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.In a school long jump competition Priya's jump is recorded as 3.8 m and Nadia's jump is recorded as 3.9 m, each correct to the nearest 0.1 m. Priya says she may have jumped further than Nadia. Decide whether Priya is right, and give a reason for your answer.
- 2.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.
- 3.Oliver drives 95 km at an average speed of 50 km/h. Work out an estimate for the time the journey takes, by rounding the distance to the nearest 100 km.
- 4.Work out the value of .
- 5.Two fair six-sided dice are rolled and their scores are added together. By listing the possible totals systematically, work out how many different totals are possible.
- 6.A garden path is measured by Jon as 12 m, correct to the nearest metre, and by Mia as 12.6 m, correct to the nearest 0.1 m. Which statement about the two measurements is correct?
- 7.A bag holds 50 counters. 2/5 of them are red and the rest are blue. 5 red counters are then taken out of the bag. Work out the new ratio of red counters to blue counters. Give your answer in its simplest form.
- 8.Work out the value of ∛125.
- 9.3.4 × 10⁵ is written as an ordinary number. Work out this number.
- 10.An allotment plot is divided into vegetables and flowers in the ratio 9 : 4. A third of the vegetable section is used for potatoes. What fraction of the whole plot is potatoes?
- 11.Work out −7 − (−3).
- 12.The rainfall in a town during April is recorded as 62.4 mm, correct to 1 decimal place. Write down the error interval for the actual rainfall, r mm.
- 13.Work out 2/3 × 3/4 exactly, giving your answer in its simplest form.
- 14.Work out the lowest common multiple of 4 and 6.
- 15.A piece of ribbon is 2.4 metres long. Kim cuts 40 cm from it. Work out the length of ribbon left, in millimetres.
Answer key
- (b) No — their possible jump lengths do not overlap — Method: each recorded jump stands for the lengths within half of 0.1 m, that is 0.05 m, of the figure recorded, and Priya is right only if the two ranges overlap. Working: Priya's jump is at least 3.8 − 0.05 = 3.75 m and below 3.85 m, because a jump of 3.85 m would have been recorded as 3.9 m; Nadia's jump is at least 3.85 m and below 3.9 + 0.05 = 3.95 m. Every length Priya could have jumped is below 3.85 m and every length Nadia could have jumped is at least 3.85 m, so Nadia jumped further whatever the exact lengths were. Answer: No — their possible jump lengths do not overlap. The distractors: the reason that a recorded jump is exactly the length jumped reaches the same verdict by treating a rounded record as exact, which is the idea this question tests; both jumps being 3.85 m would put 3.85 m inside Priya's range, when a jump of that length is recorded as 3.9 m; Priya jumping up to 3.9 m goes a whole 0.1 m above her record instead of half of it.
- (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.
- (d) 2 hours — Method: the time for a journey is the distance divided by the speed, so round the distance first and then divide by the speed. Working: 95 km rounds to 100 km, and 100 ÷ 50 = 2; the speed is in kilometres per hour, so the answer is a number of hours. Answer: 2 hours. The distractors: 1 hour comes from rounding the distance down to 50 km to match the speed, so that the journey looks like a single hour of driving; 30 minutes comes from dividing the speed by the distance, 50 ÷ 100, instead of the distance by the speed; 1 hour 54 minutes is the exact time, 95 ÷ 50 = 1.9 hours, worked out in full when the question asks for an estimate.
- (c) 0.001 — Method: a negative index means 'one over' the positive power, so $10^{-3}$ means one over $10^{3}$. Working: ten cubed is 1000, and one over 1000 is 0.001. 1000 comes from ignoring the negative sign and working out ten cubed instead of its reciprocal. −1000 comes from ignoring what the negative index does to the size, while still writing a negative sign on the large value. −0.001 comes from correctly finding the size, 0.001, but wrongly keeping a negative sign because the index was negative. Answer: 0.001.
- (a) 11 — Method: list every possible total from the smallest to the largest, and count how many different values there are. Working: the smallest total is 1+1=2 and the largest is 6+6=12, and every whole number total from 2 to 12 is possible: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 — that is 11 different totals. Answer: 11. 36 comes from counting the number of possible dice outcomes (6×6) instead of the number of different totals. 10 comes from listing the totals but missing one from the ends of the list, for example starting at 3 instead of 2. 6 comes from counting only the number of different scores on one die, not the totals of both dice together.
