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 bus leaves the station at 09:47 and arrives at its destination at 11:23. Work out the journey time.
- 2.Write 0.00072 in standard form.
- 3.At midday the temperature in the Lake District was 3 °C. By midnight it had fallen to −6 °C. Work out the fall in temperature.
- 4.Work out −(−3)⁴ + (−3)³
- 5.A length, L cm, has the error interval 24.5 ≤ L < 25.5. Write down the degree of accuracy to which the length was measured.
- 6.Round 3.947 to 2 decimal places.
- 7.Work out the value of ∛125.
- 8.Work out 4π + 2π, giving your answer as a multiple of π.
- 9.Grace drinks 1/3 of a bottle of water in the morning and another 1/3 of the same bottle in the afternoon. Work out what fraction of the bottle she has drunk altogether.
- 10.Work out 3/7 × 14/9. Give your answer as a fraction in its simplest form.
- 11.Work out 8% of £250.
- 12.A charity shop buys a coat for £24 and sells it for a profit that is 3/8 of the buying price. Work out the selling price.
- 13.Write these numbers in order, starting with the smallest: 5, −8, 0, −3, 2
- 14.The distance from the Earth to the Moon is 384,000 km. Write this distance in standard form, in kilometres.
- 15.A café offers 3 types of soup and 4 types of bread roll. Work out how many different combinations of one soup and one bread roll are possible.
Answer key
- (d) 1 hour 36 minutes — From 09:47 to 10:47 is exactly 1 hour, and from 10:47 to 11:23 is a further 36 minutes, so the journey takes 1 hour 36 minutes in total. Subtracting the hours and minutes separately without carrying across the hour boundary — hours 11 − 9 = 2, minutes 47 − 23 = 24, taking the smaller minute figure from the larger regardless of which time each came from — gives 2 hours 24 minutes. Carrying the hour correctly but then subtracting the minutes the wrong way round, 47 − 23 = 24 instead of 60 − 47 + 23 = 36, gives 1 hour 24 minutes. Counting an extra hour by treating 09:47 as if it were 08:47 gives 2 hours 36 minutes.
- (a) 7.2 × 10⁻⁴ — 0.00072 = 7.2 × 10⁻⁴, moving the decimal point 4 places to the right to reach 7.2, so the exponent is negative. Writing 7.2 × 10⁴ uses a positive exponent, which would give a number far larger than 1, not a small decimal. Writing 7.2 × 10⁻⁵ moves the point one place too many. Writing 0.72 × 10⁻³ leaves the coefficient below 1, which standard form does not allow.
- (a) 9 °C — Method: the fall is the difference between the two readings, so subtract the lower reading from the higher one; subtracting a negative number is the same as adding its positive. Working: 3 − (−6) = 3 + 6 = 9. Counting it out, the temperature drops 3 degrees to reach zero and a further 6 degrees below zero. Answer: 9 °C. The distractors: −9 °C comes from subtracting the readings the wrong way round, as −6 − 3, and reporting a fall as a negative amount; 3 °C comes from ignoring the minus sign and working out 6 − 3; 6 °C comes from counting only the part of the fall that happens below zero and forgetting the 3 degrees above it.
- (a) −108 — Method: a power is worked out before any minus sign written in front of it, while a minus sign inside the brackets is part of the base. Working: (−3)⁴ = 81, because four negative factors multiply to a positive result, so −(−3)⁴ = −81. (−3)³ = −27, because three negative factors multiply to a negative result. Adding gives −81 + (−27) = −108. Answer: −108. The distractors: 54 comes from attaching the leading minus sign to the base, working out (−(−3))⁴ = 81 and then adding −27; −54 comes from taking (−3)³ as +27, forgetting that an odd power keeps the negative sign; 108 comes from believing that any power of a negative number is positive and that the leading minus belongs to the base, giving 81 + 27.
- (a) to the nearest centimetre — Method: the error interval of a rounded measurement runs from half a unit below the stated value to half a unit above it, so the width of the interval is one whole unit of the accuracy used. Working: the interval runs from 24.5 to 25.5, a width of 25.5 − 24.5 = 1, so the unit of accuracy is 1 cm; the stated value is the midpoint, 25 cm, and 25 correct to the nearest centimetre is exactly what gives 24.5 ≤ L < 25.5. Answer: to the nearest centimetre. To the nearest 0.5 cm comes from reading the half-unit, 0.5, as the accuracy itself instead of doubling it back to the full unit. To 1 decimal place comes from seeing the bounds written with one decimal place and taking that as the accuracy, but the bounds of a value given to 1 decimal place would be only 0.05 either side. To the nearest 10 cm comes from confusing the size of the value, about 25, with the unit it was rounded to; rounding to the nearest 10 cm would give an interval 5 cm either side of the stated value.
