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.
Answer key: Number worksheet — GCSE Foundation
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- (a) 91 is not prime, because 91 = 7 × 13. — Check 91 for prime factors up to its square root, which is just under 10: 91 ÷ 7 = 13, and both 7 and 13 are prime, so 91 = 7 × 13 and 91 is not a prime number. Checking only 2, 3 and 5 misses that 7 also needs to be tried — 91 is odd, its digits do not sum to a multiple of 3 (9 + 1 = 10), and it does not end in 0 or 5, so those three checks alone wrongly suggest it is prime. Assuming any odd number ending in 1 must be prime ignores that 91 = 7 × 13 is a counterexample. Misapplying the digit-sum test for 3 by miscounting 9 + 1 as a multiple of 3 wrongly concludes 91 is divisible by 3, when the correct digit sum, 10, is not a multiple of 3. So 91 is not prime, because 91 = 7 × 13.
- (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.
- (d) 1.65 litres — Method: convert both volumes to the same unit, then add. Working: 400 ml = 400 ÷ 1000 = 0.4 litres. Total = 1.25 + 0.4 = 1.65 litres. Answer: 1.65 litres. (401.25 litres comes from adding 1.25 and 400 directly without converting millilitres to litres first. 0.85 litres comes from subtracting 0.4 from 1.25 instead of adding. 5.25 litres comes from dividing 400 by 100 instead of by 1000 — using the centimetre-to-metre factor — which turns 400 ml into 4 litres before adding.)
- (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.
- (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.)
- (d) 27/80 — Method: write the total as a fraction of a litre, then divide by the number of glasses. Working: 1.35 = 27/20, so each glass holds 27/20 ÷ 4 = 27/80 of a litre. Answer: 27/80. 27/20 comes from converting the total correctly to a fraction but forgetting to divide by the number of glasses. 27/5 comes from multiplying the total by 4 instead of dividing. 17/50 comes from rounding 1.35 ÷ 4 to 0.34 before converting to a fraction.
- (b) 5/27 — Method: multiply the numerators together and the denominators together, then simplify. Working: (5 × 2)/(6 × 9) = 10/54 = 5/27. Answer: 5/27. 7/15 comes from adding the fractions instead of multiplying: (5+2)/(6+9) = 7/15. 15/4 comes from flipping the second fraction, as if dividing: (5 × 9)/(6 × 2) = 45/12 = 15/4. 5/3 comes from cancelling the two denominators against each other, dividing both 6 and 9 by 3 to leave 5/2 × 2/3 = 10/6 = 5/3; cancelling is only valid between a numerator and a denominator, never between two denominators.
- (a) 1/5 — Work out the bracket first: 2 + 3 = 5. The reciprocal of 5 is 1/5. A candidate who forgot to take the reciprocal and just gave the value of the bracket wrote 5. A candidate who took the reciprocal but made a sign error wrote −1/5. A candidate who found the reciprocal of each number separately and added them, treating reciprocal as if it distributes over addition, worked out 1/2 + 1/3 = 5/6.
- (a) 7 — Convert the mixed number to an improper fraction: 5 1/4 = 21/4. Dividing by 3/4 means multiplying by its reciprocal, 4/3: 21/4 × 4/3 gives 84/12, which simplifies to 7. So exactly 7 complete pieces of 3/4 m can be cut. Ignoring the 1/4 m and dividing only the whole number, 5 ÷ 3/4, gives 20/3, which is 6 complete pieces with some wood left over. Multiplying by 3/4 instead of its reciprocal, 21/4 × 3/4, gives 63/16, which is 3 complete pieces. Misreading 5 1/4 as the fraction 5/4, then dividing by 3/4, gives 5/3, which is only 1 complete piece. So 7 complete pieces can be cut from the plank.
- (d) 10π cm — Method: for a circle, circumference = π × diameter. Working: circumference = π × 10 = 10π cm. Answer: 10π cm. 20π cm comes from using the formula 2 × π × radius but plugging in the diameter as if it were the radius: 2 × π × 10 = 20π. 25π cm comes from confusing circumference with area, using π × radius² with radius 5: π × 5² = 25π. 5π cm comes from halving the diameter and then multiplying by π, as if circumference were π × radius: π × 5 = 5π.
- (b) −2 — Method: the multiplication is carried out before the addition, and a negative multiplied by a positive is negative. Working: (−3) × 4 = −12, so the calculation becomes −12 + 10 = −2. Answer: −2. The distractors: 22 comes from ignoring the minus sign and working out 3 × 4 + 10 = 22; −42 comes from adding before multiplying, giving (4 + 10) × (−3) = 14 × (−3) = −42; −22 comes from adding 12 and 10 and then writing a minus sign in front of the total, instead of moving 10 places up from −12.
- (a) 360 — The inverse of ÷ 15 is × 15, so the missing number is 24 × 15 = 360. Subtracting instead of multiplying gives 24 − 15 = 9. Dividing by 15 again instead of multiplying gives 24 ÷ 15 = 1.6. Adding instead of multiplying gives 24 + 15 = 39.
- (d) 2/25 — Method: write the decimal over 100 using its two decimal places, then simplify. Working: 0.08 = 8/100 = 2/25 (dividing both numerator and denominator by 4). Answer: 2/25. The student's fraction, 8/10, comes from ignoring the zero in the tenths column and reading 0.08 as though it were 0.8; it simplifies to 4/5. 1/125 comes from writing the decimal over 1000 instead of 100, as if there were three decimal places. 25/2 comes from flipping the correct fraction upside down.
- (a) 11 — Method: list every possible total from the smallest to the largest, and count how many different values there are. Working: the smallest total is 1+1=2 and the largest is 6+6=12, and every whole number total from 2 to 12 is possible: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 — that is 11 different totals. Answer: 11. 36 comes from counting the number of possible dice outcomes (6×6) instead of the number of different totals. 10 comes from listing the totals but missing one from the ends of the list, for example starting at 3 instead of 2. 6 comes from counting only the number of different scores on one die, not the totals of both dice together.
- (d) 1/4 — Method: count the digits after the decimal point to choose a denominator that is a power of ten, then cancel the fraction by the highest common factor of the numerator and the denominator. Working: 0.25 has two digits after the point, so 0.25 is 25 hundredths and can be written as 25/100; the highest common factor of 25 and 100 is 25, and 25 ÷ 25 = 1 with 100 ÷ 25 = 4. Answer: 1/4. The distractors: 2/5 comes from reading the two digits after the point as a numerator and a denominator and writing 2 over 5; 1/25 comes from putting 1 over the digits after the point, as though 0.25 meant one twenty-fifth; 1/40 comes from miscounting the decimal places and using a denominator of 1000, giving 25/1000, which cancels by 25 to 1/40.
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