Printable · GCSE Higher · ages 14-16
Number worksheet — GCSE Higher
Fifteen questions across the number statements at Higher tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Answer key: Number worksheet — GCSE Higher
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- (a) 8,350 — Rounding to the nearest 100 means the true number can be up to half of 100, which is 50, below the recorded figure before it would round down to a lower hundred. The smallest possible number is therefore 8,400 − 50 = 8,350. Adding 50 instead of subtracting it gives 8,450, which is the upper end of the interval rather than the smallest value, and 8,450 is not itself possible because it would round up to 8,500. Subtracting a whole 100 instead of half of it gives 8,300, going too far below the recorded value. Subtracting 10 instead of half of the rounding unit gives 8,390, treating the rounding unit as 100 but the tolerance as only 10.
- (d) £6 — First find 30% of £200, which is £60, then find 10% of that: £60 × 0.1 = £6. Adding the two percentages together instead of applying them one after the other, 10% + 30% = 40%, and finding 40% of £200 gives £80. Finding 30% of £200 = £60 correctly but stopping before applying the second percentage leaves £60 as the final answer. Finding only 10% of the original £200, ignoring the 30% entirely, gives £20.
- (c) 2700 cm² — Convert both lengths to the same unit before multiplying: 0.6 m = 60 cm. The area is 45 × 60 = 2700 cm². Multiplying without converting 0.6 m to centimetres at all, treating it as 0.6 cm, gives 45 × 0.6 = 27 cm². Converting 0.6 m to centimetres by multiplying by 10 instead of 100 gives 6 cm, and 45 × 6 = 270 cm². Converting 0.6 m to centimetres by multiplying by 1000 instead of 100 gives 600 cm, and 45 × 600 = 27000 cm².
- (d) 6 + 2√3 — Multiply √3 by each term in the bracket separately. First term: √3 × 2 = 2√3. Second term: √3 × √12 = √(3 × 12) = √36 = 6. Adding the two results in the order they were found, and writing the whole-number term first, gives 6 + 2√3. Adding the numbers under the root for the second term instead of multiplying them (3 + 12 = 15) gives √15 in place of 6, leading to √15 + 2√3. Multiplying √3 by the 2 but never distributing to the √12 term at all leaves just 2√3. Treating √3 × 2 as if the 3 were multiplied by the 2 inside the root, √3 × 2 → √6, while still getting the second term correct, gives 6 + √6.
- (c) An under-estimate, by 8 — Method: work out the exact product, then compare it with the estimate; an estimate that is smaller than the exact value is an under-estimate, and the difference between them is the size of the error. Working: 48 × 21 = 48 × 20 + 48 = 960 + 48 = 1,008, and 1,008 − 1,000 = 8, so the estimate falls short. Answer: an under-estimate, by 8. The distractors: an over-estimate by 8 has the size of the error right but the direction wrong, and comes from assuming that rounding 48 up to 50 must push the estimate above the exact value, without allowing for 21 being rounded down; an over-estimate by 19 comes from working out 48 × 21 as 48 × 20 + 21 = 981, adding a 21 where another 48 belongs; the claim that the estimate is exactly right comes from arguing that one number was rounded up and the other down, so the two changes must cancel.
- (c) −28 — 6² = 36 and 4³ = 64. Work out 36 − 64 = −28. A candidate who subtracts in the wrong order gets 64 − 36 = 28. A candidate who adds instead of subtracting gets 36 + 64 = 100. A candidate who multiplies the base by the exponent instead of raising the power (6 × 2 − 4 × 3 = 12 − 12) gets 0.
- (b) £13.48 — Method: an amount of money is written to the nearest penny, which is 2 decimal places, so the calculator display has to be rounded to 2 decimal places. Working: 53.90 ÷ 4 = 13.475, and the digit in the third decimal place is 5, so the penny digit goes up from 7 to 8. Answer: £13.48. The distractors: £13.47 comes from chopping the third decimal place off instead of rounding with it; £13.50 comes from rounding to the nearest 10p rather than to the nearest penny; £13.40 comes from cutting the display short at 1 decimal place, which is both the wrong degree of accuracy and a truncation rather than a rounding.
