✏️ Paper 1 non-calculator warm-up — Foundation

Answer key

1. (a) 20% — Method: a percentage is a number of parts per hundred, so divide the numerator by the denominator and multiply by 100. Working: 1 ÷ 5 = 0.2, and 0.2 × 100 = 20; the same result comes from scaling 1/5 to 20/100. Answer: 20%. The distractors: 5% comes from reading the denominator 5 straight off as the percentage; 25% comes from confusing 1/5 with 1/4, the fraction met most often in percentage work; 50% comes from a place value slip in the division, writing 1 ÷ 5 as 0.5 instead of 0.2.
2. (c) £47.00 — First apply the 20% reduction: £65 × 0.8 = £52.00. Then take off the further £5: £52.00 − £5 = £47.00. Treating the 20% as a flat £20 rather than a percentage of the price, £65 − £20 − £5, gives £40.00. Applying the 20% reduction correctly but forgetting to take off the extra £5 leaves £52.00. Taking off the £5 first and then applying the 20% reduction to the smaller amount, (£65 − £5) × 0.8, gives £48.00.
3. (a) 18 minutes — Method: the buses leave together again after a number of minutes that is a multiple of both intervals, and the first such time is the lowest common multiple. Working: the multiples of 6 are 6, 12, 18, 24 … and the multiples of 9 are 9, 18, 27 … The first value in both lists is 18, which is 6 × 3 and 9 × 2. Answer: 18 minutes. The distractors: 54 minutes comes from multiplying 6 by 9, which does give a common multiple but not the lowest one; 3 minutes is the highest common factor of 6 and 9 rather than their lowest common multiple; 15 minutes comes from adding the two intervals together.
4. (b) 4:9 — 4 : 9 has no common factor other than 1, so it is already in its simplest form. 6 : 8 can be divided by 2 to give 3 : 4, so it is not simplest. 10 : 15 can be divided by 5 to give 2 : 3, so it is not simplest. 7 : 14 can be divided by 7 to give 1 : 2, so it is not simplest.
5. (b) 31 — Method: a prime number has exactly two factors, 1 and itself, so check each number between 30 and 40 for other factors. Working: 3 × 11 = 33, so 33 is not prime. 2 × 17 = 34, so 34 is not prime. 4 × 9 = 36, so 36 is not prime. 31 has no factors other than 1 and 31, so it is prime. Answer: 31.
6. (d) 30 — Method: 50% is one half, so 50% of a quantity is the quantity divided by 2. Working: 60 ÷ 2 = 30. Answer: 30. The distractors: 120 comes from multiplying by 2 instead of dividing; 3000 comes from multiplying by 50 without dividing by 100; 6 comes from finding 10% instead of 50%.
7. (d) 60 — Method: a percentage acts as an operator, so finding 20% of an amount means multiplying it by 20/100, which cancels to 1/5. Working: 20% = 20/100 = 1/5, and 300 ÷ 5 = 60. Answer: 60 seats. The distractors: 15 comes from reading 20% as one twentieth and working out 300 ÷ 20 = 15; 30 comes from finding 10% of 300 and stopping there instead of doubling it; 6 comes from converting 20% to 0.02 rather than 0.2, giving 0.02 × 300 = 6.
8. (d) 7:15 — Method: write the two parts in the order asked for, red first, then divide both parts by their highest common factor. Working: the factors of 7 are 1 and 7, and the factors of 15 are 1, 3, 5 and 15, so the only common factor is 1; dividing both parts by 1 leaves both counts unchanged. Answer: 7:15, which is already in its simplest form. The distractors: 15:7 is the ratio of blue to red, the reverse of the order the question asks for; 7:5 comes from cancelling the digit 1 out of 15, which is not a division by a common factor; 1:2 comes from dividing 15 by 7, rounding the result to 2 and writing the ratio as 1 to 2, but 7 is not a factor of 15 so that division is not exact.
9. (c) 150% — Percentage = (180,000 ÷ 120,000) × 100 = 150%.
10. (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.
11. (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.
