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.
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Answer key: Number worksheet — GCSE Foundation
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- (d) 5/7 — Add the parts of the ratio to find the total: 2 + 5 = 7. Oranges are 5 of those 7 parts, so the fraction of the fruit that are oranges is 5/7. 2/7 comes from finding the fraction of bananas instead of oranges. 5/2 comes from writing the ratio itself as a fraction, without adding the parts to find the total. 7/5 comes from putting the total number of parts over the number of oranges instead of the number of oranges over the total.
- (d) 11/12 — Convert both mixed numbers to improper fractions with a common denominator. 2 3/4 = 11/4, which is 33/12, and 1 5/6 = 11/6, which is 22/12. Subtracting, 33/12 − 22/12 gives 11/12, already in its simplest form. Forgetting to borrow, and instead subtracting the fraction parts the other way round to avoid a negative, 10/12 − 9/12 gives 1/12; adding that to the whole-number difference of 1 gives 13/12. Subtracting only the fraction parts, 9/12 − 10/12, and reporting just the size of that difference gives 1/12, which ignores the whole numbers altogether. Adding the two improper fractions instead of subtracting them, 33/12 + 22/12, gives 55/12. So 2 3/4 − 1 5/6 = 11/12.
- (b) 18.5 ≤ T < 18.7 — Method: the error interval reaches half the rounding unit either side of the recorded value. Working: half of 0.2 is 0.1, so the interval runs from 18.6 − 0.1 to 18.6 + 0.1. Answer: 18.5 ≤ T < 18.7. (18.4 ≤ T < 18.8 comes from using the full rounding unit, 0.2, either side instead of half of it. 18.5 ≤ T ≤ 18.7 comes from including the upper bound with ≤ instead of excluding it with <. 18.6 ≤ T < 18.8 comes from treating the recorded value as the start of the interval and adding the whole rounding unit, 0.2, above it.)
- (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) 8π cm² — A diameter of 8 cm gives a radius of 4 cm. The area of a full circle would be π × r² = π × 4² = 16π cm², and a semicircle is exactly half of this, giving 16π ÷ 2 = 8π cm². Forgetting to halve the area for the semicircle gives 16π cm², the area of the whole circle. Halving the diameter twice, using a radius of 2 instead of 4, gives π × 2² = 4π cm². Using the diameter itself as the radius, so π × 8² = 64π, and then halving that for the semicircle gives 32π cm².
- (a) 2/3 — Total parts = 6 + 4 + 5 = 15. Sopranos and altos together are not tenors: 6 + 4 = 10 parts, so the fraction is 10/15, which simplifies to 2/3. 1/3 comes from finding the fraction of tenors instead of the fraction that is not tenors. 2/5 comes from counting only the sopranos as not tenors and leaving the altos out. 4/9 comes from leaving sopranos out of the total, 4 + 5 = 9, and then using only the altos as the fraction that is not tenors.
- (d) 24 km — Method: a fraction acts as an operator, so finding 8/12 of a distance means dividing by the denominator and multiplying by the numerator. Working: 36 ÷ 12 = 3, so one twelfth of the walk is 3 km, and eight twelfths is 3 × 8 = 24 km. Answer: 24 km. The distractors: 12 km comes from working out the part of the walk still left, the other four twelfths, instead of the part already completed; 288 km comes from multiplying by the numerator without dividing by the denominator, giving 36 × 8 = 288; 4.5 km comes from dividing by the numerator instead of multiplying by it, giving 36 ÷ 8 = 4.5.
- (a) 5 — Method: 10% is 10/100, which cancels to 1/10, so finding 10% of an amount means dividing it by 10. Working: 10% = 10/100 = 1/10, and 50 ÷ 10 = 5. Answer: 5 questions. The distractors: 10 comes from reading the 10% as a count of 10 questions rather than as an operator acting on the 50; 45 comes from working out how many questions Freya answers correctly, the other 90% of the quiz, instead of how many she gets wrong; 500 comes from multiplying by 10 instead of dividing by 10.
- (d) 6 — Method: the square root of a product can be found either by multiplying first and then rooting, or by rooting each factor and multiplying the two roots together. Working: 4 × 9 = 36, and 6 × 6 = 36, so the root is 6; the same value comes from √4 × √9 = 2 × 3. Answer: 6. The distractors: 36 comes from multiplying inside the root and then leaving the root untaken; 5 comes from rooting each factor and adding the results, 2 + 3, instead of multiplying them; 18 comes from rooting the 4 only and leaving the 9 untouched, giving 2 × 9.
- (d) 36 — To find 3/8 of 96, divide by the denominator and multiply by the numerator: 96 ÷ 8 = 12, and 12 × 3 = 36. Dividing 96 by the numerator instead of the denominator, 96 ÷ 3 = 32, uses the wrong part of the fraction. Using 3/4 instead of 3/8, perhaps misreading the denominator, gives 96 × 3/4 = 72. Finding 96 ÷ 8 = 12 correctly but forgetting to multiply by the numerator 3 leaves just 12 as the final answer.
- (a) 0.08 — Method: decimal places are counted from the decimal point, including any zeros straight after it, so rounding to 2 decimal places is decided by the digit in the third decimal place. Working: 0.0759 has 7 in the second decimal place and 5 in the third, and 5 counts as rounding up, so the 7 goes up to 8. Answer: 0.08. The distractors: 0.07 comes from chopping the digits after the second decimal place off instead of rounding them; 0.076 is 0.0759 correct to 2 significant figures rather than 2 decimal places, because the zeros in front of the 7 are not significant figures; 0.1 is 0.0759 rounded to 1 decimal place, a coarser degree of accuracy than the question asks for.
- (b) £3.70 — Work in one unit throughout. 45p is £0.45, so three pens cost 3 × £0.45 = £1.35. Adding the notebook gives £1.35 + £2.35 = £3.70. £2.80 is the cost of one pen and the notebook, so the multiplication by 3 has been left out; £8.40 multiplies the notebook by 3 as well as the pens; £15.85 comes from reading 45p as £4.50.
- (a) 3 — Method: the fourth root of a number is the positive value that gives that number when it is multiplied by itself four times. Working: 2 × 2 × 2 × 2 = 16, which is too small, and 3 × 3 × 3 × 3 = 9 × 9 = 81. Answer: 3. The distractors: 9 comes from taking the square root of 81 instead of its fourth root; 4.5 comes from taking the square root and then halving it, as though a fourth root were half a square root; 20.25 comes from dividing 81 by 4, treating the root's index as a divisor.
- (b) 17 — Roots and powers are worked out first: √25 = 5 and 4² = 16. Division comes next: 12 ÷ 3 = 4. Then addition and subtraction, left to right: 5 + 16 − 4 = 17. A candidate who treated 4² as 4 × 2 = 8, multiplying the base by the exponent instead of squaring it, worked out 5 + 8 − 4 = 9. A candidate who did not evaluate the root and used 25 itself worked out 25 + 16 − 4 = 37. A candidate who ignored the priority of division and worked through 5 + 16 − 12 ÷ 3 strictly left to right got 5 + 16 = 21, then 21 − 12 = 9, then 9 ÷ 3 = 3.
- (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.)
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