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.At midday the temperature in the Lake District was 3 °C. By midnight it had fallen to −6 °C. Work out the fall in temperature.
- 2.Work out (2/3)² + (1/2)². Give your answer as a fraction.
- 3.Simplify (y³)⁴, giving your answer as a single power of y.
- 4.Write down the symbol that makes this statement true: 6/11 ___ 4/11
- 5.Work out 4368 ÷ 12.
- 6.Sam compares 0.6 and 5/8 by comparing the digit 6 with the digit 5, and says that 0.6 is the larger number. Convert 5/8 to a decimal to find the correct larger value.
- 7.A necklace is made from red beads and gold beads in the ratio 7 : 3. What fraction of the beads are gold?
- 8.Given that 2⁵ = 32, work out 2⁶.
- 9.Work out (3 × 5)²
- 10.Work out 2 × (−3 + 7)
- 11.Work out the value of ∛8.
- 12.A jacket normally costs £65. In a sale it is reduced by 20%, and the shop then takes a further £5 off at the till. Work out the final price.
- 13.Work out 0.6 × 0.04.
- 14.Work out 10% of 30% of £200.
- 15.During one night on a mountain, four hikers each recorded the temperature they felt: Ben −6 °C, Priya −2 °C, Sam −9 °C, Alex 1 °C. Work out the difference between the coldest and the warmest of these four temperatures.
Answer key
- (a) 9 °C — Method: the fall is the difference between the two readings, so subtract the lower reading from the higher one; subtracting a negative number is the same as adding its positive. Working: 3 − (−6) = 3 + 6 = 9. Counting it out, the temperature drops 3 degrees to reach zero and a further 6 degrees below zero. Answer: 9 °C. The distractors: −9 °C comes from subtracting the readings the wrong way round, as −6 − 3, and reporting a fall as a negative amount; 3 °C comes from ignoring the minus sign and working out 6 − 3; 6 °C comes from counting only the part of the fall that happens below zero and forgetting the 3 degrees above it.
- (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) y¹² — Method: when a power is raised to another power, multiply the two indices. Working: (y³)⁴ means y³ × y³ × y³ × y³, which is four lots of three y's multiplied together, so the index is 3 × 4 = 12 and (y³)⁴ = y¹². y⁷ comes from adding the indices, 3 + 4 = 7, which is the rule for multiplying two separate powers, not for raising a power to a power. y⁸¹ comes from working out 3⁴ = 81 and using that as the index, raising the inner index to the outer power instead of multiplying the two indices. 12y comes from multiplying the indices to make 12 but then treating y as a coefficient instead of a power. Answer: y¹².
- (c) > — Method: fractions with the same denominator are made of parts of the same size, so compare how many of those parts each fraction has. Working: both fractions are elevenths, and 6 elevenths is 2 more elevenths than 4 elevenths, so the fraction on the left is the larger one. The symbol must have its point facing the smaller side. Answer: >. The distractors: < comes from comparing the numerators the wrong way round, as though a larger numerator gave a smaller fraction; = comes from seeing the same denominator in both fractions and concluding that the fractions themselves are the same size; ≤ comes from working out the direction correctly but then picking the wrong symbol for it, reading it as though it meant 'is greater than or equal to'; ≤ means 'is less than or equal to', and 6 elevenths is neither less than nor equal to 4 elevenths.
- (b) 364 — Divide in stages using multiples of 12. 12 × 300 = 3600, leaving a remainder of 4368 − 3600 = 768. Then 12 × 64 = 768, so 4368 ÷ 12 = 300 + 64 = 364. Placing the decimal point as though dividing 436.8 by 12 gives 36.4. Transposing the last two digits of 364 gives 346. Working out 768 ÷ 12 as 4 instead of 64, losing the tens digit, and adding 300 + 4 gives 304. So 4368 ÷ 12 = 364.
- (a) 0.625 — Method: convert the fraction to a decimal so it can be compared properly with 0.6. Working: 5/8 = 0.625, and since 0.625 > 0.6, the larger value is 0.625. Answer: 0.625. 0.6 repeats Sam's incorrect claim, made by comparing single digits rather than full place value. 0.58 comes from converting 5/8 incorrectly, treating it as if it read 5 tenths and 8 hundredths. 0.85 comes from turning the fraction upside down and writing its digits straight after the decimal point, 8 then 5, instead of dividing.
- (d) 3/10 — Total parts = 7 + 3 = 10. Gold beads are 3 of those 10 parts, so the fraction is 3/10. 7/10 comes from finding the fraction of red beads instead of gold beads. 7/3 comes from writing the ratio itself as a fraction, without adding the parts to find the total. 3/7 comes from comparing gold beads with red beads instead of with the total number of beads.
- (d) 64 — Method: each time the power increases by 1, the value doubles, since one more 2 is multiplied in. Working: 2⁶ = 2⁵ × 2 = 32 × 2 = 64. Answer: 64. (37 comes from adding the power to the value, 32 + 5, instead of multiplying by the base. 192 comes from multiplying 32 by the new power, 6, instead of by the base, 2. 34 comes from adding the base to the value, 32 + 2, instead of multiplying.)
- (a) 225 — Method: brackets are worked out before powers, so multiply the two numbers first and then square the result. Working: 3 × 5 = 15, and 15² = 15 × 15 = 225. Answer: 225. The distractors: 30 comes from doubling 15 instead of squaring it; 75 comes from squaring only the 5 and then working out 3 × 25; 34 comes from squaring each number separately and adding, 9 + 25, instead of multiplying inside the brackets first.
- (d) 8 — Method: whatever is inside the brackets is worked out first, and the multiplication is carried out afterwards. Working: inside the brackets, −3 + 7 = 4, and then 2 × 4 = 8. Answer: 8. The distractors: 1 comes from ignoring the brackets and multiplying first, giving 2 × (−3) = −6 and then −6 + 7 = 1; −20 comes from reading −3 + 7 as −(3 + 7) = −10, so that 2 × (−10) = −20; 6 comes from working the brackets out correctly and then adding the 2 instead of multiplying by it, giving 2 + 4 = 6.
- (b) 2 — Method: the cube root of a number is the value that multiplies by itself three times to give that number. Working: 2 × 2 × 2 = 8, so ∛8 = 2. 4 comes from working out 8 ÷ 2 = 4, halving the number instead of finding its cube root. 24 comes from working out 8 × 3 = 24, multiplying by 3 instead of cube-rooting it. 64 is 8², the square of 8, not its cube root. Answer: 2.
- (c) £47.00 — First apply the 20% reduction: £65 × 0.8 = £52.00. Then take off the further £5: £52.00 − £5 = £47.00. Treating the 20% as a flat £20 rather than a percentage of the price, £65 − £20 − £5, gives £40.00. Applying the 20% reduction correctly but forgetting to take off the extra £5 leaves £52.00. Taking off the £5 first and then applying the 20% reduction to the smaller amount, (£65 − £5) × 0.8, gives £48.00.
- (c) 0.024 — Multiply the digits ignoring the decimal points: 6 × 4 = 24. Count the total number of decimal places in the two numbers being multiplied: 0.6 has 1 decimal place and 0.04 has 2, giving 3 in total. Place the decimal point in 24 so that there are 3 digits after it: 0.024. Counting only 2 decimal places instead of 3 gives 0.24. Counting 4 decimal places instead of 3 gives 0.0024. Counting only 1 decimal place instead of 3 — in effect moving the point in just one of the two numbers, as if the calculation were 6 × 0.4 — gives 2.4. So 0.6 × 0.04 = 0.024.
- (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.
- (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.
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