Printable · GCSE Foundation · ages 14-16
The four operations and place value worksheet — GCSE Foundation
Fifteen questions on "the four operations and place value" — DfE statement N2. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
The four operations and place value worksheet — GCSE Foundation
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- 1.Work out 4368 ÷ 12.
- 2.Meera says that 0.7 ÷ 0.1 = 0.07. Work out the correct value of 0.7 ÷ 0.1.
- 3.Work out 1 − 1/2 − 1/4 − 1/8 − 1/16. Give your answer as a fraction.
- 4.A recipe for one cake needs 2/3 of a cup of sugar. Priya has 3 1/2 cups of sugar. Work out how many complete cakes she can make.
- 5.Work out 2 3/4 − 1 5/6. Give your answer as a fraction in its simplest form.
- 6.A café buys 18 boxes of teabags at £3.45 each, and sells all the teabags for £108 in total. Work out the café's profit.
- 7.Work out 3.7 × 24.
- 8.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.
- 9.Work out −7 − (−3).
- 10.Work out 3/7 × 14/9. Give your answer as a fraction in its simplest form.
- 11.Grace drinks 1/3 of a bottle of water in the morning and another 1/3 of the same bottle in the afternoon. Work out what fraction of the bottle she has drunk altogether.
- 12.Work out −4.5 + 2.8.
- 13.Work out (−2/5) × (−10/3). Give your answer as a fraction in its simplest form.
- 14.Jack buys 3 books, each costing £4.25, and pays with a £20 note. Work out how much change he receives.
- 15.Work out 250 ÷ 1000.
Answer key
- (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) 7 — Dividing by 0.1 is the same as multiplying by 10, so 0.7 ÷ 0.1 = 7. Meera's answer of 0.07 comes from dividing 0.7 by 10 instead of by 0.1, the wrong way round. A candidate who confuses 0.1 with 0.01 multiplies by 100 instead of 10 and gets 70. A candidate who thinks dividing by a number less than 1 does not change the value gets 0.7.
- (a) 1/16 — Method: terms can only be subtracted once they share a denominator, so write every term over the largest denominator, 16, and then subtract the numerators in order from left to right. Working: 1 = 16/16, 1/2 = 8/16, 1/4 = 4/16 and 1/8 = 2/16, so the numerators give 16 − 8 − 4 − 2 − 1 = 1, over a denominator of 16. Answer: 1/16. The distractors: 1/8 comes from stopping one term early, after 16 − 8 − 4 − 2 = 2; 3/16 comes from a sign slip on the last term, adding it instead of subtracting it, which gives 2 + 1 = 3; 15/16 comes from working from the right-hand end as though the last four terms were bracketed together, so that only a single sixteenth is taken away from 1.
- (b) 5 — Method: divide the total amount of sugar by the amount needed for one cake, then round down because a part-used amount of sugar cannot make an extra whole cake. Working: 3 1/2 ÷ 2/3 = 7/2 × 3/2 = 21/4 = 5.25; only 5 complete cakes can be made, since the leftover 0.25 of a portion is not enough for a 6th cake. Answer: 5. 5.25 gives the exact result of the division without rounding down to a whole number of cakes. 7 comes from multiplying 3.5 by 2 and ignoring the need to also divide by 3 as part of dividing by the fraction 2/3. 6 comes from rounding 5.25 up to the nearest whole number instead of down, wrongly assuming a 6th cake could be made from the leftover sugar.
- (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) £45.90 — The cost is 18 × £3.45 = £62.10. Profit = £108 − £62.10 = £45.90. A candidate who does not borrow in the tenths column, doing 1 − 0 = 1 instead of borrowing to make 10 − 1 = 9 and so leaving the units as 8 − 2 = 6, gets £46.10. A candidate who adds the cost to the selling price instead of subtracting gets £108 + £62.10 = £170.10. A candidate who gives the cost instead of the profit gets £62.10.
- (b) 88.8 — Multiply as whole numbers first, ignoring the decimal point: 37 × 24. Split it as 37 × 20 = 740 and 37 × 4 = 148, so 37 × 24 = 740 + 148 = 888. 3.7 has 1 decimal place and 24 has none, so the answer needs 1 decimal place: 88.8. Counting the 2 digits in "3.7" as though that were the number of decimal places gives 8.88 instead of 1 decimal place. Leaving the decimal point out altogether gives 888. Misreading 37 × 4 as 138 rather than 148 gives a running total of 878, placed with 1 decimal place as 87.8. So 3.7 × 24 = 88.8.
