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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Number worksheet — GCSE Foundation
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- 1.A newspaper reports that 48,000 people attended a festival. The figure is correct to the nearest thousand. Which statement about the number of people who attended must be true?
- 2.Work out √49
- 3.A garden has flowers, herbs and vegetables in the ratio 2 : 3 : 4. What fraction of the garden is not vegetables?
- 4.A circle has a diameter of 10 cm. Work out the exact circumference of the circle, in terms of π.
- 5.A pair of trainers is priced at £80 online. The shop takes 25% off the price and then adds £10 for next-day delivery. Work out the total cost.
- 6.Work out 5⁰ + 5¹ + 5²
- 7.Work out (−6) + (−4) × 3
- 8.Work out 4368 ÷ 12.
- 9.Work out (−2)² − 3
- 10.Insert one pair of brackets into 2 + 3 × 5 − 1 so that the calculation is equal to 24. Which calculation is correct?
- 11.Write 0.325 as a fraction in its simplest form.
- 12.An allotment is divided into two plots in the ratio 2:3. The larger plot has an area of 18 m². Work out the fraction of the total area taken up by the smaller plot.
- 13.Work out the highest common factor of 12 and 18.
- 14.A rectangular field measures 19.6 m by 48.3 m. Work out an estimate for the area of the field, by rounding each length to 1 significant figure.
- 15.Work out ∛(−27) + ∛8
Answer key
- (a) Fewer than 48,500 people attended — Method: a figure given to the nearest thousand lies within half of 1,000, that is 500, of the figure printed, so the attendance is at least 47,500 and below 48,500. Working: 48,000 − 500 = 47,500 and 48,000 + 500 = 48,500, and an attendance of 48,500 would have been reported as 49,000, so every possible attendance is below 48,500. Answer: Fewer than 48,500 people attended. The distractors: more than 48,500 turns the upper limit of the range into a minimum; fewer than 47,500 uses the lower limit as though it were the upper one; more than 48,000 assumes the printed figure was rounded down, when it could just as well have been rounded up from a smaller attendance.
- (c) 7 — Method: a square root asks which positive number multiplied by itself gives the number under the root sign, so work up through the square numbers until one of them is 49. Working: 5 × 5 = 25, 6 × 6 = 36 and 7 × 7 = 49. Answer: 7. The distractors: 9 comes from recalling the wrong square fact and pairing 49 with 9, when 9 × 9 = 81; 24.5 comes from treating a square root as a halving and working out 49 ÷ 2; 2401 comes from squaring 49 instead of square-rooting it, applying the inverse operation the wrong way round.
- (d) 5/9 — Total parts = 2 + 3 + 4 = 9. Flowers and herbs together make up 2 + 3 = 5 parts, so the fraction that is not vegetables is 5/9. 4/9 comes from finding the fraction of vegetables instead of the fraction that is not vegetables. 5/7 comes from leaving flowers out of the total and using only herbs and vegetables, 3 + 4 = 7, as the total. 3/4 comes from writing the ratio of herbs to vegetables directly as a fraction.
- (d) 10π cm — Method: for a circle, circumference = π × diameter. Working: circumference = π × 10 = 10π cm. Answer: 10π cm. 20π cm comes from using the formula 2 × π × radius but plugging in the diameter as if it were the radius: 2 × π × 10 = 20π. 25π cm comes from confusing circumference with area, using π × radius² with radius 5: π × 5² = 25π. 5π cm comes from halving the diameter and then multiplying by π, as if circumference were π × radius: π × 5 = 5π.
- (d) £70 — Method: the discount is a percentage of the original price only, so work it out, subtract it, and add the fixed delivery charge afterwards. Working: 25% of £80 is £80 ÷ 4 = £20, so the discounted price is £80 − £20 = £60, and the total is £60 + £10 = £70. Answer: £70. The distractors: £60 comes from working out the discounted price and stopping there, leaving the delivery charge out of the total; £65 comes from taking £25 off the price instead of 25% of it, giving £80 − £25 = £55 and then £55 + £10 = £65; £67.50 comes from adding the delivery charge before the discount and reducing the whole amount, giving 75% of £90 = £67.50.
- (c) 31 — Method: work out each power separately, remembering that any non-zero base raised to the power 0 is 1 and a base raised to the power 1 is itself, then add the three values. Working: 5⁰ = 1, 5¹ = 5 and 5² = 25, so the total is 1 + 5 + 25 = 31. Answer: 31. The distractors: 30 comes from taking 5⁰ as 0 instead of 1; 35 comes from taking 5⁰ as 5, treating a zero index as leaving the base unchanged; 125 comes from adding the indices first, as though the three terms were being multiplied, and working out 5³.
