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.
Non-calculator
Answer key: Number worksheet — GCSE Foundation
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- (b) 0 > −1 — Method: test each statement by placing both of its numbers on a number line; the greater number is the one further to the right. Working: every negative number lies to the left of zero, so zero is greater than −1. Among the negatives, −3 lies to the right of −5, so −5 is not greater than −3. Answer: 0 > −1. The distractors: −5 > −3 comes from ordering the negatives by the size of their digits, so that 5 makes −5 look the larger; −3 > 0 comes from ignoring the minus sign and comparing 3 with 0; 0 < −1 comes from the belief that zero is the smallest number there is, so that even a negative number is above it.
- (a) 2.5 × 10⁷ — Method: place the decimal point so that the coefficient is at least 1 and less than 10, then count the places it has moved. Working: the digits give a coefficient of 2.5, and the decimal point travels from the end of 25,000,000 until it sits between the 2 and the 5, a move of 7 places. Answer: 2.5 × 10⁷. The distractors: 25 × 10⁶ is the same area but not in standard form, because the coefficient must be less than 10; 2.5 × 10⁸ comes from counting the eight digits of 25,000,000 instead of the seven places the decimal point moves; 2.5 × 10⁻⁷ comes from making the index negative because the decimal point was carried to the left.
- (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.
- (a) 8 — To undo 'multiply by 4, then add 8', reverse the operations in reverse order: subtract 8 first, then divide by 4. 40 − 8 = 32, and 32 ÷ 4 = 8, so the number is 8. A candidate who added 8 again instead of subtracting worked out 40 + 8 = 48, then 48 ÷ 4 = 12. A candidate who divided before subtracting, doing the inverse operations in the wrong order, worked out 40 ÷ 4 = 10, then 10 − 8 = 2. A candidate who multiplied instead of dividing at the last step worked out (40 − 8) × 4 = 32 × 4 = 128.
- (c) 3 or −5 — Method: a point a fixed distance from another can lie on either side of it, so move the given distance in each direction from the starting point. Working: moving 4 units to the right gives −1 + 4 = 3, and moving 4 units to the left gives −1 − 4 = −5. Answer: 3 or −5. The distractors: 5 or −3 comes from starting at 1 instead of −1, giving 1 + 4 and 1 − 4; 3 only comes from moving to the right and forgetting that the point could lie to the left as well; 4 or −4 comes from measuring the distance from zero instead of from point A, which just repeats the given distance.
- (a) 24 — Method: rearrange the relationship so that the lowest common multiple stands alone; it is the product of the two numbers divided by their highest common factor. Working: 48 = 2 × the lowest common multiple, so the lowest common multiple is 48 ÷ 2 = 24. Checking, 24 is in the 6 times table and in the 8 times table. Answer: 24. The distractors: 48 comes from giving the product of the two numbers and never dividing by the highest common factor; 96 comes from multiplying by the highest common factor instead of dividing by it; 12 comes from dividing by the highest common factor twice, once for each of the two numbers.
- (b) 25 — Method: check the calculator answer by following the order of operations — each power is worked out before the addition. Working: 4² = 4 × 4 = 16 and 3² = 3 × 3 = 9, and 16 + 9 = 25. Answer: 25. The distractors: 49 is the value Freya wrote down and comes from adding first and then squaring, working out (4 + 3)² instead of 4² + 3²; 14 comes from doubling each number instead of squaring it, adding 8 and 6; 12 comes from multiplying 4 by 3 instead of squaring each number and adding the results.
- (d) 6 × 10⁷ — Method: the coefficients and the powers of ten are handled separately — multiply the coefficients, and add the indices because the powers share the base 10. Working: 2 × 3 = 6 for the coefficients, and 10³ × 10⁴ = 10⁷ for the powers; 6 lies between 1 and 10, so the coefficient needs no adjustment. Answer: 6 × 10⁷. The distractors: 5 × 10⁷ comes from adding the coefficients, 2 + 3, instead of multiplying them; 6 × 10¹² comes from multiplying the indices, 3 × 4, instead of adding them; 6 × 10¹ comes from subtracting the indices, 4 − 3, which is the rule for dividing rather than for multiplying.
- (a) 10π cm — The circumference of a circle is found from C = 2 × π × r. With a radius of 5 cm, this gives C = 2 × π × 5 = 10π cm. Using the radius directly in the formula without doubling it gives 5π cm, missing the factor of 2. Using the formula for area, π × r², instead of circumference gives 25π cm, which is also the wrong units for a length. Doubling the radius to get a diameter of 10 and then applying the circumference formula a second time gives 20π cm, doubling the answer that is already correct.
- (c) 8,000,000,000 — Method: a power of ten is 1 followed by that many zeros, so 8 × 10⁹ is 8 multiplied by 1 followed by nine zeros. Working: 10⁹ = 1,000,000,000, and 8 × 1,000,000,000 puts an 8 in front of those nine zeros. Answer: 8,000,000,000. The distractors: 800,000,000 comes from writing nine digits altogether instead of nine zeros after the 8; 720 comes from reading 10⁹ as 10 × 9 = 90 and then working out 8 × 90; 0.000000008 comes from treating the index as negative and carrying the decimal point nine places to the left.
- (c) 2 — First write 1 1/2 as an improper fraction, 3/2. To divide by 3/4, multiply by its reciprocal, 4/3: 3/2 × 4/3 = 12/6 = 2. Dropping the whole number and dividing only the fractional part, 1/2 ÷ 3/4 = 1/2 × 4/3, gives 2/3. Multiplying by 3/4 directly instead of using its reciprocal, 3/2 × 3/4, gives 9/8. Using the reciprocal of the first fraction instead of the second, 2/3 × 3/4, gives 1/2.
- (c) 693 — Method: multiplying a bracket by a number multiplies every term inside it, so 7 × (100 − 1) = 7 × 100 − 7 × 1. Working: 7 × 100 = 700 and 7 × 1 = 7, so the calculation becomes 700 − 7 = 693. Answer: 693. The distractors: 699 comes from subtracting the 1 itself rather than 7 lots of it, giving 700 − 1 = 699; 707 comes from adding the second product instead of subtracting it, giving 700 + 7 = 707; 700 comes from rounding 99 up to 100 and then offering the estimate 7 × 100 as an exact value.
- (c) −1 — Method: find each cube root separately, keeping its sign, and then add the two results. Working: (−3) × (−3) × (−3) = −27, so ∛(−27) = −3, and 2 × 2 × 2 = 8, so ∛8 = 2. Adding gives −3 + 2 = −1. Answer: −1. The distractors: 5 comes from taking the cube root of a negative number as positive, giving 3 + 2; −5 comes from reading the minus sign as applying to the whole sum and working out −(3 + 2); −6 comes from multiplying the two roots, −3 × 2, instead of adding them.
- (c) 1/4 — A negative index means the reciprocal of the positive power, so 2⁻² = 1 ÷ 2² = 1/4. Treating the negative sign as making the answer negative instead gives −(2²) = −4. Ignoring the negative sign altogether gives just 2² = 4. Finding the reciprocal correctly but then also applying a negative sign gives −1/4.
- (c) 6 — Listing systematically: red-black, red-grey, blue-black, blue-grey, green-black, green-grey gives 6 different outfits, matching 3 × 2 = 6. 5 comes from adding the number of T-shirts and shorts instead of multiplying them. 3 comes from counting only the T-shirts. 2 comes from counting only the pairs of shorts.
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