Printable · GCSE Foundation · ages 14-16
Number worksheet — GCSE Foundation
Fifteen questions across the number statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
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Answer key: Number worksheet — GCSE Foundation
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- (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.
- (a) 3/5 — The question compares the rock albums directly with the pop albums, so write the rock part over the pop part: 3/5. 5/3 comes from writing the ratio the wrong way round, comparing pop with rock instead of rock with pop. 3/8 comes from comparing the rock albums with the total number of albums, 3 + 5 = 8, instead of with the pop albums. 5/8 comes from comparing the pop albums with the total number of albums instead of the rock albums with the pop albums.
- (c) −2 — On a number line, negative numbers get smaller as their size (ignoring the sign) gets bigger, so −2 is closest to zero and is the largest of the four. A candidate who ignores the negative signs and orders the numbers as if they were positive, largest digit first, would pick −9 as the 'largest'. Continuing that same reversed ranking, the next number in digit order is −7. The number one step further along that same reversed ranking is −4, still short of the true largest value, −2.
- (a) 4/9 — Method: write the number of parts for the photos being compared over the number of parts for the photos they are being compared to. Working: black-and-white photos are 4 parts and colour photos are 9 parts, so black-and-white photos as a fraction of colour photos is 4/9. Answer: 4/9. 9/4 comes from writing the ratio the wrong way round, comparing colour to black-and-white instead of black-and-white to colour. 4/13 comes from comparing black-and-white photos to the total number of photos (4+9=13) instead of to the colour photos. 9/13 comes from comparing colour photos to the total number of photos instead of black-and-white to colour.
- (a) 40 — Multiply the number of choices for each item: 4 × 5 × 2 = 40. 11 comes from adding the three numbers instead of multiplying them. 20 comes from multiplying only the crisps and chocolate bars, 4 × 5, and forgetting the drink. 10 comes from multiplying only the chocolate bars and drinks, 5 × 2, and forgetting the crisps.
- (d) 40 miles — Method: round each number to 1 significant figure first, then divide to estimate the daily distance. Working: 830 rounds to 800, and 19 rounds to 20, and 800 ÷ 20 = 40, so the estimate is 40 miles per day. 41.5 miles comes from rounding only the number of days and working out 830 ÷ 20 = 41.5, without rounding the distance too. 830 miles is the total distance for the whole trek, given as the answer without dividing by the number of days at all. 4 miles comes from working out 80 ÷ 20 = 4, misplacing a digit in the rounded distance. Answer: 40 miles.
- (a) −22 — Method: both multiplications are carried out before the addition, and a positive multiplied by a negative is negative. Working: (−3) × 4 = −12 and 2 × (−5) = −10, so the calculation becomes −12 + (−10) = −22. Answer: −22. The distractors: 22 comes from ignoring the minus signs and working out 3 × 4 + 2 × 5 = 22; 50 comes from working from left to right with no priority at all, giving −12 + 2 = −10 and then −10 × (−5) = 50; −2 comes from taking 2 × (−5) as +10, so that −12 + 10 = −2.
- (a) 360 — The inverse of ÷ 15 is × 15, so the missing number is 24 × 15 = 360. Subtracting instead of multiplying gives 24 − 15 = 9. Dividing by 15 again instead of multiplying gives 24 ÷ 15 = 1.6. Adding instead of multiplying gives 24 + 15 = 39.
- (b) 4 m — Blue fabric is 3 parts and this equals 1.5 m, so one part is 1.5 ÷ 3 = 0.5 m. Yellow fabric is 5 parts, so it is 5 × 0.5 = 2.5 m. The total length is 1.5 + 2.5 = 4 m. 2.5 m is the length of yellow fabric only, without adding the blue fabric back in. 1.5 m is just the given length of blue fabric, with the yellow fabric never worked out. 2.4 m comes from swapping the ratio, treating blue as 5 parts and yellow as 3 parts, giving one part as 1.5 ÷ 5 = 0.3 m and yellow as 3 × 0.3 = 0.9 m, then adding 1.5 + 0.9 = 2.4.
- (a) 300 — Method: round each number to 1 significant figure, then divide the rounded values. Working: 8,900 rounds to 9,000 and 29 rounds to 30; cancelling a zero from each gives 900 ÷ 3. Answer: 300. The distractors: 450 comes from rounding 29 down to 20 instead of to the nearest ten, giving 9,000 ÷ 20; 3,000 comes from rounding 29 to 3 rather than to 30, a place-value slip that divides by a number ten times too small; 307 is the exact quotient rounded to the nearest whole number, worked out in full when the question asks for an estimate.
- (a) 1/5 — Work out the bracket first: 2 + 3 = 5. The reciprocal of 5 is 1/5. A candidate who forgot to take the reciprocal and just gave the value of the bracket wrote 5. A candidate who took the reciprocal but made a sign error wrote −1/5. A candidate who found the reciprocal of each number separately and added them, treating reciprocal as if it distributes over addition, worked out 1/2 + 1/3 = 5/6.
- (c) £7.05 — Tom is correct — Method: multiply the cost of one jar by the number of jars, then compare the total to £7. Working: 3 × £2.35 = £7.05, and since £7.05 is more than £7, Tom's claim is correct. Answer: £7.05 — Tom is correct. "£7.05 — Tom is incorrect" comes from reaching the right total but reading the 05 after the point as making the amount less than £7. "£6.95 — Tom is incorrect" comes from working out 3 × 35p as 3 × 30p + 5p = 95p, multiplying only the tens digit, and adding it to 3 × £2 = £6. "£7.20 — Tom is correct" comes from rounding £2.35 up to £2.40 before multiplying: 3 × £2.40 = £7.20.
- (a) 1/16 — Method: terms can only be subtracted once they share a denominator, so write every term over the largest denominator, 16, and then subtract the numerators in order from left to right. Working: 1 = 16/16, 1/2 = 8/16, 1/4 = 4/16 and 1/8 = 2/16, so the numerators give 16 − 8 − 4 − 2 − 1 = 1, over a denominator of 16. Answer: 1/16. The distractors: 1/8 comes from stopping one term early, after 16 − 8 − 4 − 2 = 2; 3/16 comes from a sign slip on the last term, adding it instead of subtracting it, which gives 2 + 1 = 3; 15/16 comes from working from the right-hand end as though the last four terms were bracketed together, so that only a single sixteenth is taken away from 1.
- (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.
- (d) 1.65 litres — Method: convert both volumes to the same unit, then add. Working: 400 ml = 400 ÷ 1000 = 0.4 litres. Total = 1.25 + 0.4 = 1.65 litres. Answer: 1.65 litres. (401.25 litres comes from adding 1.25 and 400 directly without converting millilitres to litres first. 0.85 litres comes from subtracting 0.4 from 1.25 instead of adding. 5.25 litres comes from dividing 400 by 100 instead of by 1000 — using the centimetre-to-metre factor — which turns 400 ml into 4 litres before adding.)
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