- (c) 3.95 — The digit after the second decimal place is 7, which is 5 or more, so round the second decimal place up: 3.947 rounds to 3.95. A candidate who truncated instead of rounding, simply cutting off after 2 decimal places, wrote 3.94. A candidate who rounded to 1 decimal place instead of 2 wrote 3.9. A candidate who rounded up but mishandled the carry wrote 4.0.
- (a) 5 — Method: the cube root of a number is the value that multiplies by itself three times to give that number. Working: 5 × 5 × 5 = 125, so ∛125 = 5. 62.5 comes from working out 125 ÷ 2 = 62.5, halving the number instead of finding its cube root. 375 comes from working out 125 × 3 = 375, multiplying by 3 instead of cube-rooting it. 15625 comes from working out 125², squaring the number instead of finding its cube root. Answer: 5.
- (d) 6π — 4π and 2π are like terms, both multiples of π, so they combine by adding their coefficients: 4 + 2 = 6, giving 6π. Multiplying the coefficients instead of adding them, 4 × 2 = 8, gives 8π. Treating the combination as if the two π's multiplied together as well as the coefficients gives 6π². Dropping the π altogether and adding only the coefficients gives 6.
- (c) 2/3 — Method: fractions with the same denominator are added by adding the numerators and leaving the denominator alone, because the parts are already the same size. Working: 1/3 + 1/3 has numerators 1 + 1 = 2 and the denominator stays as 3, giving 2/3. Answer: 2/3. The distractors: 2/6 comes from adding the denominators as well as the numerators, 1 + 1 over 3 + 3; 2/9 comes from adding the numerators but multiplying the denominators, 1 + 1 over 3 × 3; 1/9 comes from multiplying throughout instead of adding, 1 × 1 over 3 × 3.
- (c) 2/3 — Method: multiply the numerators together and the denominators together, then divide both parts of the result by their highest common factor. Working: 3 × 14 = 42 and 7 × 9 = 63, giving 42/63; the highest common factor of 42 and 63 is 21, and 42 ÷ 21 = 2 with 63 ÷ 21 = 3. Answer: 2/3. The distractors: 17/16 comes from adding the numerators and adding the denominators, giving (3 + 14)/(7 + 9); 27/98 comes from turning the second fraction upside down and multiplying, which divides instead of multiplying and gives 3/7 × 9/14; 2/21 comes from cancelling the 7 into the 14 in the numerator but leaving the 7 in the denominator, giving 6/63.
- (d) £20 — 1% of £250 = £2.50, so 8% = 8 × £2.50 = £20. A candidate who misplaces the decimal point and finds 0.8% instead gets £2. A candidate who confuses 8% with 80% gets £200. A candidate who rounds 8% up to the nearby 10% gets £25.
- (d) £33.00 — The profit is 3/8 of £24 = (£24 ÷ 8) × 3 = £3 × 3 = £9.00. Selling price = £24 + £9.00 = £33.00. A candidate who gives the profit instead of the selling price gets £9.00. A candidate who subtracts the profit instead of adding it gets £24 − £9 = £15.00. A candidate who works out one eighth of £24 and adds that on, forgetting to multiply by the numerator 3, gets £24 + £3 = £27.00.
- (d) −8, −3, 0, 2, 5 — Method: place the numbers on a number line and read them from left to right. Working: the negative numbers −8 and −3 come before 0, with −8 further left than −3 because it is further from zero in the negative direction; then 0, then the positive numbers 2 and 5 in increasing size. Answer: −8, −3, 0, 2, 5. −3, −8, 0, 2, 5 swaps −3 and −8, putting the negative number closer to zero first. 5, 2, 0, −3, −8 lists the numbers from largest to smallest instead of smallest to largest. −8, −3, 2, 0, 5 puts 2 before 0, treating positive numbers as if they always come before zero.
- (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.
- (b) 12 — Each of the 3 soups can be paired with each of the 4 bread rolls, so multiply: 3 × 4 = 12. 7 comes from adding the two numbers instead of multiplying them. 3 comes from using only the number of soups and ignoring the bread rolls. 4 comes from using only the number of bread rolls and ignoring the soups.
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