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.Which of these numbers lies between −8 and −5 on a number line?
- 2.In a school, 3/5 of the students study French. Of these students, 2/3 also study Spanish. What fraction of all the students study both French and Spanish?
- 3.A coach journey has a 55 minute first leg, a 20 minute break and a 40 minute second leg. The coach leaves at 13:25. Work out the arrival time, using the 24-hour clock.
- 4.A recipe for one cake needs 2/3 of a cup of sugar. Priya has 3 1/2 cups of sugar. Work out how many complete cakes she can make.
- 5.A carpenter has a plank of wood 4.8 m long. She cuts off 3 pieces, each 0.9 m long, to make shelves. Work out the length of wood remaining.
- 6.In a class, 1/3 of the pupils are girls. There are 12 girls in the class. Work out how many pupils are in the class.
- 7.Work out 15% of £40, using 10% and 5%.
- 8.Write these numbers in order, starting with the largest: 7/8, 0.8, 78%, 17/20
- 9.The distance from the Earth to the Moon is 384,000 km. Write this distance in standard form, in kilometres.
- 10.Work out 1 − 1/2 − 1/4 − 1/8 − 1/16. Give your answer as a fraction.
- 11.Work out 100 − 4 × 5²
- 12.A semicircle has a diameter of 8 cm. Work out the exact area of the semicircle, in terms of π.
- 13.Ava measures a metal rod and records its length as 15 cm, correct to the nearest centimetre. Ben measures the same rod and records its length as 14.8 cm, correct to the nearest 0.1 cm. Decide whether both records can be correct, and give a reason for your answer.
- 14.Two fair six-sided dice are rolled and their scores are added together. By listing the possible totals systematically, work out how many different totals are possible.
- 15.The population of a planet is 8 × 10⁹. Write this population as an ordinary number.
Answer key
- (d) −6 — Method: check each option to see if it falls strictly between −8 and −5, remembering that numbers become smaller as they get more negative. Working: −8 < −6 < −5, so −6 lies between them. Answer: −6. −9 comes from a value that is more negative than −8, so it lies outside the range, below −8. −4 comes from a value that is less negative than −5, so it lies outside the range, above −5. 6 comes from ignoring the negative signs and picking a positive number between 5 and 8.
- (c) 2/5 — To find a fraction of a fraction, multiply them together: 3/5 × 2/3 = 6/15, which simplifies to 2/5. Multiplying only the numerators, 3 × 2 = 6, but adding the denominators, 5 + 3 = 8, instead of multiplying them gives 6/8, which simplifies to 3/4. Using only the fraction who study French, 3/5, and ignoring that a further fraction of them also study Spanish gives 3/5. Dividing by 2/3 instead of multiplying by it, using its reciprocal 3/2, gives 3/5 × 3/2 = 9/10.
- (d) 15:20 — Method: add all three journey segments to the departure time, converting the total number of minutes into hours and minutes. Working: 55 + 20 + 40 = 115 minutes = 1 hour 55 minutes. 13:25 + 1 hour 55 minutes = 15:20. Answer: 15:20. (15:00 comes from leaving out the 20 minute break and adding only 55 + 40 = 95 minutes. 14:25 comes from leaving out the 55 minute first leg and adding only 20 + 40 = 60 minutes. 14:40 comes from leaving out the 40 minute second leg and adding only 55 + 20 = 75 minutes.)
- (b) 5 — Method: divide the total amount of sugar by the amount needed for one cake, then round down because a part-used amount of sugar cannot make an extra whole cake. Working: 3 1/2 ÷ 2/3 = 7/2 × 3/2 = 21/4 = 5.25; only 5 complete cakes can be made, since the leftover 0.25 of a portion is not enough for a 6th cake. Answer: 5. 5.25 gives the exact result of the division without rounding down to a whole number of cakes. 7 comes from multiplying 3.5 by 2 and ignoring the need to also divide by 3 as part of dividing by the fraction 2/3. 6 comes from rounding 5.25 up to the nearest whole number instead of down, wrongly assuming a 6th cake could be made from the leftover sugar.
- (b) 2.1 m — The three pieces use 3 × 0.9 = 2.7 m of wood. Remaining wood = 4.8 − 2.7 = 2.1 m. A candidate who miscounts and only subtracts 2 pieces instead of 3 gets 4.8 − 1.8 = 3.0 m. A candidate who adds instead of subtracting gets 4.8 + 2.7 = 7.5 m. A candidate who gives the length used instead of the length remaining gets 2.7 m.
