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.
Number worksheet — GCSE Higher
MathsUKwww.geekhero.co.uk
- 1.Work out √225 ÷ 3.
- 2.Simplify x⁽³⁄⁴⁾ ÷ x⁽¹⁄⁴⁾
- 3.A board game has 5 different character pieces and 4 different colour tokens. The dragon piece can only be used with the gold token. Work out how many different combinations of one character piece and one colour token are possible.
- 4.A digital timer truncates every time to 1 decimal place. It shows a swimmer's time for one length as 12.3 seconds. Using t for the swimmer's actual time in seconds, write down the error interval for t.
- 5.Work out (5 + 2) × 3²
- 6.Simplify 3² ÷ 3⁵, giving your answer as a single power of 3.
- 7.A quantity surveyor calculates a length as 24.6851 m. Round this length to 2 decimal places.
- 8.A lift has a safe working load of 500 kg. Four people get in the lift and the lift's display records their total mass as 493 kg, correct to the nearest kg. Decide whether the four people are definitely within the safe working load.
- 9.A café meal deal is one sandwich, one snack and one drink. There are 4 different sandwiches, 3 different snacks and 2 different drinks to choose from. Work out how many different meal deals are possible.
- 10.In a music collection, the ratio of pop albums to rock albums is 5 : 3. Work out the number of rock albums as a fraction of the number of pop albums.
- 11.Simplify (3a²)² × (2a)³
- 12.A charity raises money from a raffle and a cake sale in the ratio 5 : 3. Altogether the charity raises £320. Work out how much money the cake sale raised.
- 13.Two-digit numbers are formed using the digits 2, 5, 7 and 8, and each digit may be used only once in a number. Work out how many of these two-digit numbers are even.
- 14.A circle has a radius of 3 cm. Which of these is the exact area of the circle?
- 15.Ten players enter a chess tournament. Every player plays every other player exactly once. Work out how many games are played in the tournament.
Answer key
- (b) 5 — Method: find the square root first, then divide. Working: √225 = 15, and 15 ÷ 3 = 5. Answer: 5. (75 comes from dividing 225 by 3 first and forgetting to take the square root at all. 8.7 comes from dividing 225 by 3 inside the root, √(225 ÷ 3) ≈ 8.7, instead of taking the root first. 45 comes from misreading the divisor as 5 instead of 3, working out 225 ÷ 5 = 45.)
- (d) x⁽¹⁄²⁾ — Method: dividing two powers of the same letter subtracts the index of the divisor from the index of the term being divided, and fractional indices are subtracted like any other fractions. Working: 3/4 − 1/4 = 2/4, which simplifies to 1/2, so the result is x⁽¹⁄²⁾. Answer: x⁽¹⁄²⁾. The distractors: x comes from adding the indices, 3/4 + 1/4 = 1, as though the powers were being multiplied; x³ comes from dividing the indices, so that 3/4 divided by 1/4 gives 3; x⁽³⁄¹⁶⁾ comes from multiplying the indices, 3/4 × 1/4.
- (d) 17 — Without the restriction there would be 5 × 4 = 20 combinations. The dragon piece can only be paired with the gold token, so of the 4 tokens, 3 are not allowed with the dragon piece, giving 20 − 3 = 17 valid combinations. 20 comes from ignoring the restriction completely. 19 comes from subtracting only 1 of the 3 invalid dragon combinations instead of all 3, 20 − 1 = 19. 16 comes from multiplying only the 4 non-dragon pieces by the 4 tokens, 4 × 4 = 16, and forgetting to add back the one valid combination of the dragon piece with the gold token.
- (c) 12.3 ≤ t < 12.4 — Method: truncating cuts the later digits off instead of rounding them, so nothing is ever pushed upwards. The displayed value is therefore the smallest the time can be, and the time can run up to, but not reach, the next value the display can show. Working: the display reads 12.3, so the actual time is at least 12.3 seconds; as soon as the time reaches 12.3 + 0.1 = 12.4 seconds the display would read 12.4, so 12.4 is not included. Answer: 12.3 ≤ t < 12.4. The distractors: 12.25 ≤ t < 12.35 is the interval for a time rounded to 1 decimal place, and this display does not round; 12.3 < t ≤ 12.4 excludes the one value the display certainly allows and includes the one it rules out; 12.3 ≤ t ≤ 12.4 treats 12.4 seconds as possible, but at 12.4 seconds the display would no longer read 12.3.
- (b) 63 — 5 + 2 = 7, then 3² = 9, then 7 × 9 = 63. Ignoring the brackets and applying BIDMAS as if the expression were unbracketed gives 3² = 9, then 2 × 9 = 18, then 5 + 18 = 23. Squaring the bracket instead of the 3 gives 7² = 49, then 49 × 3 = 147 — the power belongs to the 3 alone. Multiplying by 3 before squaring the whole product gives 7 × 3 = 21, then 21² = 441.
- (c) 3⁻³ — Method: when dividing powers of the same base, subtract the index of the number you are dividing by from the index of the number being divided, keeping them in the order the question writes them. Working: 2 − 5 = −3, so 3² ÷ 3⁵ = 3⁻³. It is worth checking this against the numbers: 3² = 9 and 3⁵ = 243, and 9 ÷ 243 = 1/27, which is 3⁻³. 3³ comes from subtracting the other way round, 5 − 2 = 3, which reverses the sign of the index and gives 27 instead of 1/27. 3⁷ comes from working out 2 + 5 = 7, which is the rule for multiplying powers, not dividing them. 3¹⁰ comes from multiplying the indices, 2 × 5 = 10, instead of subtracting them. Answer: 3⁻³.
