⏱️ Ten-minute starters — Foundation

Answer key

1. (b) −1% — Method: write each change as a multiplier and multiply them. A 10% fall is × 0.9 and a 10% rise is × 1.1. Working: 0.9 × 1.1 = 0.99, so the final price is 99% of the original, which is 1% less. Answer: an overall change of −1%. The distractors: 0% comes from assuming a 10% fall and a 10% rise cancel — they do not, because the rise is 10% of a smaller amount; +1% has the size right but the sign wrong, from reading the multiplier 0.99 as 1% above 1 instead of 1% below it; −2% comes from finding the 1% fall and then counting it once for each of the two changes.
2. (d) 1 — Method: find the step in the outputs for each step of 1 in the input, write the rule from that step and from one pair of values, then apply the rule to the last input. Working: the outputs −5, −3, −1 rise by 2 while x rises in ones, so x is multiplied by 2; at x = 1, 2 × 1 = 2 while y = −5, so 7 is subtracted, giving y = 2x − 7; at x = 4 the rule gives 2 × 4 = 8 and 8 − 7 = 1. Answer: y = 1. The distractors: 3 comes from carrying the outputs on one step too far, to x = 5; 0 comes from assuming the outputs −5, −3, −1 carry on by adding 1 rather than by adding 2; 8 comes from doubling the input and forgetting to subtract the 7.
3. (b) 11/12 — Method: raining and not raining are the only two outcomes, so their probabilities add to 1; subtract the given probability from 1. Working: writing 1 as 12/12 gives 12/12 − 1/12, and only the numerators are subtracted, 12 − 1 = 11. Answer: 11/12, close to the right-hand end of the 0 to 1 scale because rain is unlikely. The distractors: 1/12 comes from giving back the probability that it does rain; 1/11 comes from subtracting the 1 from the denominator instead of subtracting the fraction from 1; 11/11 comes from subtracting 1 from the numerator and from the denominator of 12/12 rather than from the numerator alone.
4. (d) 450 g — Find the amount of rice per person: 300 ÷ 4 = 75 g. Multiply by the new number of people: 75 × 6 = 450 g. Giving 200 g swaps which number the rice is divided and multiplied by (300 ÷ 6 × 4 = 200), scaling the wrong way. Giving 180 g uses 4 + 6 = 10 as the base instead of the original 4 people (300 × 6 ÷ 10 = 180). Giving 400 g assumes each of the 2 extra people needs 300 ÷ 6 = 50 g on top of the original 300 g (300 + 2 × 50 = 400), rather than scaling the whole amount in proportion.
5. (c) −5 ≤ −5 — The symbol ≤ means 'less than or equal to', and −5 is equal to −5, so this statement is true. −3 ≥ −1 is false: a candidate who ignores the negative signs and compares 3 with 1 would wrongly think −3 is the bigger number, but on the number line −3 is smaller than −1. 0.4 < 2/5 is false because 2/5 converts to exactly 0.4, so the two values are equal, not one strictly less than the other — a candidate who assumes a fraction is automatically bigger than a similar-looking decimal without converting it would miss this. 7/10 ≤ 0.6 is false because 7/10 converts to 0.7, which is bigger than 0.6; a candidate who misplaces the decimal point and converts 7/10 as 0.07 would wrongly believe this statement is true.
6. (a) Leeds has a higher median and a greater range than York. — In order, Leeds's temperatures are 14, 16, 18, 19 and 23, so the median is the middle value, 18, and the range is 23 − 14 = 9. York's temperatures in order are 15, 17, 17, 18 and 18, so the median is 17, and the range is 18 − 15 = 3. Since 18 is higher than 17, and 9 is greater than 3, Leeds has both the higher median and the greater range. Choosing 'Leeds has a higher median but a smaller range than York' gets the median comparison right but the range comparison backwards — Leeds's range of 9 is actually greater than York's range of 3. Choosing 'York has a higher median and a greater range than Leeds' reverses both comparisons. Choosing 'York has a higher median but a smaller range than Leeds' reverses the median comparison; York's median of 17 is lower than Leeds's 18, even though it is correct that York's range is the smaller one.
7. (a) 36 km/h — Speed is distance ÷ time. Convert 25 minutes to hours: 25 ÷ 60 = 5/12 hours. Then 15 ÷ (5/12) = 15 × 12/5 = 36 km/h. Dividing 15 by 25 without converting minutes to hours gives 0.6 km/h. Multiplying 15 by 25/60 instead of dividing by it gives 6.25 km/h. Working out the speed in km per minute, 0.6, and then multiplying by 24 instead of 60 gives 14.4 km/h.
8. (a) 4:3:1 — Convert every part to the same unit: 2 m = 200 cm, so the ratio is 200 : 150 : 50. Dividing all three parts by 50 gives 4 : 3 : 1. Writing 2 : 150 : 50 has not converted 2 m into centimetres, so the units do not match. Writing 3 : 4 : 1 has the first two parts the wrong way round. Writing 4 : 3 : 2 comes from an arithmetic slip on the last part: 50 ÷ 50 = 1, not 2.
9. (b) 25 — Method: check the calculator answer by following the order of operations — each power is worked out before the addition. Working: 4² = 4 × 4 = 16 and 3² = 3 × 3 = 9, and 16 + 9 = 25. Answer: 25. The distractors: 49 is the value Freya wrote down and comes from adding first and then squaring, working out (4 + 3)² instead of 4² + 3²; 14 comes from doubling each number instead of squaring it, adding 8 and 6; 12 comes from multiplying 4 by 3 instead of squaring each number and adding the results.
10. (a) SSS – all three corresponding sides are equal — All three pairs of corresponding sides are stated as equal — LM = XY, MN = YZ and LN = XZ — with no angle mentioned. This matches the SSS condition, so triangle LMN is congruent to triangle XYZ.
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. (b) 12 years — Method: call the number of years t, add t to both ages, and form an equation from the comparison at that future time. Working: in t years the leader will be 44 + t and the scout will be 16 + t, so 44 + t = 2(16 + t); expanding gives 44 + t = 32 + 2t, and subtracting t and 32 from both sides gives t = 12. Checking: in 12 years the leader will be 56 and the scout 28, and 56 = 2 × 28. Answer: 12 years. The distractors: 6 years comes from halving the leader's present age instead, so that the scout has to reach 22, which takes 6 years; 14 years comes from halving the 28-year gap between the two ages; 28 years comes from giving the age gap itself as the number of years.