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.
Non-calculator
Number worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- 1.A circle has a diameter of 10 cm. Work out the exact circumference of the circle, in terms of π.
- 2.A car manufacturer offers a car in 6 colours and 4 trim levels. Two colour-and-trim combinations are not available: red with sport trim, and white with sport trim. Work out how many different colour-and-trim combinations are available.
- 3.A jug holds 1.25 litres of water. Ella pours in another 400 ml. Work out the total volume, in litres.
- 4.Work out (−2)³ + (−3)² − (−4)
- 5.Work out the total cost of 3 pens at 45p each and one notebook at £2.35. Give your answer in pounds.
- 6.A lorry travels 105 miles in 1 hour 45 minutes. Work out its average speed in miles per hour.
- 7.A window display has red baubles, gold baubles and green baubles in the ratio 4 : 5 : 6. What fraction of the baubles are not green?
- 8.A number line runs from 0 to 1 and is divided into 4 equal parts. Which mark on the line represents 1/4?
- 9.Work out 3 1/5 + 1 2/3. Give your answer as a mixed number in its simplest form.
- 10.Which of these numbers rounds to 0.048 when rounded to 2 significant figures?
- 11.Write down the symbol that makes this statement true: 3/10 ___ 0.3
- 12.A circle has a radius of 3 cm. Which of these is the exact area of the circle?
- 13.Which of these units would be most appropriate to use when stating the mass of a small car?
- 14.Write 0.325 as a fraction in its simplest form.
- 15.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.
Answer key
- (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.
- (c) 22 — Without restriction there are 6 × 4 = 24 combinations. Two specific combinations are not available, so subtract 2: 24 − 2 = 22. 24 comes from ignoring the restriction completely. 23 comes from subtracting only 1 of the 2 excluded combinations. 18 comes from removing the whole sport trim level, 6 × 3 = 18, instead of removing just the two excluded combinations.
- (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.)
- (a) 5 — Method: each index is worked out first, and subtracting a negative number is the same as adding the positive. Working: (−2)³ = (−2) × (−2) × (−2) = −8 and (−3)² = (−3) × (−3) = 9, while − (−4) becomes + 4, so the calculation becomes −8 + 9 + 4 = 5. Answer: 5. The distractors: −13 comes from taking (−3)² as −9, giving −8 − 9 + 4 = −13; −3 comes from reading − (−4) as − 4, giving −8 + 9 − 4 = −3; 21 comes from treating every power of a negative number as positive, so that (−2)³ is taken as 8 and the calculation becomes 8 + 9 + 4 = 21.
- (b) £3.70 — Work in one unit throughout. 45p is £0.45, so three pens cost 3 × £0.45 = £1.35. Adding the notebook gives £1.35 + £2.35 = £3.70. £2.80 is the cost of one pen and the notebook, so the multiplication by 3 has been left out; £8.40 multiplies the notebook by 3 as well as the pens; £15.85 comes from reading 45p as £4.50.
- (d) 60 mph — Average speed = distance ÷ time, with the time measured in hours. 45 minutes is 45/60 of an hour, which is 0.75 of an hour, so the journey takes 1.75 hours. Speed = 105 ÷ 1.75 = 60 mph. 52.5 mph rounds the time up to 2 hours, 72.4 mph writes 1 hour 45 minutes as 1.45 hours, and 183.75 mph multiplies the distance by the time instead of dividing.
- (b) 3/5 — Method: add all three parts for the total, add together the parts that are not green, then write this over the total. Working: total parts = 4 + 5 + 6 = 15. Not green = 4 + 5 = 9. Fraction = 9/15 = 3/5. Answer: 3/5. 2/5 comes from finding the fraction that IS green (6/15 = 2/5) instead of not green. 4/15 comes from only counting the red baubles as 'not green' and forgetting the gold ones. 9/10 comes from adding only two of the three ratio parts to find the total (4+6=10), missing out the gold part, while still using 9 for the numerator.
- (d) the first mark after 0 — Method: when a unit length is split into equal parts, each gap is one part of the whole, so 1/4 is one gap along from 0. Working: four equal parts means each gap measures 1/4, so the marks after 0 stand for 1/4, 2/4, 3/4 and 4/4. One gap along from 0 is therefore the mark for 1/4. Answer: the first mark after 0. The distractors: the second mark after 0 comes from counting 0 itself as the first mark; the third mark after 0 comes from counting back from the 1 end instead of forward from 0; the fourth mark after 0 comes from reading the 4 in the denominator as the number of the mark rather than the number of parts.
- (d) 4 13/15 — Convert to fifteenths: 1/5 is equivalent to 3/15 (multiply by 3/3), and 2/3 is equivalent to 10/15 (multiply by 5/5), so 3 1/5 is equivalent to 3 3/15 and 1 2/3 is equivalent to 1 10/15. Add the whole numbers (3 + 1 = 4) and the fractions (3/15 + 10/15 = 13/15), giving 4 13/15. A candidate who adds the numerators and denominators straight across, treating 1/5 + 2/3 as (1+2)/(5+3), gets a fraction part of 3/8, giving 4 3/8. A candidate who adds the fraction parts correctly but forgets to add the second whole number gets 3 13/15. A candidate who adds the whole numbers but copies the first fraction across without ever adding 2/3 to it gets 4 1/5.
- (c) 0.0479 — Method: round each option to 2 significant figures and check which one gives 0.048. Working: for 0.0479, the first two significant figures are 4 and 7; the next digit is 9, so 7 rounds up to 8, giving 0.048. For 0.0485, the first two significant figures are 4 and 8; the next digit is 5, so 8 rounds up to 9, giving 0.049, not 0.048. 0.052 already has exactly 2 significant figures, 5 and 2, so it stays as 0.052 and does not round to 0.048 at all. 0.04 has only 1 significant figure, so it is already less precise than the 2 significant figures asked for. Answer: 0.0479.
- (a) = — Method: convert the fraction to a decimal so both numbers are in the same form. Working: 3/10 means 3 divided by 10, which is 0.3. Answer: 3/10 = 0.3, so the symbol is =. < comes from treating the fraction as smaller than the decimal without converting it first. > comes from the same mistake the other way round. ≠ comes from assuming a fraction and a decimal can never be equal.
- (c) 9π cm² — The area of a circle is π × r². With a radius of 3 cm this is π × 3² = 9π cm², and this is exact because π has not been replaced by any approximation. Writing 28.3 cm² replaces π with a rounded decimal value, 3.14, and then rounds the result again, so it is only an approximation. Writing 28.26 cm² uses π ≈ 3.14 without a final rounding step, but this is still only an approximation of 9π, not the exact value. Writing 27 cm² comes from replacing π with the rough approximation 3, which is even further from the true value.
- (a) Kilograms — A small car has a mass of roughly 1000–1500 kilograms, so kilograms gives a sensible number to work with — neither too small nor too large. Grams would give an enormous number, over a million, since grams suit much lighter objects. Litres measures capacity, not mass, so it is the wrong type of unit for a car's mass entirely. Metres measures length, not mass, so it is also the wrong type of unit.
- (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.
- (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.)
Build your own mix at the worksheet builder.