- (d) 24.69 — To round to 2 decimal places, look only at the third decimal digit to decide whether the second decimal digit rounds up. In 24.6851 the third decimal digit is 5, and since 5 is 5 or more, the second decimal digit rounds up from 8 to 9, giving 24.69. Rounding to 1 decimal place instead of 2 gives 24.7, one place value too coarse. Keeping the third decimal digit rather than dropping it gives 24.685, which is 3 decimal places. Looking at the fourth decimal digit, 1, instead of the third one, and wrongly deciding that no rounding is needed, leaves the length unrounded at 24.68.
- (c) Yes — the greatest possible total is 493.5 kg, under 500 kg — 493 kg correct to the nearest kg means the true total mass, m, satisfies 492.5 kg ≤ m < 493.5 kg. The greatest possible total is 493.5 kg, which is under the 500 kg safe working load, so the four people are definitely within it. 'The true total could be as high as 498 kg' comes from treating 'nearest kg' as an error of ±5 kg instead of ±0.5 kg. 'Cannot be decided without the exact total' overlooks that the error interval already gives the greatest possible total, so the decision can be made without knowing the exact figure. '493 kg is only an estimate, so it may be over 500 kg' ignores that the error interval is bounded — the true total cannot exceed 493.5 kg, well under 500 kg.
- (d) 24 — Method: the three parts of the deal are chosen independently, so every sandwich can be taken with every snack and every one of those pairs with every drink; the product rule multiplies the number of choices in each part. Working: there are 4 choices of sandwich and each can be taken with any of the 3 snacks, giving 4 × 3 = 12 sandwich-and-snack pairs; each of those pairs can be completed with either of the 2 drinks, so the number of meal deals is 12 × 2 = 24. Answer: 24. The distractors: 9 comes from adding the choices, 4 + 3 + 2, instead of multiplying them, and a candidate who adds writes that total down as the count; 12 comes from multiplying the sandwiches by the snacks and never bringing the drink into the count at all; 27 comes from adding the items on the menu to get 9 and then multiplying that by the 3 parts of the deal, which counts the menu rather than the combinations.
- (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.
- (d) 72a⁷ — Method: a power outside brackets applies to every factor inside them, and multiplying two powers of the same letter adds their indices. Working: (3a²)² = 3² × a⁴ = 9a⁴, and (2a)³ = 2³ × a³ = 8a³. Multiplying the two results gives 9 × 8 = 72 for the number and 4 + 3 = 7 for the index of a. Answer: 72a⁷. The distractors: 36a⁷ comes from squaring the 2 in (2a)³ instead of cubing it, giving 4a³ and then 9 × 4; 72a¹² comes from multiplying the indices 4 and 3 when the two terms are multiplied, instead of adding them; 17a⁷ comes from adding the coefficients 9 and 8 rather than multiplying them.
- (d) £120 — Total parts = 5 + 3 = 8, so one part is worth 320 ÷ 8 = 40 pounds. The cake sale is 3 parts, so it raised 3 × 40 = 120 pounds. £200 comes from working out the raffle's share, 5 × 40, instead of the cake sale's share. £40 comes from finding the value of one part but forgetting to multiply by 3. £192 comes from dividing the total by 5 instead of 8 to find the value of one part, 320 ÷ 5 = 64, then multiplying by 3, 3 × 64 = 192.
- (c) 6 — The units digit must be even, so it can be 2 or 8, giving 2 choices. The tens digit can then be any of the remaining 3 digits, since one digit has been used for the units. Multiply: 2 × 3 = 6. 12 comes from working out how many two-digit numbers can be made in total, 4 × 3 = 12, ignoring the requirement that the number is even. 8 comes from choosing the units digit from 2 options and then wrongly allowing any of the 4 digits again for the tens digit, 2 × 4 = 8, which lets a digit repeat. 2 comes from counting only the choices for the units digit and forgetting the tens digit.
- (c) 9π cm² — The area of a circle is π × r². With a radius of 3 cm this is π × 3² = 9π cm², and this is exact because π has not been replaced by any approximation. Writing 28.3 cm² replaces π with a rounded decimal value, 3.14, and then rounds the result again, so it is only an approximation. Writing 28.26 cm² uses π ≈ 3.14 without a final rounding step, but this is still only an approximation of 9π, not the exact value. Writing 27 cm² comes from replacing π with the rough approximation 3, which is even further from the true value.
- (c) 45 — Method: count the ordered pairings with the product rule and then correct for the fact that a game between two players is the same game whichever player it is counted from. Working: each of the 10 players meets 9 opponents, so 10 × 9 = 90 pairings are counted; every game has been counted twice, once from each player's side, so the number of games is 90 ÷ 2 = 45. Answer: 45. The distractors: 90 comes from stopping at 10 × 9 and never halving, so that each game is counted once for each of its two players; 55 comes from adding 10 + 9 + 8 + ... + 1 instead of 9 + 8 + ... + 1, which counts one extra round of games; 20 comes from multiplying the 10 players by the 2 players in each game rather than pairing the players with one another.
Build your own mix at the worksheet builder.