Printable · GCSE Foundation · ages 14-16
Multiplicative relationships as ratios or fractions worksheet — GCSE Foundation
Fifteen questions on "multiplicative relationships as ratios or fractions" — DfE statement R6. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Multiplicative relationships as ratios or fractions worksheet — GCSE Foundation
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- 1.x is 2/3 of y. Write the ratio x : y in its simplest form.
- 2.A recipe for 8 people needs 320 g of rice. Sanjay scales the recipe down to serve 5 people, then wants to scale his new amount back up to serve 8 people again. Work out the multiplier he should use to scale his amount for 5 people back up to the amount for 8 people.
- 3.Ravi has 5 pears. He has 4 times as many apples as pears. Write the number of pears as a fraction of the number of apples.
- 4.y is always the same multiple of x. When x = 6, y = 15. Work out the value of y when x = 10.
- 5.A rope of length A is 1.5 times as long as a rope of length B. Write the ratio A : B in its simplest form.
- 6.A photo is enlarged so that its new width is 1.4 times its original width. Write the multiplier that would scale the new width back down to the original width, as a fraction in its simplest form.
- 7.In a mix of mortar the ratio of the mass of sand to the mass of cement is 5 : 2. Write the mass of cement as a fraction of the total mass of the mix.
- 8.A tray holds 8 muffins and 20 cupcakes. Write the ratio of the number of muffins to the number of cupcakes in its simplest form.
- 9.Cheng is paid an hourly rate that is proportional to the number of hours she works. She earns £52.50 for 7 hours. Work out how much she earns for 11 hours, assuming the same rate.
- 10.A shop's number of online orders is proportional to its number of in-store orders. Last month there were 45 in-store orders and 180 online orders. Write the ratio of online orders to in-store orders, in its simplest form.
- 11.£1 is worth 1.25 US dollars. Write the ratio of pounds to dollars in its simplest form, using whole numbers.
- 12.To make orange paint, red and yellow paint are mixed in the ratio 6:15. Write the amount of red paint as a fraction of the amount of yellow paint, in its simplest form.
- 13.The table shows values of x and y. x: 3, 5, 8. y: 12, 20, 32. Which statement correctly describes the relationship between x and y?
- 14.A scale model of a bus is built so that 5 cm on the model represents 60 cm on the real bus. Write the ratio of the length of the model to the length of the real bus, in its simplest form.
- 15.5 tins of soup cost £8 in total. Work out the multiplier that changes the number of tins into the total cost in pounds.
Answer key
- (d) 2 : 3 — x is 2/3 of y means for every 3 parts of y, x is 2 parts, so x : y = 2 : 3. 3 : 2 comes from writing the ratio the wrong way round. 2 : 5 comes from comparing x with the total of x and y (2 parts out of 5), instead of with y alone. 3 : 5 comes from comparing y with the total of x and y (3 parts out of 5), instead of with x.
- (a) 8/5 — Scaling down from 8 people to 5 people uses the multiplier 5/8. To reverse this and scale back up from 5 people to 8 people, use the reciprocal of that multiplier: flip 5/8 to get 8/5. 5/8 comes from using the forward (scaling down) multiplier again, instead of reversing it. 3/5 comes from writing the difference in people (8 − 5 = 3) over 5, instead of using the reciprocal of 5/8. 25/64 comes from multiplying the forward multiplier by itself (5/8 × 5/8), instead of finding its reciprocal.
- (d) 1/4 — Method: work out both amounts first, then put the amount named before the word 'of' on top and the amount named after it on the bottom. Working: there are 4 × 5 = 20 apples, so the fraction is 5/20; dividing the numerator and the denominator by 5 gives 1/4. Answer: 1/4. The distractors: 4/1 comes from quoting the multiplier given in the question, which compares the apples with the pears rather than the pears with the apples; 4/5 comes from writing the 20 apples over the 25 fruits altogether, 20/25, a part-to-whole fraction; 5/9 comes from reading '4 times as many' as '4 more than', giving 5 + 4 = 9 apples and the fraction 5/9.
- (c) 25 — Find the constant multiplier from the given pair: 15 ÷ 6 = 2.5, so y is always 2.5 times x. When x = 10, y = 10 × 2.5 = 25. 19 comes from assuming an additive relationship instead of a multiplicative one — adding the difference 15 − 6 = 9 onto 10. 4 comes from using the multiplier the wrong way round (6 ÷ 15 = 0.4) and then multiplying by 10. 15 comes from simply repeating the given value of y, without applying the multiplier to the new value of x at all.
- (c) 3 : 2 — Method: 'A is k times B' means that if B is taken as 1 part then A is k parts, and a ratio in its simplest form is written with whole numbers that have no common factor. Working: taking B as 1, A is 1.5, so A : B = 1.5 : 1; multiplying both parts by 2 clears the decimal and gives 3 : 2, and 3 and 2 share no factor. Answer: 3 : 2. The distractors: 2 : 3 comes from writing B before A, reversing the order the question asks for; 1.5 : 1 is the right relationship but is not in its simplest form, because a ratio in its simplest form uses whole numbers; 3 : 5 comes from comparing A with the combined length rather than with B, since A : (A + B) = 1.5 : 2.5 = 3 : 5.
