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.
Non-calculator
Number worksheet — GCSE Higher
MathsUKwww.geekhero.co.uk
- 1.A lorry travels 105 miles in 1 hour 45 minutes. Work out its average speed in miles per hour.
- 2.At a bus station one bus leaves every 6 minutes and another leaves every 9 minutes. The two buses leave together at the start of the day. Work out how long it is until they next leave together.
- 3.A bag holds 5 different sweets. Noah takes 2 of the sweets out of the bag together, so the order in which he takes them does not matter. Work out how many different pairs of sweets he could take.
- 4.Four students each estimate √70 without a calculator. Ben says: '8² = 64 and 9² = 81, and 70 is about a third of the way from 64 to 81, so √70 ≈ 8.3.' Chloe says: '8.5² = 72.25, which is close to 70 but a little too big, so I'll bring it down slightly to √70 ≈ 8.4.' Dan says: '9² = 81 is closer to 70 than 8² = 64 is, so √70 ≈ 8.9.' Ella says: '8² = 64 and 9² = 81, so I'll just take the midpoint of 8 and 9: √70 ≈ 8.5.' Given that √70 = 8.3666... to 4 decimal places, whose estimate is closest?
- 5.Write 0.325 as a fraction in its simplest form.
- 6.A rectangular patio measures 90 cm by 120 cm. Ben wants to cover it exactly with identical square tiles, as large as possible, with no tiles cut. Work out the side length of the largest square tile he can use.
- 7.A tank contains 120 litres of water. Water is drained out at a rate of 8 litres per minute for 6 minutes, and then a hose adds 15 litres. Work out how much water is left in the tank.
- 8.Work out (2/3)² + (1/2)². Give your answer as a fraction.
- 9.The density of a type of plastic is 0.9 g/cm³. Work out this density in kg/m³.
- 10.A plank has length a = 12 cm and a second plank has length b = 7 cm, each correct to the nearest centimetre. Work out the upper bound of a + b.
- 11.Insert one pair of brackets into 2 + 3 × 5 − 1 so that the calculation is equal to 24. Which calculation is correct?
- 12.Light travels at 3 × 10⁸ metres per second. Work out how far light travels in 2 × 10⁻⁶ seconds. Give your answer in standard form.
- 13.Write 0.06 as a fraction in its simplest form.
- 14.Work out ((−3) + 5) × (−4) − (−6) ÷ 2
- 15.Two bags of mixed nuts are combined. Bag A has peanuts and cashews in the ratio 2 : 3. Bag B has peanuts and cashews in the ratio 1 : 4. Both bags contain the same total number of nuts. Work out the fraction of the combined mixture that is peanuts.
Answer key
- (d) 60 mph — Average speed = distance ÷ time, with the time measured in hours. 45 minutes is 45/60 of an hour, which is 0.75 of an hour, so the journey takes 1.75 hours. Speed = 105 ÷ 1.75 = 60 mph. 52.5 mph rounds the time up to 2 hours, 72.4 mph writes 1 hour 45 minutes as 1.45 hours, and 183.75 mph multiplies the distance by the time instead of dividing.
- (a) 18 minutes — Method: the buses leave together again after a number of minutes that is a multiple of both intervals, and the first such time is the lowest common multiple. Working: the multiples of 6 are 6, 12, 18, 24 … and the multiples of 9 are 9, 18, 27 … The first value in both lists is 18, which is 6 × 3 and 9 × 2. Answer: 18 minutes. The distractors: 54 minutes comes from multiplying 6 by 9, which does give a common multiple but not the lowest one; 3 minutes is the highest common factor of 6 and 9 rather than their lowest common multiple; 15 minutes comes from adding the two intervals together.
- (d) 10 — Method: list the pairs systematically, taking each sweet in turn and pairing it only with the sweets that come after it, so that no pair is written down twice. Working: numbering the sweets 1 to 5, the first sweet pairs with 4 others, the second pairs with 3 sweets that come after it, the third with 2 and the fourth with 1, so the total is 4 + 3 + 2 + 1 = 10. Answer: 10. The distractors: 20 comes from working out 5 × 4 and never halving, which counts each pair twice, once in each order; 25 comes from working out 5 × 5, which allows the same sweet to be chosen twice; 9 comes from adding the 5 choices and the 4 remaining choices instead of combining them as a selection of two.
