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
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- 1.Write down the decimal that is equal to 3/5.
- 2.The attendance at a football match is 23,000, correct to the nearest thousand. Which of these attendances is not possible for the match?
- 3.In a fruit crate the ratio of oranges to lemons is 5 : 6. What fraction of the fruit in the crate are lemons?
- 4.Write these numbers in order, starting with the smallest: 3, −1, 0, −5
- 5.On a number line, point A is at −1. Point B is 4 units from point A. Write down the two possible positions of point B.
- 6.A student always writes the digits after the decimal point over a denominator of 100, whatever the number of decimal places. Using this wrong rule, they convert 0.9 to a fraction. Work out the correct fraction that 0.9 is equal to, giving your answer in its simplest form.
- 7.The length of a nail is 12 cm, correct to the nearest centimetre. Using L for the length of the nail in centimetres, write down the error interval for L.
- 8.A shop buys boxes of pens for £4 each. It sells each box on to a school for £3 more than it paid, and charges an extra £2 delivery fee for the whole order. A school orders 5 boxes. Work out the total cost of the order.
- 9.A market trader buys 8 melons at £1.35 each. He sells all of them for a total of £15.60. Work out his profit.
- 10.Put these numbers in order, starting with the smallest: 3.2 × 10⁴, 2.9 × 10⁵, 4.1 × 10³
- 11.A charity raises 8 × 10⁶ pounds. This total is shared equally among 2 × 10² local projects. Work out how much each project receives, in standard form.
- 12.Write down the symbol that makes this statement true: −9 ___ −4
- 13.Write 60 as a product of its prime factors, using index notation.
- 14.A number, n, is a multiple of both 6 and 9. Work out the smallest possible value of n that is greater than 20.
- 15.Write 90 as a product of its prime factors.
Answer key
- (b) 0.6 — Method: convert the fraction to an equivalent fraction with denominator 10, then read off the decimal. Working: 3/5 = 6/10 (multiplying numerator and denominator by 2) = 0.6. Answer: 0.6. 0.35 comes from combining the digits 3 and 5 directly after the decimal point instead of converting the fraction. 0.53 comes from writing the numerator and denominator digits in the wrong order. 1.67 comes from flipping the fraction to 5/3 before converting to a decimal.
- (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.
- (a) 6/11 — Method: add the parts of the ratio to find the total, then write the required part over the total. Working: 5 + 6 = 11 parts in total; lemons make up 6 of the 11 parts, so the fraction of lemons is 6/11, which is already in its simplest form. Answer: 6/11. 5/11 comes from finding the fraction of oranges instead of lemons. 5/6 comes from writing the ratio of oranges to lemons directly as a fraction instead of comparing lemons to the total. 6/5 comes from writing the ratio of lemons to oranges directly as a fraction instead of comparing lemons to the total.
- (d) −5, −1, 0, 3 — Method: order the numbers by their position on a number line, smallest (furthest left) first. Working: both −5 and −1 lie to the left of 0, and 3 lies to the right of 0. Of the two negatives, −5 is 5 units from zero and −1 is 1 unit from zero, so −5 is further left. Answer: −5, −1, 0, 3. The distractors: 3, 0, −1, −5 is the correct order written the wrong way round, starting with the largest; −1, −5, 0, 3 comes from ordering the two negatives by the size of their digits, so that −1 is treated as the smaller; 0, −1, −5, 3 comes from believing that zero is the smallest number there is and then listing the negatives by their digits.
- (c) 3 or −5 — Method: a point a fixed distance from another can lie on either side of it, so move the given distance in each direction from the starting point. Working: moving 4 units to the right gives −1 + 4 = 3, and moving 4 units to the left gives −1 − 4 = −5. Answer: 3 or −5. The distractors: 5 or −3 comes from starting at 1 instead of −1, giving 1 + 4 and 1 − 4; 3 only comes from moving to the right and forgetting that the point could lie to the left as well; 4 or −4 comes from measuring the distance from zero instead of from point A, which just repeats the given distance.
- (c) 9/10 — Method: write the decimal over 10 using its one decimal place. Working: 0.9 = 9/10, which is already in its simplest form since 9 and 10 share no common factor. Answer: 9/10. The student's fraction, 9/100, comes from always writing the denominator as 100, regardless of how many decimal places the number actually has. 9 comes from dropping the decimal point altogether and treating 0.9 as the whole number 9. 1/9 comes from flipping the correct fraction upside down.
