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.Write down the symbol that makes this statement true: 0.45 ___ 0.5
- 2.Which one of these statements about negative numbers is true?
- 3.Write 0.25 as a fraction in its simplest form.
- 4.Write these numbers in order, starting with the smallest: −1.4, 5/4, −6/5, 1.3, 0
- 5.A photography studio offers 4 backdrops and 3 outfits for a portrait session. The plain grey backdrop, one of the 4 backdrops, cannot be used with the formal suit outfit, one of the 3 outfits. Work out how many different backdrop-and-outfit combinations are possible.
- 6.A restaurant offers a lunch deal of one starter from 5 options, one main from 6 options and one dessert from 3 options. Work out how many different lunch deals are possible.
- 7.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.
- 8.A garden has flowers, herbs and vegetables in the ratio 2 : 3 : 4. What fraction of the garden is not vegetables?
- 9.A charity bake sale sells 187 cakes at £2.95 each. By rounding each number to 1 significant figure, work out an estimate for the total amount raised.
- 10.Work out the value of √64.
- 11.Sara has a red T-shirt, a blue T-shirt and a green T-shirt. She has black shorts and grey shorts. By listing the combinations systematically, work out how many different outfits of one T-shirt and one pair of shorts she can make.
- 12.Round 24,681 to the nearest 1,000.
- 13.A car travels 180 km using 6 litres of fuel. Work out the car's fuel consumption in kilometres per litre, then work out how many kilometres it can travel on a full tank of 12 litres at this rate.
- 14.In standard form, 2,000 is written as 2 × 10ⁿ. Write down the value of n.
- 15.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.
Answer key
- (c) < — Compare the two decimals by their value, not by how many digits they have: 0.45 is worth less than half, while 0.5 is exactly half, so 0.45 is smaller. The correct symbol is <, since 0.45 is less than 0.5. Choosing > treats 0.45 as bigger because it has more digits after the decimal point than 0.5 — extra decimal digits do not make a number bigger. Choosing = comes from rounding 0.45 to 1 decimal place, 0.5, and then treating the rounded value as if it were the original number. Choosing ≥ would mean 0.45 is greater than or equal to 0.5, which is false in both parts, since 0.45 is neither equal to nor bigger than 0.5. So 0.45 < 0.5.
- (b) 0 > −1 — Method: test each statement by placing both of its numbers on a number line; the greater number is the one further to the right. Working: every negative number lies to the left of zero, so zero is greater than −1. Among the negatives, −3 lies to the right of −5, so −5 is not greater than −3. Answer: 0 > −1. The distractors: −5 > −3 comes from ordering the negatives by the size of their digits, so that 5 makes −5 look the larger; −3 > 0 comes from ignoring the minus sign and comparing 3 with 0; 0 < −1 comes from the belief that zero is the smallest number there is, so that even a negative number is above it.
- (d) 1/4 — Method: count the digits after the decimal point to choose a denominator that is a power of ten, then cancel the fraction by the highest common factor of the numerator and the denominator. Working: 0.25 has two digits after the point, so 0.25 is 25 hundredths and can be written as 25/100; the highest common factor of 25 and 100 is 25, and 25 ÷ 25 = 1 with 100 ÷ 25 = 4. Answer: 1/4. The distractors: 2/5 comes from reading the two digits after the point as a numerator and a denominator and writing 2 over 5; 1/25 comes from putting 1 over the digits after the point, as though 0.25 meant one twenty-fifth; 1/40 comes from miscounting the decimal places and using a denominator of 1000, giving 25/1000, which cancels by 25 to 1/40.
- (d) −1.4, −6/5, 0, 5/4, 1.3 — Method: convert the fractions 5/4 and −6/5 to decimals so every number is written the same way, then compare all five decimals. Working: 5/4 = 1.25 and −6/5 = −1.2. Comparing −1.4, −1.2, 0, 1.25 and 1.3 in size gives the order −1.4, −1.2, 0, 1.25, 1.3. Answer: −1.4, −6/5, 0, 5/4, 1.3. −6/5, −1.4, 0, 5/4, 1.3 swaps the two negative numbers, treating −6/5 as more negative than −1.4 even though −1.2 is closer to zero than −1.4. 1.3, 5/4, 0, −6/5, −1.4 lists the numbers from largest to smallest instead of smallest to largest. −1.4, −6/5, 0, 1.3, 5/4 swaps 5/4 and 1.3, comparing the numerator 5 directly with 1.3 instead of converting 5/4 to the decimal 1.25 first.
