500 g, since 1 kg = 1000 g. This gives the ratio 400 : 1500. Divide both parts by their highest common factor, 100, to get 4 : 15. Giving 40 : 150 divides by 10 only, which is not the highest common factor, so it is not fully simplified. Giving 15 : 4 swaps the order. Giving 4 : 1.5 has not converted 1.5 kg into grams, so the two parts are not measured in the same unit.
8. (d) 19 — 2² = 4, then 4 × 4 = 16, then 3 + 16 = 19. Adding before multiplying gives 3 + 4 = 7, then 7 × 4 = 28 — multiplication comes before addition. Squaring the product instead of just the 2 gives 4 × 2 = 8, then 8² = 64, then 3 + 64 = 67. Working strictly left to right throughout gives 3 + 4 = 7, then 7 × 2 = 14, then 14² = 196.
9. (c) £80 — Method: a unit fraction acts as an operator, so finding 1/9 of a price means dividing that price by 9. Working: £720 ÷ 9 = £80. Answer: £80. The distractors: £6480 comes from multiplying by the denominator instead of dividing by it, giving £720 × 9 = £6480; £640 comes from working out what is left of the £720 once the case is paid for, £720 − £80, instead of the cost of the case itself; £72 comes from dividing by 10 instead of 9, treating one ninth as one tenth.
10. (b) 8:12 — Method: two ratios are equivalent when one is obtained from the other by multiplying, or dividing, both parts by the same number. Working: multiplying both parts of 2:3 by 4 gives 2 × 4 = 8 and 3 × 4 = 12, and the check runs the other way too, since the highest common factor of 8 and 12 is 4 and dividing both parts by 4 returns 2:3. Answer: 8:12. The distractors: 8:3 comes from multiplying only the first part by 4 and leaving the second part alone; 12:8 comes from multiplying both parts by 4 correctly but then writing the two parts the wrong way round; 4:5 comes from adding 2 to each part instead of multiplying, and adding the same amount to both parts changes the ratio.
11. (c) £70 — Method: 20% of an amount is 20/100 of it; a reliable route is to find 10% by dividing by 10 and then double it. Working: 10% of £350 is £350 ÷ 10 = £35, and 20% is twice as much, £35 × 2 = £70. Answer: £70. The distractors: £35 comes from finding 10% and stopping there; £17.50 comes from reading 20% as one twentieth and working out £350 ÷ 20 = £17.50; £280 comes from taking 20% off the money raised rather than finding 20% of it, giving £350 ÷ 5 = £70 and then £350 − £70 = £280.
12. (c) 36 — Add the parts of the ratio: 3 + 4 = 7. Divide the total by the number of parts: 84 ÷ 7 = 12, so one part is worth 12. The smaller share has 3 parts: 3 × 12 = 36. (48 is the larger share, using 4 parts instead of 3. 42 comes from splitting 84 in half, treating the ratio as if it were 1:1. 28 comes from dividing 84 by 3 — one of the ratio numbers — instead of dividing by the total number of parts, 7.)
13. (a) 4 — Method: a unit fraction acts as an operator, so finding 1/5 of an amount means dividing that amount by 5. Working: 20 ÷ 5 = 4, so Noah gives away 4 stickers. Answer: 4 stickers. The distractors: 100 comes from multiplying by the denominator instead of dividing by it, giving 20 × 5 = 100; 16 comes from working out how many stickers Noah keeps, the other four fifths of the pack, instead of how many he gives away; 5 comes from writing down the denominator, which is the number of equal groups the pack is split into rather than the size of one group.
14. (c) 693 — Method: multiplying a bracket by a number multiplies every term inside it, so 7 × (100 − 1) = 7 × 100 − 7 × 1. Working: 7 × 100 = 700 and 7 × 1 = 7, so the calculation becomes 700 − 7 = 693. Answer: 693. The distractors: 699 comes from subtracting the 1 itself rather than 7 lots of it, giving 700 − 1 = 699; 707 comes from adding the second product instead of subtracting it, giving 700 + 7 = 707; 700 comes from rounding 99 up to 100 and then offering the estimate 7 × 100 as an exact value.
15. (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.
16. (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.
17. (d) 2/25 — Method: write the decimal over 100 using its two decimal places, then simplify. Working: 0.08 = 8/100 = 2/25 (dividing both numerator and denominator by 4). Answer: 2/25. The student's fraction, 8/10, comes from ignoring the zero in the tenths column and reading 0.08 as though it were 0.8; it simplifies to 4/5. 1/125 comes from writing the decimal over 1000 instead of 100, as if there were three decimal places. 25/2 comes from flipping the correct fraction upside down.
18. (b) 80 — Method: the difference between 40% and 25% of the number is 15% of the number, and that difference is 12. Working: 15% of the number is 12, so 1% of the number is 12 ÷ 15 = 0.8, and the number is 0.8 × 100 = 80. Check: 40% of 80 is 32, 25% of 80 is 20, and 32 − 20 = 12. Answer: 80. The distractors: 30 comes from solving 40% of the number = 12; 48 comes from solving 25% of the number = 12; 15 is the percentage difference written as the answer.
19. (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.
20. (d) 3,200,000 — 1 million = 1,000,000, so 3.2 million = 3.2 × 1,000,000 = 3,200,000. A candidate who moves the decimal point one place too many gets 32,000,000. A candidate who moves it one place too few gets 320,000. A candidate who writes the .2 as extra thousands instead of hundred-thousands gets 3,002,000.