- (a) 11.5 ≤ L < 12.5 — Rounding to the nearest centimetre means L can be up to half a centimetre below or above 12 before it would round to a different whole number. Half of 1 cm is 0.5 cm, so the lower bound is 12 − 0.5 = 11.5 and the upper bound is 12 + 0.5 = 12.5. A value exactly at the upper bound, 12.5, would round up to 13, not 12, so 12.5 itself is excluded, giving 11.5 ≤ L < 12.5. Writing 11.5 ≤ L ≤ 12.5 wrongly includes 12.5 on both ends. Writing 11 ≤ L < 13 uses a whole centimetre either side instead of half a centimetre. Writing 11.5 < L < 12.5 wrongly excludes the lower bound, which is a value that does round to 12.
- (b) £37 — One box costs £4 + £3 = £7. Five boxes cost 5 × £7 = £35. Adding the single £2 delivery fee gives £35 + £2 = £37. A candidate who added the £2 delivery fee to each box instead of once for the whole order worked out 5 × (£7 + £2) = 5 × £9 = £45. A candidate who forgot the £3 markup and used the shop's buying price worked out 5 × £4 + £2 = £22. A candidate who added the £3 markup only once, after multiplying the buying price by 5, worked out 5 × £4 + £3 + £2 = £25.
- (a) £4.80 — The total cost of the melons is 8 × £1.35 = £10.80. Profit is the selling total minus the cost, so £15.60 − £10.80 = £4.80. Subtracting the price of just one melon from the selling total instead of the total cost of all eight, £15.60 − £1.35, gives £14.25. Miscalculating the cost of the melons as 8 × £1.30 = £10.40, dropping the 5p from each price, gives a profit of £15.60 − £10.40 = £5.20. Adding the selling total and the cost together instead of subtracting them, £15.60 + £10.80, gives £26.40.
- (c) 4.1 × 10³, 3.2 × 10⁴, 2.9 × 10⁵ — The exponent decides the size first: 10³ is smaller than 10⁴, which is smaller than 10⁵, so the order is 4.1 × 10³, then 3.2 × 10⁴, then 2.9 × 10⁵. Reversing the whole list gives largest to smallest instead of smallest to largest. Comparing 3.2 × 10⁴ and 4.1 × 10³ by their coefficients alone, 3.2 against 4.1, and swapping them ignores that 10⁴ is bigger than 10³ regardless of the coefficient. Comparing 2.9 × 10⁵ and 3.2 × 10⁴ by their coefficients alone and swapping them makes the same mistake at the top of the list.
- (c) 4 × 10⁴ — 8 ÷ 2 = 4, and 6 − 2 = 4, so each project receives 4 × 10⁴ pounds. Multiplying the exponents instead of subtracting them gives 6 × 2 = 12, so 4 × 10¹². Adding the exponents instead of subtracting them gives 6 + 2 = 8, so 4 × 10⁸. Subtracting the coefficients instead of dividing them gives 8 − 2 = 6, so 6 × 10⁴.
- (c) < — Method: compare the two negative numbers by their position on a number line, not by the size of the digit. Working: −9 is nine places to the left of zero and −4 is four places to the left of zero, so −9 is further from zero in the negative direction, which makes it the smaller number. Answer: −9 < −4, so the symbol is <. > comes from comparing 9 and 4 as if both numbers were positive, ignoring the negative signs. = comes from treating both numbers as equal simply because they are both negative. ≥ combines > (false) with = (also false).
- (b) 2² × 3 × 5 — Repeatedly divide 60 by prime numbers: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, and 5 is itself prime. So 60 is 2 × 2 × 3 × 5, which in index notation is 2² × 3 × 5. Stopping the factor tree after only three divisions and writing 2 × 3 × 5 misses that the 2 divides in twice, and gives only 30, not 60. Squaring the 3 as well as the 2 gives 2² × 3² × 5, which comes to 180, far too big. Squaring the 5 instead of the 2 gives 2 × 3 × 5², which comes to 150, also too big. So 60 = 2² × 3 × 5.
- (a) 36 — Method: find the lowest common multiple of 6 and 9, then move up the list of common multiples until one is greater than 20. Working: the common multiples of 6 and 9 are 18, 36, 54 …. 18 is not greater than 20, so the next one, 36, is the smallest value of n that is greater than 20. 18 is the lowest common multiple itself, but it fails the 'greater than 20' condition. 54 is the common multiple after 36, one step too far. 27 is a multiple of 9 but not of 6, since 27 ÷ 6 is not a whole number. Answer: 36.
- (a) 2 × 3² × 5 — Method: divide repeatedly by the smallest prime number until only prime factors remain. Working: 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 is prime, so 90 = 2 × 3 × 3 × 5, written as 2 × 3² × 5. 2 × 3 × 15 stops before the 15 is broken down into 3 × 5, so it is not fully factorised. 3 × 3 × 10 stops before the 10 is broken down into 2 × 5. 2 × 45 stops after only one division. Answer: 2 × 3² × 5.
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