- (b) Chloe's estimate — √70 = 8.3666... to 4 decimal places. Comparing each estimate against this: Ben's 8.3 is 0.0666 away; Chloe's 8.4 is only 0.0334 away; Dan's 8.9 is 0.5334 away; Ella's 8.5 is 0.1334 away. Chloe's estimate is closest, because her method tests an actual calculation, 8.5² = 72.25, sees that it overshoots 70, and corrects slightly downward from it, rather than only comparing which perfect square is nearer. Ben's sharing-out is not a bad idea in itself — 70 sits 6 of the way along the 17 from 64 to 81, so 'about a third of the way' from 8 to 9 points at roughly 8.35 — but he then rounds that position down to 8.3, and it is that rounding, not the sharing-out, that leaves him twice as far from √70 as Chloe. Dan's claim that 81 is closer to 70 than 64 is is backwards: 70 − 64 = 6, while 81 − 70 = 11, so 64 is in fact the nearer square, which makes his estimate of 8.9 the furthest from the truth of all four. Ella's plain midpoint of 8 and 9 tests nothing at all: √70 does not sit halfway between 8 and 9, and her 8.5 lands further from √70 than Chloe's checked estimate does.
- (a) 13/40 — Method: write the decimal over 1000 using its three decimal places, then simplify. Working: 0.325 = 325/1000 = 13/40 (dividing both numerator and denominator by 25). Answer: 13/40. 13/4 comes from writing the decimal over 100 instead of 1000, as if there were only two decimal places. 8/25 comes from rounding 0.325 down to 0.32 before converting. 3/8 comes from recalling the learned conversion 3/8 = 0.375 and matching it to 0.325 because both are three-place decimals beginning with 3, instead of converting the decimal given.
- (a) 30 cm — The tile's side length must be a common factor of 90 and 120. The factors of 90 include 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90; the factors of 120 include 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. The highest number common to both lists is 30, so the largest square tile has a side length of 30 cm. Picking 15 cm, a common factor but not the largest, gives tiles that are smaller than necessary. Picking 10 cm, also a common factor but smaller still, wastes even more of the possible tile size. Working out the lowest common multiple instead of the highest common factor gives 360 cm, a length far bigger than either side of the patio. So the largest square tile Ben can use has a side length of 30 cm.
- (c) 87 litres — Work out how much water is drained: 8 × 6 = 48 litres. Subtract this from the starting amount: 120 − 48 = 72 litres. Then add the 15 litres from the hose: 72 + 15 = 87 litres. Subtracting the 15 litres instead of adding it, as though the hose also removed water, gives 120 − 48 − 15 = 57 litres. Stopping after the drain step, without adding the hose water back in, leaves the working at 72 litres. Adding the rate and the time instead of multiplying them, 8 + 6 = 14 litres drained, and then working from there gives 120 − 14 + 15 = 121 litres. So 87 litres of water is left in the tank.
- (c) 25/36 — Method: square a fraction by squaring its numerator and its denominator separately, then add the two results over a common denominator. Working: (2/3)² = 4/9 and (1/2)² = 1/4; the lowest common denominator of 9 and 4 is 36, so 4/9 = 16/36 and 1/4 = 9/36, and 16 + 9 = 25. Answer: 25/36. The distractors: 49/36 comes from adding the two fractions first and squaring the total, giving (7/6)²; 5/13 comes from squaring correctly but then adding the numerators and the denominators, as (4 + 1)/(9 + 4); 7/3 comes from doubling each fraction instead of squaring it, giving 4/3 + 1.
- (a) 900 kg/m³ — Both parts of the compound unit change. There are 100 cm in a metre, so 1 m³ = 100³ = 1 000 000 cm³, and there are 1000 g in a kilogram. So 1 g/cm³ = 1 000 000 ÷ 1000 = 1000 kg/m³, and the plastic is 0.9 × 1000 = 900 kg/m³. 900 000 kg/m³ converts the volume but leaves the mass in grams, 0.0009 kg/m³ converts the mass but leaves the volume in cm³, and 0.09 kg/m³ uses the 100 cm in a metre without cubing it.