- (a) They cannot both be describing the same path — Jon's measurement means the true length, l, satisfies 11.5 m ≤ l < 12.5 m. Mia's measurement means the true length satisfies 12.55 m ≤ l < 12.65 m. These two ranges do not overlap, so the two measurements cannot both be describing the same path. 'They must both be describing the same path' ignores that the two ranges do not overlap at all. 'Jon's measurement must be wrong' wrongly assumes Jon is the one at fault, when the mismatch does not show which measurement, if either, is wrong. 'Mia's measurement must be wrong' makes the same unjustified assumption in the other direction.
- (b) 1 : 2 — 2/5 of 50 is 20, so there are 20 red counters and 50 − 20 = 30 blue counters. Taking 5 red counters out leaves 15 red and 30 blue, so red : blue = 15 : 30. Dividing both parts by 15 gives 1 : 2. 2 : 3 is the ratio before any counters are removed, 2 : 1 has the two parts the wrong way round, and 4 : 5 comes from taking the 5 counters out of the blue instead of the red.
- (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) 340,000 — 3.4 × 10⁵ means moving the decimal point in 3.4 five places to the right, giving 340,000. Moving it only four places gives 34,000, one place short. Moving it six places gives 3,400,000, one place too many. Treating the exponent as negative instead of positive moves the point the wrong way, giving 0.000034.
- (a) 3/13 — Vegetables are 9 of the 9 + 4 = 13 parts, so vegetables are 9/13 of the plot. Potatoes are a third of the vegetable section, so potatoes are 1/3 of 9/13, which is 9/39, simplifying to 3/13, of the whole plot. 9/13 comes from stopping after finding the fraction of the plot that is vegetables, without taking the further third for potatoes. 1/3 gives the fraction of the vegetable section that is potatoes, not the fraction of the whole plot. 4/39 comes from taking a third of the flowers' fraction, 4/13, instead of the vegetables' fraction.
- (b) −4 — To subtract a negative number, add its positive equivalent: −7 − (−3) becomes −7 + 3. Work out −7 + 3 to get −4. Treating "− (−3)" as simply "−3" without flipping the sign gives the wrong working −7 − 3, which is −10. Ignoring the negative sign on −7 and just subtracting the values, 7 − 3, gives 4, which loses the sign of the starting number. Flipping the sign of both numbers, 7 + 3, gives 10, which changes more than the double negative allows. So −7 − (−3) = −4.
- (c) 62.35 ≤ r < 62.45 — Method: with a value rounded to 1 decimal place, the error interval reaches half of 0.1 either side. Working: half of 0.1 is 0.05, so the interval runs from 62.4 − 0.05 to 62.4 + 0.05. Answer: 62.35 ≤ r < 62.45. (62 ≤ r < 63 comes from rounding to the nearest whole number instead of 1 decimal place. 62.35 ≤ r ≤ 62.45 comes from including the upper bound with ≤ instead of excluding it with <. 62.3 ≤ r < 62.5 comes from using 0.1 either side instead of half of it.)
- (b) 1/2 — To multiply fractions, multiply the numerators together and multiply the denominators together: 2 × 3 = 6 and 3 × 4 = 12, giving 6/12, which simplifies to 1/2. Adding the fractions instead of multiplying them, using a common denominator of 12, gives 8/12 + 9/12 = 17/12. Dividing by 3/4 instead of multiplying by it, so multiplying by its reciprocal 4/3, gives 2/3 × 4/3 = 8/9. Multiplying only the numerators, 2 × 3 = 6, and keeping the first denominator, 3, unchanged gives 6/3 = 2.
- (a) 12 — List the multiples of each number: multiples of 4 are 4, 8, 12, 16, 20, 24; multiples of 6 are 6, 12, 18, 24. The lowest number that appears in both lists is 12. Picking 24, a common multiple but not the lowest one, gives an answer that is too big. Picking 6, the larger of the two original numbers rather than a common multiple, ignores that the lowest common multiple must appear in both lists. Working out the highest common factor instead of the lowest common multiple gives 2. So the lowest common multiple of 4 and 6 is 12.
- (c) 2000 mm — Put both lengths into the same unit first. There are 1000 mm in a metre, so the ribbon is 2.4 × 1000 = 2400 mm, and there are 10 mm in a centimetre, so the piece cut off is 40 × 10 = 400 mm. The length left is 2400 − 400 = 2000 mm. 2360 mm subtracts 40 mm instead of 400 mm, 200 mm works in centimetres and then labels the result as millimetres, and 2800 mm adds the piece that was cut off instead of subtracting it.
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