- (b) 5/7 — The enlargement multiplier is 1.4, which as a fraction is 7/5. To reverse an enlargement, use the reciprocal of the multiplier: flip 7/5 to get 5/7. 7/5 comes from using the enlargement multiplier again, instead of reversing it. 3/5 comes from treating the reverse as 'give back the extra amount', working out 1 − (1.4 − 1) = 0.6, instead of using the reciprocal. 5/2 comes from ignoring the whole number in 1.4 and inverting only the decimal part, 0.4, as if it were the whole multiplier.
- (c) 2/7 — Method: each number in a ratio counts parts, so the whole is all the parts added together, and the fraction wanted is the cement parts over that total. Working: the mix has 5 + 2 = 7 parts altogether, and 2 of those parts are cement, so the cement is 2/7 of the mix. Answer: 2/7. The distractors: 2/5 comes from writing the cement parts over the sand parts, comparing one part with the other part instead of with the whole; 5/7 comes from writing the sand parts over the total, which answers about the wrong material; 7/2 comes from writing the total over the cement parts, turning the fraction upside down.
- (c) 2:5 — Divide both numbers by their highest common factor, 4: 8 ÷ 4 = 2 and 20 ÷ 4 = 5, giving the ratio 2:5. Choosing 5:2 comes from writing the ratio the wrong way round, as cupcakes to muffins. Choosing 2:3 comes from using the difference between the two amounts (20 − 8 = 12) as the second part of the ratio instead of the number of cupcakes, then simplifying 8:12 by dividing by 4. Choosing 2:7 comes from comparing the muffins with the total number of items on the tray (8 out of 28) instead of comparing them with the cupcakes.
- (d) £82.50 — Find the hourly rate: £52.50 ÷ 7 = £7.50 per hour. For 11 hours: 11 × £7.50 = £82.50. £30 comes from working out the pay for only the extra 4 hours (4 × £7.50), and forgetting to include the original £52.50. £99 comes from misremembering the hourly rate as £9 instead of £7.50, then 11 × £9. £56.50 comes from adding the extra number of hours (4) straight onto the pay in pounds (52.5 + 4), confusing hours with pounds.
- (d) 4:1 — Write the ratio online : in-store using the numbers in the question: 180 : 45. Divide both parts by their highest common factor, 45, to give 4 : 1. (1:4 comes from writing the ratio the wrong way round, in-store : online. 3:1 comes from subtracting the in-store orders from the online orders, 180 − 45 = 135, and comparing that to the in-store orders, 135:45, instead of dividing. 4:5 comes from comparing the online orders to the total number of orders, 180:225.)
- (d) 4:5 — Write the ratio pounds : dollars as 1 : 1.25. Multiply both parts by 4 to clear the decimal: 1 × 4 = 4 and 1.25 × 4 = 5, giving 4 : 5. (5:4 comes from writing the ratio the wrong way round, dollars to pounds. 1:1 comes from rounding 1.25 dollars down to the nearest whole dollar. 1:5 comes from multiplying only the dollars by 4 to clear the decimal and leaving the pounds as 1 — both parts of a ratio must be multiplied by the same number.)
- (b) 2/5 — The ratio red : yellow is 6:15, so write red over yellow: 6/15. Divide both numbers by their highest common factor, 3: 6÷3 = 2, 15÷3 = 5, giving 2/5. (5/2 comes from writing the ratio the wrong way round, yellow over red, 15/6, which simplifies to 5/2. 2/7 comes from comparing the red paint to the total amount of paint, 6 parts out of 21. 5/7 comes from comparing the yellow paint to the total amount of paint, 15 parts out of 21.)
- (a) y is always 4 times x. — Check the multiplier for each pair: 12 ÷ 3 = 4, 20 ÷ 5 = 4, 32 ÷ 8 = 4 — the same multiplier every time, so y is always 4 times x. 'x is always 4 times y' comes from writing the multiplier the wrong way round. 'y is always x plus 9' only works for the first pair (3 + 9 = 12); it gives 14 for x = 5 and 17 for x = 8, not 20 and 32. 'y is always double x, plus 6' also only works for the first pair (3 × 2 + 6 = 12); it gives 16 for x = 5, not 20.
- (d) 1:12 — Write the ratio model : real bus using the numbers in the question: 5 : 60. Divide both parts by their highest common factor, 5: 5÷5 = 1, 60÷5 = 12, giving 1 : 12. (12:1 comes from writing the ratio the wrong way round, real bus : model. 1:55 comes from dividing the first part correctly, 5÷5 = 1, but subtracting on the second part, 60 − 5 = 55, instead of dividing it by 5 as well. 5:12 comes from dividing only the real length by 5 and leaving the model length unchanged — both parts of a ratio must be divided by the same number.)
- (c) 1.6 — The multiplier from tins to pounds is found by dividing the cost by the number of tins: £8 ÷ 5 = 1.6. 0.625 comes from dividing the wrong way round, tins ÷ cost (5 ÷ 8), which is the multiplier from pounds back to tins. 3 comes from subtracting the number of tins from the cost (8 − 5), instead of dividing. 40 comes from multiplying the cost by the number of tins (8 × 5), instead of dividing.
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