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.
Number worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- 1.Write these numbers in order, starting with the largest: 7/8, 0.8, 78%, 17/20
- 2.A recipe needs 0.485 kg of flour per cake. A bakery estimates its flour order by rounding this amount to 1 significant figure, then multiplying by the 60 cakes it plans to bake. Work out the bakery's estimate for the total flour needed, in kg.
- 3.The number 24 can be written as 2³ × 3, and the number 60 can be written as 2² × 3 × 5. Work out the lowest common multiple of 24 and 60.
- 4.Work out 2 × (−3 + 7)
- 5.A number is divided by 5, then 6 is subtracted, giving the result −1. Work out the number.
- 6.Write down the symbol that makes this statement true: −2/3 ___ −5/6
- 7.A sponsored walk is 36 km long. Aisha has completed 8/12 of the walk. Work out how far she has walked.
- 8.The attendance at a football match is 23,000, correct to the nearest thousand. Which of these attendances is not possible for the match?
- 9.Work out 3 + 4 × (−2).
- 10.Write 0.36 as a fraction in its simplest form.
- 11.Jack buys 3 books, each costing £4.25, and pays with a £20 note. Work out how much change he receives.
- 12.Work out 100 − 4 × 5²
- 13.A cinema has 21 rows of seats with 29 seats in each row. Work out an estimate for the number of people the cinema can hold, by rounding each number to 1 significant figure.
- 14.A tin of soup has a mass of 450 g. A bag of potatoes has a mass of 2.6 kg. Work out the total mass, in kilograms.
- 15.Work out the highest common factor of 20 and 32.
Answer key
- (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.
- (d) 30 kg — Round 0.485 kg to 1 significant figure: 0.5 kg. Multiply by the 60 cakes: 0.5 × 60 = 30 kg. A candidate who rounded to 2 significant figures instead of 1 used 0.49 kg, giving 0.49 × 60 = 29.4 kg. A candidate who used the unrounded amount instead of the estimate worked out 0.485 × 60 = 29.1 kg. A candidate who rounded 0.485 down to 0.4 kg instead of up to 0.5 kg worked out 0.4 × 60 = 24 kg.
- (a) 120 — For the lowest common multiple, take each prime that appears in either factorisation, raised to the higher power. In 2³ × 3 and 2² × 3 × 5, the prime 2 appears with power 3 in one and power 2 in the other — take the higher, 2³; the prime 3 appears with the same power in both, 3¹; and the prime 5 appears only in the second factorisation, so use 5¹. Multiplying these, 2³ × 3 × 5, gives 120. Taking the lower power of 2 instead of the higher, and leaving out 5 altogether, gives the highest common factor, 12, instead. Multiplying the two original numbers together, 24 × 60, gives 1440, which double-counts every shared prime factor. Assuming the lowest common multiple is simply the larger of the two numbers gives 60, but 60 is not a multiple of 24 — 60 ÷ 24 does not divide exactly. So the lowest common multiple of 24 and 60 is 120.
- (d) 8 — Method: whatever is inside the brackets is worked out first, and the multiplication is carried out afterwards. Working: inside the brackets, −3 + 7 = 4, and then 2 × 4 = 8. Answer: 8. The distractors: 1 comes from ignoring the brackets and multiplying first, giving 2 × (−3) = −6 and then −6 + 7 = 1; −20 comes from reading −3 + 7 as −(3 + 7) = −10, so that 2 × (−10) = −20; 6 comes from working the brackets out correctly and then adding the 2 instead of multiplying by it, giving 2 + 4 = 6.
- (a) 25 — Reverse the operations in reverse order: undo the subtraction by adding 6, then undo the division by multiplying by 5. −1 + 6 = 5, so the number divided by 5 equals 5, and 5 × 5 = 25 — checking, 25 ÷ 5 − 6 = 5 − 6 = −1. A candidate who subtracted 6 again instead of adding worked out −1 − 6 = −7, then −7 × 5 = −35. A candidate who multiplied by 5 before undoing the subtraction, doing the inverse operations in the wrong order, worked out −1 × 5 = −5, then −5 + 6 = 1. A candidate who multiplied by 5 but forgot to undo the subtraction at all worked out −1 × 5 = −5 and stopped there.
- (c) > — Method: write both fractions with a common denominator so they can be compared directly. Working: −2/3 = −4/6. Comparing −4/6 and −5/6, −4 is greater than −5, so −4/6 is greater than −5/6. Answer: −2/3 > −5/6, so the symbol is >. < comes from treating the fraction with the larger numerator and denominator, 5/6, as the greater number, ignoring that both fractions are negative. = comes from assuming the fractions must be equal because they look similar in size. ≤ combines < (false) with = (false).