- (b) −18 — Method: the multiplication is carried out before the addition, and a negative multiplied by a positive is negative. Working: (−4) × 3 = −12, so the calculation becomes (−6) + (−12) = −18. Answer: −18. The distractors: −30 comes from adding first and multiplying afterwards, giving (−6 + −4) × 3 = −10 × 3 = −30; 6 comes from treating (−4) × 3 as +12 on the grounds that a minus sign makes a product positive, giving −6 + 12 = 6; 18 comes from ignoring both minus signs and working out 6 + 4 × 3 = 18.
- (b) 364 — Divide in stages using multiples of 12. 12 × 300 = 3600, leaving a remainder of 4368 − 3600 = 768. Then 12 × 64 = 768, so 4368 ÷ 12 = 300 + 64 = 364. Placing the decimal point as though dividing 436.8 by 12 gives 36.4. Transposing the last two digits of 364 gives 346. Working out 768 ÷ 12 as 4 instead of 64, losing the tens digit, and adding 300 + 4 gives 304. So 4368 ÷ 12 = 364.
- (b) 1 — Method: BIDMAS deals with the index before the subtraction, and a negative number multiplied by itself gives a positive result. Working: (−2)² = (−2) × (−2) = 4, so the calculation becomes 4 − 3 = 1. Answer: 1. The distractors: −7 comes from squaring only the 2 and leaving the minus sign outside the index, giving −(2²) − 3 = −4 − 3 = −7; −1 comes from subtracting the square from 3 instead of 3 from the square, giving 3 − 4 = −1; 25 comes from carrying out the subtraction before the index, giving (−2 − 3)² = (−5)² = 25.
- (b) (2 + 3) × 5 − 1 — 2 + 3 = 5, then 5 × 5 = 25, then 25 − 1 = 24, so the brackets belong around 2 + 3. Placing them around 5 − 1 instead gives 5 − 1 = 4, then 3 × 4 = 12, then 2 + 12 = 14. Leaving the multiplication bracketed instead changes nothing, because it already had priority: 3 × 5 = 15, then 2 + 15 = 17, then 17 − 1 = 16. Bracketing both 2 + 3 and 5 − 1 uses two pairs instead of the one asked for: 2 + 3 = 5, 5 − 1 = 4, then 5 × 4 = 20.
- (a) 13/40 — Method: write the decimal over 1000 using its three decimal places, then simplify. Working: 0.325 = 325/1000 = 13/40 (dividing both numerator and denominator by 25). Answer: 13/40. 13/4 comes from writing the decimal over 100 instead of 1000, as if there were only two decimal places. 8/25 comes from rounding 0.325 down to 0.32 before converting. 3/8 comes from recalling the learned conversion 3/8 = 0.375 and matching it to 0.325 because both are three-place decimals beginning with 3, instead of converting the decimal given.
- (a) 2/5 — The ratio 2:3 has 2 + 3 = 5 parts in total, and the larger plot is 3 of those parts. Since the larger plot is 18 m², each part is 18 ÷ 3 = 6 m², so the total area is 5 × 6 = 30 m² and the smaller plot is 2 × 6 = 12 m². The fraction of the total area taken up by the smaller plot is 12/30, which simplifies to 2/5. Giving the fraction for the larger plot instead of the smaller one gives 3/5. Comparing the smaller plot to the larger plot instead of to the total area gives 2/3. Assuming the two plots split the area evenly, ignoring the given ratio altogether, gives 1/2.
- (b) 6 — List the factors of each number: the factors of 12 are 1, 2, 3, 4, 6 and 12; the factors of 18 are 1, 2, 3, 6, 9 and 18. The common factors are 1, 2, 3 and 6, and the highest of these is 6. Picking 2, a common factor but not the largest, gives an answer that is too small. Picking 3, also a common factor but still not the largest, gives another answer that is too small. Working out the lowest common multiple instead of the highest common factor gives 36. So the highest common factor of 12 and 18 is 6.
- (c) 1,000 m² — Method: round each length to 1 significant figure, then use area of a rectangle = length × width on the rounded lengths. Working: 19.6 m rounds to 20 m and 48.3 m rounds to 50 m, so the estimate is 20 × 50 = 1,000 and the area is about 1,000 m². Answer: 1,000 m². The distractors: 800 m² comes from rounding 48.3 down to 40 when the digit after its first significant figure is 8 and sends it up to 50, giving 20 × 40 = 800; 140 m² is the perimeter of the rounded rectangle, 2 × 20 + 2 × 50 = 140, not its area; 70 m² comes from adding the rounded lengths, 20 + 50 = 70, instead of multiplying them.
- (c) −1 — Method: find each cube root separately, keeping its sign, and then add the two results. Working: (−3) × (−3) × (−3) = −27, so ∛(−27) = −3, and 2 × 2 × 2 = 8, so ∛8 = 2. Adding gives −3 + 2 = −1. Answer: −1. The distractors: 5 comes from taking the cube root of a negative number as positive, giving 3 + 2; −5 comes from reading the minus sign as applying to the whole sum and working out −(3 + 2); −6 comes from multiplying the two roots, −3 × 2, instead of adding them.
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