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.
Answer key: Number worksheet — GCSE Foundation
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- (b) 0 > −1 — Method: test each statement by placing both of its numbers on a number line; the greater number is the one further to the right. Working: every negative number lies to the left of zero, so zero is greater than −1. Among the negatives, −3 lies to the right of −5, so −5 is not greater than −3. Answer: 0 > −1. The distractors: −5 > −3 comes from ordering the negatives by the size of their digits, so that 5 makes −5 look the larger; −3 > 0 comes from ignoring the minus sign and comparing 3 with 0; 0 < −1 comes from the belief that zero is the smallest number there is, so that even a negative number is above it.
- (a) 4.5 × 10⁴ — 45,000 = 4.5 × 10,000 = 4.5 × 10⁴, with the coefficient between 1 and 10 as standard form requires. Writing 45 × 10³ keeps the coefficient too large — 45 is not between 1 and 10. Writing 4.5 × 10³ undercounts the places moved, giving only 4,500. Writing 4.5 × 10⁵ overcounts the places moved, giving 450,000.
- (b) 750,000,000 — Method: multiplying by 10⁸ moves the decimal point eight places to the right, and every empty place is filled with a zero. Working: 10⁸ = 100,000,000, and moving the decimal point in 7.5 eight places to the right gives 7.5 × 100,000,000. Answer: 750,000,000. The distractors: 7,500,000,000 comes from removing the decimal point first to make 75 and then writing eight zeros after it, which carries the digits one place too far; 600 comes from reading 10⁸ as 10 × 8 = 80 and working out 7.5 × 80; 0.000000075 comes from moving the decimal point eight places to the left, as though the index were negative.
- (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.
- (b) 3/7 — Total parts = 3 + 4 = 7. Boys are 3 of the 7 parts, so the fraction is 3/7. 4/7 comes from finding the fraction of girls instead of boys. 3/4 comes from writing the ratio itself as a fraction, without adding the parts to find the total. 7/3 comes from putting the total number of parts over the number of boys instead of the number of boys over the total.
- (c) 6 — Method: the first significant figure is the first non-zero digit; round using the digit after it to decide whether to round up or down. Working: the first significant figure of 6.283 is the 6; the next digit is 2, which rounds down, so 6.283 rounds to 6. 6.3 comes from rounding to 2 significant figures instead of 1. 10 comes from rounding up to the nearest 10 instead of finding 1 significant figure of the number itself. 0.6 comes from misplacing the decimal point after rounding. Answer: 6.
- (b) £8 — Rounding to 1 significant figure: £1.85 rounds to £2, and 3.6 kg rounds to 4 kg. The estimate is £2 × 4 = £8. A candidate who used the unrounded values instead of estimating worked out 1.85 × 3.6 = £6.66. A candidate who rounded only the mass and used the exact price worked out 1.85 × 4 = £7.40. A candidate who rounded the price to the nearest 10p instead of 1 significant figure worked out 1.9 × 4 = £7.60.
- (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.
- (b) 8π cm² — A diameter of 8 cm gives a radius of 4 cm. The area of a full circle would be π × r² = π × 4² = 16π cm², and a semicircle is exactly half of this, giving 16π ÷ 2 = 8π cm². Forgetting to halve the area for the semicircle gives 16π cm², the area of the whole circle. Halving the diameter twice, using a radius of 2 instead of 4, gives π × 2² = 4π cm². Using the diameter itself as the radius, so π × 8² = 64π, and then halving that for the semicircle gives 32π cm².
- (b) £220 — Method: round each ticket price to the nearest £5, multiply each rounded price by the number of tickets, then add the two totals. Working: the adult price £38.50 rounds to £40, and 4 × £40 = £160; the child price £19.75 rounds to £20, and 3 × £20 = £60; £160 + £60 = £220. Answer: £220. £213.25 is the exact total cost, found without rounding the prices first, so it is not an estimate. £160 comes from including the cost of the adult tickets only and forgetting the three children's tickets. £200 comes from swapping the two ticket quantities, using 3 adults and 4 children instead of 4 adults and 3 children.
- (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.
- (d) 7/20 — Method: write the decimal over the power of ten that matches the number of digits after the point, then divide the numerator and the denominator by their highest common factor. Working: 0.35 has two digits after the point, so it is 35 hundredths and can be written as 35/100; the highest common factor of 35 and 100 is 5, and 35 ÷ 5 = 7 with 100 ÷ 5 = 20. Answer: 7/20. The distractors: 3/10 comes from reading only the first digit after the point and converting 0.3; 7/25 comes from dividing the numerator by 5 but the denominator by 4, using a different factor on the top and on the bottom; 35/10 comes from counting one decimal place instead of two and writing the digits over 10.
- (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.
- (a) 20 — Method: round each number to 1 significant figure, then divide. Working: 397 rounds to 400 and 21 rounds to 20, so the estimate is 400 ÷ 20 = 20. Answer: 20. The distractors: 19 comes from not estimating at all, since 397 ÷ 21 = 18.9 to 1 decimal place, which rounds to 19, while the question asks for an estimate from rounded numbers; 200 comes from dividing 400 by 2 instead of by 20, losing the place value of the rounded divisor; 2 comes from dividing the leading digits only and losing the place value of both numbers.
- (c) 43 — Method: work out each power separately before adding. Working: 3³ = 27 and 2⁴ = 16, so 3³ + 2⁴ = 27 + 16 = 43. Answer: 43. (25 comes from using 3² instead of 3³, giving 9 + 16. 35 comes from working out 2⁴ as 2 × 4 = 8 instead of 2 × 2 × 2 × 2, giving 27 + 8. 432 comes from multiplying the two powers together instead of adding them.)
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