- (d) 24 km — Method: a fraction acts as an operator, so finding 8/12 of a distance means dividing by the denominator and multiplying by the numerator. Working: 36 ÷ 12 = 3, so one twelfth of the walk is 3 km, and eight twelfths is 3 × 8 = 24 km. Answer: 24 km. The distractors: 12 km comes from working out the part of the walk still left, the other four twelfths, instead of the part already completed; 288 km comes from multiplying by the numerator without dividing by the denominator, giving 36 × 8 = 288; 4.5 km comes from dividing by the numerator instead of multiplying by it, giving 36 ÷ 8 = 4.5.
- (d) 23,600 — 23,000 correct to the nearest thousand means the true attendance, a, satisfies 22,500 ≤ a < 23,500. 23,600 is outside this range, so it is not possible. 22,500 is the lower bound of the range, so it is possible. 23,499 is just below the upper bound, so it is possible. 23,000 is the given value itself, so it is possible.
- (b) −5 — Using the order of operations, work out the multiplication first: 4 × (−2) = −8. Then 3 + (−8) = −5. A candidate who adds before multiplying gets (3 + 4) × (−2) = −14. A candidate who drops the negative sign on the multiplication gets 3 + 4 × 2 = 11. A candidate who works out the multiplication correctly but gives that as the final answer, forgetting to combine it with the 3, gets −8.
- (b) 9/25 — Method: write the decimal over the matching power of ten, then divide the numerator and the denominator by their highest common factor. Working: 0.36 has two digits after the point, so 0.36 = 36/100; the highest common factor of 36 and 100 is 4, and 36 ÷ 4 = 9 with 100 ÷ 4 = 25; since 9 and 25 share no factor greater than 1, the fraction is fully cancelled. Answer: 9/25. The distractors: 3/10 comes from reading only the first digit after the point and converting 0.3; 9/50 comes from dividing the numerator by 4 but the denominator by only 2; 36/10 comes from counting one decimal place instead of two and writing the digits over 10.
- (c) £7.25 — Find the total cost of the books first: 3 × 4.25 = 12.75, so the books cost £12.75 in total. Subtract this from the £20 note: 20.00 − 12.75 = 7.25, so the change is £7.25. Stopping after finding the cost and not subtracting it from £20 gives £12.75, which is the amount spent, not the change. Borrowing correctly in the pence column but forgetting to reduce the pounds column by 1 gives £8.25 instead of £7.25. Multiplying 3 × 4.25 as 12.25 instead of 12.75, a multiplication slip, makes the change come out £0.50 too high, at £7.75. So Jack receives £7.25 change.
- (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.
- (d) 600 — Method: the number of seats is the number of rows multiplied by the number of seats in each row, so round each number to 1 significant figure and then multiply the rounded values, which is quick because a product of two multiples of ten is found by multiplying the non-zero digits and attaching the zeros. Working: 21 rounds to 20 and 29 rounds to 30; 2 × 3 = 6, and 20 and 30 carry one zero each, so two zeros follow the 6. Answer: about 600 seats. The distractors: 50 comes from adding the two rounded numbers instead of multiplying them, 20 + 30; 60 comes from multiplying 20 by the 3 of 30 and forgetting the zero in 30; 6,000 comes from attaching three zeros to 2 × 3 when 20 and 30 provide only two between them.
- (c) 3.05 kg — There are 1000 g in 1 kg, so 450 g = 450 ÷ 1000 = 0.45 kg. The total mass is 0.45 + 2.6 = 3.05 kg. 7.1 kg divides the grams by 100 instead of 1000, 2.645 kg divides them by 10 000, and 452.6 kg adds the two numbers without converting the grams at all.
- (b) 4 — Method: list the factors of each number and compare them; the highest common factor is the largest number that appears in both lists. Working: the factors of 20 are 1, 2, 4, 5, 10, 20; the factors of 32 are 1, 2, 4, 8, 16, 32. The numbers that appear in both lists are 1, 2 and 4, and the largest of these is 4. 2 is a common factor of 20 and 32 but not the largest one. 8 is a factor of 32 but not of 20, since 20 ÷ 8 is not a whole number. 160 is the lowest common multiple of 20 and 32, not their highest common factor. Answer: 4.
Build your own mix at the worksheet builder.