- (c) 11 — Method: work out the total number of combinations as if there were no restriction, then subtract the one combination that is not allowed. Working: without any restriction there are 4 backdrops × 3 outfits = 12 combinations. The grey backdrop with the formal suit is not allowed, removing 1 combination: 12 − 1 = 11. Answer: 11. 12 comes from forgetting to remove the combination that is not allowed. 8 comes from removing the entire formal suit outfit from the count instead of just the one combination with the grey backdrop. 10 comes from removing two combinations instead of just the one that is not allowed.
- (c) 90 — Multiply the number of choices for each course: 5 × 6 × 3 = 90. 14 comes from adding the three numbers instead of multiplying them. 30 comes from multiplying only the starters and mains, 5 × 6, and forgetting the dessert. 18 comes from multiplying only the mains and desserts, 6 × 3, and forgetting the starter.
- (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.
- (d) 5/9 — Total parts = 2 + 3 + 4 = 9. Flowers and herbs together make up 2 + 3 = 5 parts, so the fraction that is not vegetables is 5/9. 4/9 comes from finding the fraction of vegetables instead of the fraction that is not vegetables. 5/7 comes from leaving flowers out of the total and using only herbs and vegetables, 3 + 4 = 7, as the total. 3/4 comes from writing the ratio of herbs to vegetables directly as a fraction.
- (a) £600 — Method: round the number of cakes and the price of each cake to 1 significant figure, then multiply the rounded values. Working: 187 rounds to 200, and £2.95 rounds to £3 (the digit after the first, 9, rounds the 2 up to 3), so the estimate is 200 × £3 = £600. £400 comes from rounding £2.95 down to £2 instead of up to £3, giving 200 × £2 = £400. £561 comes from rounding only the price and using the exact number of cakes, 187 × £3 = £561. £570 comes from rounding 187 to the nearest 10 as 190 instead of to 1 significant figure as 200, giving 190 × £3 = £570. Answer: £600.
- (d) 8 — Method: the square root of a number is the value that multiplies by itself to give that number. Working: 8 × 8 = 64, so √64 = 8. 32 comes from working out 64 ÷ 2 = 32, halving the number instead of finding its square root. 4 is the cube root of 64, since 4 × 4 × 4 = 64, not the square root. 16 comes from working out 8 × 2 = 16, doubling the correct answer by mistake. Answer: 8.
- (c) 6 — Listing systematically: red-black, red-grey, blue-black, blue-grey, green-black, green-grey gives 6 different outfits, matching 3 × 2 = 6. 5 comes from adding the number of T-shirts and shorts instead of multiplying them. 3 comes from counting only the T-shirts. 2 comes from counting only the pairs of shorts.
- (d) 25,000 — Method: to round to the nearest 1,000, look at the hundreds digit; 5 or more sends the thousands digit up, less than 5 leaves it where it is, and every digit below the thousands becomes zero. Working: 24,681 has 4 in the thousands place and 6 in the hundreds place. As 6 is 5 or more, the 4 thousands go up to 5 thousands and the hundreds, tens and units are replaced by zeros. Answer: 25,000. The distractors: 24,000 comes from cutting the last three digits off instead of rounding, which leaves the thousands digit untouched; 24,700 is 24,681 rounded to the nearest 100, a finer degree of accuracy than the question asks for; 20,000 is 24,681 rounded to the nearest 10,000, a coarser degree of accuracy.
- (d) 360 km — Method: first find the kilometres per litre by dividing distance by fuel used, then multiply this rate by the new tank size. Working: 180 ÷ 6 = 30 km per litre; 30 × 12 = 360 km. Answer: 360 km. 30 km comes from finding the correct fuel consumption but stopping there, without scaling it up to the full tank. 2160 km comes from multiplying the original distance (180) by the tank size (12) directly, skipping the unit rate. 90 km comes from pairing the numbers the wrong way round: dividing the distance by the new tank size, 180 ÷ 12 = 15, and then multiplying by the original 6 litres, 15 × 6 = 90.
- (c) 3 — Method: the index counts how many times the coefficient has been multiplied by 10, which is the number of places the decimal point moves from the end of the number to just after the first significant digit. Working: 2,000 = 2 × 1,000, and 1,000 = 10 × 10 × 10, which is three tens. Answer: 3. The distractors: 4 comes from counting the four digits of 2,000 rather than the three places the decimal point moves; 2 comes from copying the coefficient 2 into the index; −3 comes from making the index negative, which would describe a number smaller than 1 rather than two thousand.
- (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.
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