- (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.
- (b) −4 — To subtract a negative number, add its positive equivalent: −7 − (−3) becomes −7 + 3. Work out −7 + 3 to get −4. Treating "− (−3)" as simply "−3" without flipping the sign gives the wrong working −7 − 3, which is −10. Ignoring the negative sign on −7 and just subtracting the values, 7 − 3, gives 4, which loses the sign of the starting number. Flipping the sign of both numbers, 7 + 3, gives 10, which changes more than the double negative allows. So −7 − (−3) = −4.
- (c) 2/3 — Method: multiply the numerators together and the denominators together, then divide both parts of the result by their highest common factor. Working: 3 × 14 = 42 and 7 × 9 = 63, giving 42/63; the highest common factor of 42 and 63 is 21, and 42 ÷ 21 = 2 with 63 ÷ 21 = 3. Answer: 2/3. The distractors: 17/16 comes from adding the numerators and adding the denominators, giving (3 + 14)/(7 + 9); 27/98 comes from turning the second fraction upside down and multiplying, which divides instead of multiplying and gives 3/7 × 9/14; 2/21 comes from cancelling the 7 into the 14 in the numerator but leaving the 7 in the denominator, giving 6/63.
- (c) 2/3 — Method: fractions with the same denominator are added by adding the numerators and leaving the denominator alone, because the parts are already the same size. Working: 1/3 + 1/3 has numerators 1 + 1 = 2 and the denominator stays as 3, giving 2/3. Answer: 2/3. The distractors: 2/6 comes from adding the denominators as well as the numerators, 1 + 1 over 3 + 3; 2/9 comes from adding the numerators but multiplying the denominators, 1 + 1 over 3 × 3; 1/9 comes from multiplying throughout instead of adding, 1 × 1 over 3 × 3.
- (d) −1.7 — Since the numbers have different signs, find the difference between their sizes: 4.5 − 2.8 = 1.7, then keep the sign of the number further from zero. So −4.5 + 2.8 = −1.7. A candidate who drops the negative sign gets 1.7. A candidate who adds the magnitudes instead of finding the difference gets −(4.5 + 2.8) = −7.3. A candidate who takes the smaller digit from the larger in the tenths column, doing 8 − 5 = 3 instead of borrowing to make 15 − 8 = 7, gets 2.3 and so −2.3.
- (d) 4/3 — Method: the product of two negative numbers is positive, so work with 2/5 × 10/3 and then simplify. Multiply the numerators together and the denominators together. Working: 2 × 10 = 20 and 5 × 3 = 15, giving 20/15; both 20 and 15 divide by 5, so 20/15 = 4/3. Answer: 4/3. The distractors: −4/3 has the arithmetic right but keeps a minus sign, from treating negative × negative as negative; 3/25 comes from turning the second fraction upside down and multiplying, which divides instead of multiplying and gives 2/5 × 3/10 = 6/50; −56/15 comes from adding the two fractions instead of multiplying them, giving −6/15 − 50/15.
- (c) £7.25 — Find the total cost of the books first: 3 × 4.25 = 12.75, so the books cost £12.75 in total. Subtract this from the £20 note: 20.00 − 12.75 = 7.25, so the change is £7.25. Stopping after finding the cost and not subtracting it from £20 gives £12.75, which is the amount spent, not the change. Borrowing correctly in the pence column but forgetting to reduce the pounds column by 1 gives £8.25 instead of £7.25. Multiplying 3 × 4.25 as 12.25 instead of 12.75, a multiplication slip, makes the change come out £0.50 too high, at £7.75. So Jack receives £7.25 change.
- (d) 0.25 — Dividing by 1000 moves every digit three place-value columns, so 250 ÷ 1000 = 0.25. A candidate who divides by 100 instead of 1000 gets 2.5. A candidate who divides by 10,000 instead of 1000 gets 0.025. A candidate who divides by 10 instead of 1000 gets 25.
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