- (b) 20 — Each length has its own error interval: 11.5 ≤ a < 12.5 and 6.5 ≤ b < 7.5. The upper bound of a sum is found by adding the upper bounds of both quantities: 12.5 + 7.5 = 20. Bounding only one of the two lengths and adding the other quantity's given value unbounded, 12.5 + 7 = 19.5, misses that both measurements carry their own uncertainty. Adding the lower bounds instead of the upper bounds, 11.5 + 6.5 = 18, gives the lower bound of the sum, not the upper one. Using a whole centimetre of error either side instead of half a centimetre, (12 + 1) + (7 + 1) = 21, comes from forgetting the error is half the rounding unit.
- (b) (2 + 3) × 5 − 1 — 2 + 3 = 5, then 5 × 5 = 25, then 25 − 1 = 24, so the brackets belong around 2 + 3. Placing them around 5 − 1 instead gives 5 − 1 = 4, then 3 × 4 = 12, then 2 + 12 = 14. Leaving the multiplication bracketed instead changes nothing, because it already had priority: 3 × 5 = 15, then 2 + 15 = 17, then 17 − 1 = 16. Bracketing both 2 + 3 and 5 − 1 uses two pairs instead of the one asked for: 2 + 3 = 5, 5 − 1 = 4, then 5 × 4 = 20.
- (a) 6 × 10² metres — Method: distance = speed × time, so multiply the coefficients and add the indices. Working: 3 × 2 = 6 for the coefficients, and 8 + (−6) = 2 for the indices; 6 lies between 1 and 10, so the coefficient needs no adjustment. Answer: 6 × 10² metres, which is 600 metres. The distractors: 5 × 10² metres comes from adding the coefficients, 3 + 2, instead of multiplying them; 6 × 10¹⁴ metres comes from subtracting the indices, 8 − (−6), which is the rule for dividing rather than for multiplying; 6 × 10⁻⁴⁸ metres comes from multiplying the indices, 8 × (−6), instead of adding them.
- (b) 3/50 — Method: write the decimal over the power of ten that matches the number of digits after the point, counting every digit including a zero, then divide the numerator and the denominator by their highest common factor. Working: 0.06 has two digits after the point, so it is 6 hundredths and can be written as 6/100; the highest common factor of 6 and 100 is 2, and 6 ÷ 2 = 3 with 100 ÷ 2 = 50. Answer: 3/50. The distractors: 3/5 comes from ignoring the zero straight after the point and converting 0.6 instead, giving 6/10, which cancels to 3/5; 3/500 comes from counting three decimal places instead of two and writing 6/1000, which cancels to 3/500; 1/6 comes from putting 1 over the digits after the point, as though 0.06 meant one sixth.
- (c) −5 — Method: the bracket is worked out first, then the multiplication and the division, which stand as separate parts, and the subtraction that joins them is carried out last; subtracting a negative is the same as adding. Working: (−3) + 5 = 2, so the product is 2 × (−4) = −8; the division gives (−6) ÷ 2 = −3; joining them gives −8 − (−3) = −8 + 3 = −5. Answer: −5. The distractors: −11 comes from taking away 3 instead of taking away −3, giving −8 − 3 = −11; −1 comes from working from left to right once the bracket is done, giving −8 − (−6) = −2 and then −2 ÷ 2 = −1; 11 comes from treating the first product as positive because it was worked out from a bracket, giving 8 − (−3) = 11.
- (b) 3/10 — Since each bag's ratio has 5 parts and both bags contain the same total number of nuts, imagine each bag has 5 nuts: Bag A has 2 peanuts and Bag B has 1 peanut, so together there are 2 + 1 = 3 peanuts out of a combined 5 + 5 = 10 nuts, giving 3/10. 1/5 comes from using only Bag A's peanuts, 2 out of 10, without adding Bag B's peanuts. 1/10 comes from using only Bag B's peanut, without adding Bag A's peanuts. 3/5 comes from writing the combined peanuts over the number of parts in one bag instead of the combined total number of nuts.
Build your own mix at the worksheet builder.