- (d) 36 — Method: the fraction is acting as an operator on the whole class, so one third of the class equals 12; the operation has to be reversed, and the inverse of dividing by 3 is multiplying by 3. Working: 1/3 × (number of pupils) = 12, so the number of pupils = 12 × 3 = 36. Answer: 36 pupils. The distractors: 4 comes from applying the operator instead of reversing it, working out 12 ÷ 3 = 4; 18 comes from reading the 12 girls as two thirds of the class, giving 12 ÷ 2 × 3 = 18; 24 comes from working out the number of boys, the other two thirds, as 2 × 12 = 24 and giving that instead of the size of the class.
- (a) £6 — 10% of £40 is £4, and 5% of £40 is half of that, £2. Adding these gives 15% of £40 = £4 + £2 = £6. Finding only the 10% part and stopping there gives £4. Finding only the 5% part and stopping there gives £2. Multiplying 40 by 15 without dividing by 100 gives £600, which treats the percentage as if it were a whole number multiplier.
- (a) 7/8, 17/20, 0.8, 78% — Method: convert every value to a decimal, then order the decimals from largest to smallest. Working: 7/8 = 0.875, 17/20 = 0.85, 0.8 = 0.8, 78% = 0.78. Ordering from largest to smallest gives 7/8, 17/20, 0.8, 78%. Answer: 7/8, 17/20, 0.8, 78%. '78%, 0.8, 17/20, 7/8' comes from ordering the converted decimals from smallest to largest instead of largest to smallest. '17/20, 7/8, 0.8, 78%' comes from converting 17/20 incorrectly as larger than 7/8, for example treating 17/20 as 0.87 instead of 0.85, swapping the top two. '7/8, 0.8, 17/20, 78%' comes from converting 17/20 incorrectly as equal to 0.8, swapping the middle two.
- (a) 3.84 × 10⁵ — 384,000 = 3.84 × 100,000 = 3.84 × 10⁵, with the decimal point moved five places and the coefficient kept between 1 and 10. Moving the point six places instead of five gives 3.84 × 10⁶, ten times too large. Leaving the coefficient as 38.4 gives 38.4 × 10⁴, which is not between 1 and 10. Using a negative exponent instead of a positive one gives 3.84 × 10⁻⁵, a number far smaller than 1.
- (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.
- (a) 0 — Method: BIDMAS works through the index first, then the multiplication, then the subtraction. Working: 5² = 25, then 4 × 25 = 100, and finally 100 − 100 = 0. Answer: 0. The distractors: 2400 comes from working from left to right and subtracting first, giving (100 − 4) × 25 = 96 × 25 = 2400; −300 comes from multiplying before applying the index, giving (4 × 5)² = 20² = 400 and then 100 − 400 = −300; 60 comes from reading 5² as 5 × 2 = 10, so that 4 × 10 = 40 and 100 − 40 = 60.
- (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) Yes, because 14.8 cm rounds to 15 cm to the nearest cm — Method: a recorded measurement is not an exact length; it stands for every length that rounds to it, so the two records agree if one rod can produce both. Working: Ben's record of 14.8 cm to the nearest 0.1 cm means the rod is between 14.75 cm and 14.85 cm, and 14.8 is nearer to 15 than to 14, so a rod of that length is recorded as 15 cm to the nearest centimetre. Both records can therefore come from the same rod. Answer: Yes, because 14.8 cm rounds to 15 cm to the nearest cm. The distractors: the claim that 14.8 cm rounds to 15.0 cm to 1 decimal place is false, since 14.8 cm is already written to 1 decimal place and stays 14.8 cm; the claim that it rounds to 14 cm is false, because 14.8 is 0.2 away from 15 and 0.8 away from 14; the claim that the two lengths are not the same treats each record as an exact length, when each is only a rounded record of one rod.
- (a) 11 — Method: list every possible total from the smallest to the largest, and count how many different values there are. Working: the smallest total is 1+1=2 and the largest is 6+6=12, and every whole number total from 2 to 12 is possible: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 — that is 11 different totals. Answer: 11. 36 comes from counting the number of possible dice outcomes (6×6) instead of the number of different totals. 10 comes from listing the totals but missing one from the ends of the list, for example starting at 3 instead of 2. 6 comes from counting only the number of different scores on one die, not the totals of both dice together.
- (c) 8,000,000,000 — Method: a power of ten is 1 followed by that many zeros, so 8 × 10⁹ is 8 multiplied by 1 followed by nine zeros. Working: 10⁹ = 1,000,000,000, and 8 × 1,000,000,000 puts an 8 in front of those nine zeros. Answer: 8,000,000,000. The distractors: 800,000,000 comes from writing nine digits altogether instead of nine zeros after the 8; 720 comes from reading 10⁹ as 10 × 9 = 90 and then working out 8 × 90; 0.000000008 comes from treating the index as negative and carrying the decimal point nine places to the left.
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