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.Meera says that 0.7 ÷ 0.1 = 0.07. Work out the correct value of 0.7 ÷ 0.1.
- 2.A tank contains 120 litres of water. Water is drained out at a rate of 8 litres per minute for 6 minutes, and then a hose adds 15 litres. Work out how much water is left in the tank.
- 3.Work out 250 ÷ 1000.
- 4.Work out 3 + 4 × (−2).
- 5.Work out −4.5 + 2.8.
- 6.Work out (−2/5) × (−10/3). Give your answer as a fraction in its simplest form.
- 7.Work out 3/7 × 14/9. Give your answer as a fraction in its simplest form.
- 8.Work out −7 − (−3).
- 9.Work out 3 1/5 + 1 2/3. Give your answer as a mixed number in its simplest form.
- 10.In the number 3.472, work out the value of the digit 7.
- 11.A carpenter has a plank of wood 4.8 m long. She cuts off 3 pieces, each 0.9 m long, to make shelves. Work out the length of wood remaining.
- 12.Write '3.2 million' as a number in figures.
- 13.Work out (2/3)² + (1/2)². Give your answer as a fraction.
- 14.Work out −6 × (−3).
- 15.Work out 2 3/4 − 1 5/6. Give your answer as a fraction in its simplest form.
Answer key
- (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.
- (c) 87 litres — Work out how much water is drained: 8 × 6 = 48 litres. Subtract this from the starting amount: 120 − 48 = 72 litres. Then add the 15 litres from the hose: 72 + 15 = 87 litres. Subtracting the 15 litres instead of adding it, as though the hose also removed water, gives 120 − 48 − 15 = 57 litres. Stopping after the drain step, without adding the hose water back in, leaves the working at 72 litres. Adding the rate and the time instead of multiplying them, 8 + 6 = 14 litres drained, and then working from there gives 120 − 14 + 15 = 121 litres. So 87 litres of water is left in the tank.
- (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.
- (b) −5 — Using the order of operations, work out the multiplication first: 4 × (−2) = −8. Then 3 + (−8) = −5. A candidate who adds before multiplying gets (3 + 4) × (−2) = −14. A candidate who drops the negative sign on the multiplication gets 3 + 4 × 2 = 11. A candidate who works out the multiplication correctly but gives that as the final answer, forgetting to combine it with the 3, gets −8.
- (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) 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.
- (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.
- (d) 4 13/15 — Convert to fifteenths: 1/5 is equivalent to 3/15 (multiply by 3/3), and 2/3 is equivalent to 10/15 (multiply by 5/5), so 3 1/5 is equivalent to 3 3/15 and 1 2/3 is equivalent to 1 10/15. Add the whole numbers (3 + 1 = 4) and the fractions (3/15 + 10/15 = 13/15), giving 4 13/15. A candidate who adds the numerators and denominators straight across, treating 1/5 + 2/3 as (1+2)/(5+3), gets a fraction part of 3/8, giving 4 3/8. A candidate who adds the fraction parts correctly but forgets to add the second whole number gets 3 13/15. A candidate who adds the whole numbers but copies the first fraction across without ever adding 2/3 to it gets 4 1/5.
- (c) 0.07 — Each digit after the decimal point has a place value: the first digit is tenths, the second is hundredths, the third is thousandths. In 3.472, the 4 is in the tenths place and the 7 is in the hundredths place, so it is worth 0.07. Reading it as 7 ignores place value altogether, treating it as if it were a whole number. Reading it as 0.7 puts it one place too big, in the tenths place. Reading it as 0.007 puts it one place too small, in the thousandths place. The digit 7 in 3.472 is worth 0.07.
- (b) 2.1 m — The three pieces use 3 × 0.9 = 2.7 m of wood. Remaining wood = 4.8 − 2.7 = 2.1 m. A candidate who miscounts and only subtracts 2 pieces instead of 3 gets 4.8 − 1.8 = 3.0 m. A candidate who adds instead of subtracting gets 4.8 + 2.7 = 7.5 m. A candidate who gives the length used instead of the length remaining gets 2.7 m.
- (d) 3,200,000 — 1 million = 1,000,000, so 3.2 million = 3.2 × 1,000,000 = 3,200,000. A candidate who moves the decimal point one place too many gets 32,000,000. A candidate who moves it one place too few gets 320,000. A candidate who writes the .2 as extra thousands instead of hundred-thousands gets 3,002,000.
- (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.
- (b) 18 — Method: multiply the two numbers ignoring their signs, then apply the rule that a negative number multiplied by a negative number gives a positive answer. Working: 6 × 3 = 18; since both −6 and −3 are negative, the product is positive. Answer: 18. −18 comes from keeping the answer negative, as if only one of the two negative signs affects the sign of the product. −9 comes from adding the two numbers instead of multiplying them, −6 + (−3) = −9. 9 comes from adding 6 and 3 as if both numbers were positive, ignoring the negative signs entirely.
- (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.
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