12. (d) 600 — Method: the number of seats is the number of rows multiplied by the number of seats in each row, so round each number to 1 significant figure and then multiply the rounded values, which is quick because a product of two multiples of ten is found by multiplying the non-zero digits and attaching the zeros. Working: 21 rounds to 20 and 29 rounds to 30; 2 × 3 = 6, and 20 and 30 carry one zero each, so two zeros follow the 6. Answer: about 600 seats. The distractors: 50 comes from adding the two rounded numbers instead of multiplying them, 20 + 30; 60 comes from multiplying 20 by the 3 of 30 and forgetting the zero in 30; 6,000 comes from attaching three zeros to 2 × 3 when 20 and 30 provide only two between them.
13. (a) 8:1 — Multiply both parts of the ratio by 4 to clear the fraction: 2 × 4 = 8 and 1/4 × 4 = 1, giving 8 : 1. Getting 1 : 8 has the two parts the wrong way round. Getting 2 : 4 comes from writing down the denominator of the fraction (4) as the second part instead of multiplying through by it. Getting 8 : 4 comes from multiplying only the first part of the ratio by 4 and leaving the second part as the fraction's denominator.
14. (a) 3:2 — There are 24 green sweets and 16 orange sweets. The highest common factor of 24 and 16 is 8. Divide both numbers by 8: 24 ÷ 8 = 3 and 16 ÷ 8 = 2, so the ratio is 3 : 2. Dividing by 4 instead of 8 gives 6 : 4, which still has a common factor of 2, so it is not fully simplified. Writing green sweets to the total number of sweets, 24 : 40, simplifies to 3 : 5 — that compares green to everything, not green to orange, so it answers a different question. Swapping the order gives 2 : 3, green and orange the wrong way round.
15. (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.
16. (d) 3 : 7 — There are 9 apples and 21 oranges, giving the ratio 9 : 21. The highest common factor of 9 and 21 is 3: 9 ÷ 3 = 3 and 21 ÷ 3 = 7, so the simplest form is 3 : 7. Giving 9 : 21 has not been simplified. Giving 7 : 3 swaps the order, comparing oranges to apples instead of apples to oranges. Giving 9 : 12 compares the number of apples with the difference between the two amounts (21 − 9 = 12), not with the number of oranges.
17. (a) 0.35, 3/8, 0.4 — Method: convert the fraction to a decimal so all three values can be compared directly. Working: 3/8 = 0.375, so the three values are 0.35, 0.375 and 0.4. In order from smallest to largest: 0.35, 3/8 (0.375), 0.4. Answer: 0.35, 3/8, 0.4. "3/8, 0.35, 0.4" comes from assuming a fraction must be smaller than any decimal, without converting it first. "0.4, 3/8, 0.35" comes from writing the numbers in reverse order. "0.35, 0.4, 3/8" comes from converting 3/8 by flipping it to 8/3, making it seem larger than both decimals.
18. (b) 3/50 — Method: write the decimal over the power of ten that matches the number of digits after the point, counting every digit including a zero, then divide the numerator and the denominator by their highest common factor. Working: 0.06 has two digits after the point, so it is 6 hundredths and can be written as 6/100; the highest common factor of 6 and 100 is 2, and 6 ÷ 2 = 3 with 100 ÷ 2 = 50. Answer: 3/50. The distractors: 3/5 comes from ignoring the zero straight after the point and converting 0.6 instead, giving 6/10, which cancels to 3/5; 3/500 comes from counting three decimal places instead of two and writing 6/1000, which cancels to 3/500; 1/6 comes from putting 1 over the digits after the point, as though 0.06 meant one sixth.
19. (c) 1.05 litres — Method: find the value of one part of the ratio from the total volume, then find the share for pineapple juice. Working: the ratio 5:3:2 has 5 + 3 + 2 = 10 parts, so one part is 3.5 ÷ 10 = 0.35 litres, and the pineapple juice is 3 × 0.35 = 1.05 litres. So 1.05 litres of pineapple juice is needed. Distractor 1.75 litres is the volume of orange juice, not pineapple juice. Distractor 0.7 litres is the volume of lemonade, not pineapple juice. Distractor 0.35 litres is the value of one part, found correctly but never multiplied by 3.
20. (a) 100 — The difference between the parts of the ratio is 7 − 3 = 4 parts, and this is worth 40 beads. Divide to find one part: 40 ÷ 4 = 10. The total number of parts is 7 + 3 = 10, so the total number of beads is 10 × 10 = 100. (40 is just the given difference between gold and silver, not the total. 70 is the number of gold beads only, using 7 parts. 30 is the number of silver beads only, using 3 parts.)