- (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).
- (b) 30 — Method: the smallest matching total is the lowest common multiple of the two pack sizes. Working: multiples of 6 are 6, 12, 18, 24, 30 …; multiples of 10 are 10, 20, 30 …. The lowest common multiple is 30. 60 comes from working out 6 × 10 = 60, the product of the pack sizes rather than their lowest common multiple. 16 comes from working out 6 + 10 = 16, which is not a common multiple at all. 2 is the highest common factor of 6 and 10, not a total of tickets. Answer: 30.
- (a) 3/5 — The question compares the rock albums directly with the pop albums, so write the rock part over the pop part: 3/5. 5/3 comes from writing the ratio the wrong way round, comparing pop with rock instead of rock with pop. 3/8 comes from comparing the rock albums with the total number of albums, 3 + 5 = 8, instead of with the pop albums. 5/8 comes from comparing the pop albums with the total number of albums instead of the rock albums with the pop albums.
- (a) 33.2 — The radius was rounded to 1 decimal place, so its error interval is 3.15 ≤ r < 3.25. The upper bound for the area uses the upper bound of the radius, squared: area = π × 3.25² ≈ 33.183, which rounds to 33.2 m² (3 s.f.). Using the given value of the radius directly instead of its upper bound, π × 3.2² ≈ 32.2, ignores that the radius itself has a range of possible values. Bounding the radius correctly but forgetting to square it, using area = π × 3.25 ≈ 10.2 instead of π × 3.25², drops the whole squaring step from the area formula. Using the LOWER bound of the radius instead of the upper one, π × 3.15² ≈ 31.2, finds the lower bound of the area, not the upper one.
- (a) 10 °C — Method: work out the coldest and warmest of the four temperatures, then subtract to find the difference. Working: the coldest temperature is Sam's, −9 °C, and the warmest is Alex's, 1 °C. The difference is 1 − (−9) = 1 + 9 = 10. Answer: 10 °C. 8 °C comes from working out 1 − 9 = −8 and reporting 8, dropping the negative sign on −9 instead of turning the subtraction into an addition. 3 °C comes from comparing the wrong pair, Sam's −9 °C and Ben's −6 °C, instead of the coldest and the warmest: −6 − (−9) = 3. 7 °C comes from comparing Ben's −6 °C with Alex's 1 °C, mistakenly treating Ben's reading as the coldest instead of Sam's.
- (c) 0.5, 0.45, −0.09, −0.15, −0.6 — Method: compare the decimals by their position on a number line, remembering that with negative decimals the one closer to zero is larger. Working: 0.5 and 0.45 are positive, so they come first, with 0.5 the larger of the two. Among the negatives, −0.09 is closest to zero, then −0.15, then −0.6 is furthest from zero and so the smallest. Answer: 0.5, 0.45, −0.09, −0.15, −0.6. 0.5, 0.45, −0.15, −0.09, −0.6 swaps −0.09 and −0.15, treating the negative decimal with more digits after the point as closer to zero. −0.6, −0.15, −0.09, 0.45, 0.5 lists the numbers from smallest to largest instead of largest to smallest. 0.5, 0.45, −0.6, −0.15, −0.09 orders the negative decimals by the size of the digit (0.6 > 0.15 > 0.09) as if they were positive, instead of recognising that a bigger negative decimal is further from zero and so smaller.
- (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.
- (a) 18 — Method: round each number to the nearest whole number, then square each rounded number and add the results. Working: 2.9 rounds to 3 and 3.1 rounds to 3, so the estimate is 3² + 3² = 9 + 9. Answer: 18. The distractors: 36 comes from adding before squaring, working out (3 + 3)² instead of 3² + 3²; 12 comes from doubling each rounded number instead of squaring it, adding 6 and 6; 6 comes from adding the two rounded numbers and forgetting to square them at all.
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