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.
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Multiplicative relationships as ratios or fractions worksheet — GCSE Foundation
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- 1.In a fruit drink, cranberry juice and apple juice are mixed in the ratio 3:7. Write the amount of apple juice as a fraction of the amount of cranberry juice, in its simplest form.
- 2.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.
- 3.To make a shade of green paint, the amount of yellow paint used is always 0.75 times the amount of blue paint used. Write the amount of blue paint as a fraction of the amount of yellow paint, in its simplest form.
- 4.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.
- 5.A baker uses 450 g of flour and 300 g of butter in a batch of pastry. Write the mass of butter as a fraction of the total mass of flour and butter, in its simplest form.
- 6.£1 is worth 1.25 US dollars. Write the ratio of pounds to dollars in its simplest form, using whole numbers.
- 7.In one batch of shortbread, a bakery uses 240 g of butter and 160 g of sugar. Another batch, made in the same ratio, uses 90 g of sugar. Work out how much butter is needed for the second batch.
- 8.A market stall sells apples at a fixed price per kilogram. Priya buys 3 kg for £5.40. Write the ratio of the mass in kilograms to the cost in pounds, in its simplest form.
- 9.y is always the same multiple of x. When x = 6, y = 15. Work out the value of y when x = 10.
- 10.5 tins of soup cost £8 in total. Work out the multiplier that changes the number of tins into the total cost in pounds.
- 11.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.
- 12.In a car park there are 35 cars and 15 vans. Write the ratio of the number of vans to the number of cars in its simplest form.
- 13.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.
- 14.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.
- 15.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.
Answer key
- (d) 7/3 — The ratio cranberry : apple is 3:7, so apple juice is 7 parts and cranberry juice is 3 parts. Write apple over cranberry: 7/3. (3/7 comes from writing the ratio the wrong way round, cranberry over apple. 7/10 comes from comparing the apple juice to the total amount of the mixture, 7 parts out of 10. 3/10 comes from comparing the cranberry juice to the total amount of the mixture, 3 parts out of 10.)
- (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.)
- (d) 4/3 — The amount of yellow paint is 0.75 times the amount of blue paint. As a fraction, 0.75 = 3/4, so yellow = 3/4 of blue. To write blue as a fraction of yellow, use the reciprocal: flip 3/4 to get 4/3. 3/4 comes from keeping the original fraction without inverting it. 1/4 comes from computing 1 − 3/4 = 1/4, which is not how a fraction reverses. 40/3 comes from converting 0.75 to a fraction as 75/1000 = 3/40 (misplacing the decimal point), then inverting that.
- (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.
- (b) 2/5 — Method: a fraction taken 'of the total' has the whole batch as its denominator, so add the two masses first and then write the butter over that total. Working: the total mass is 450 g + 300 g = 750 g, so the fraction is 300/750; the highest common factor of 300 and 750 is 150, and 300 ÷ 150 = 2 while 750 ÷ 150 = 5. Answer: 2/5 of the batch. The distractors: 2/3 comes from comparing the butter with the flour, 300/450, a part-to-part fraction when the question asks for a part compared with the whole; 3/5 comes from writing the flour over the total, 450/750, which answers about the wrong ingredient; 3/2 comes from writing the flour over the butter, 450/300, which both uses the wrong denominator and reverses the order.
- (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.)
- (c) 135 g — Find the ratio of butter to sugar in the first batch: 240:160, which simplifies to 3:2. For the second batch, sugar = 90 g, so butter = 90 × 3/2 = 135 g. (60 g comes from using the ratio the wrong way round, 90 × 2/3. 170 g comes from subtracting the drop in sugar, 160 − 90 = 70 g, from the original butter amount, 240 − 70, instead of scaling. 240 g comes from not scaling the butter amount at all.)
- (c) 5 : 9 — Write the ratio mass : cost = 3 : 5.4. Multiply both parts by 10 to clear the decimal: 30 : 54. Both numbers share a factor of 6, so 30 ÷ 6 = 5 and 54 ÷ 6 = 9, giving 5 : 9. 3 : 5 comes from ignoring the decimal point and treating £5.40 as £5. 9 : 5 comes from writing the ratio the wrong way round, cost to mass instead of mass to cost. 1 : 18 comes from multiplying only the cost by 10 instead of both parts, giving 3 : 54, and then cancelling that correctly to 1 : 18 — the cancelling is fine, but the ratio being cancelled is not the right one.
- (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) 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.
- (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) 3 : 7 — Method: a ratio 'of A to B' is written with A first, and it is simplified by dividing both parts by their highest common factor. Working: vans are named first, so the ratio starts as 15 : 35; the highest common factor of 15 and 35 is 5, and 15 ÷ 5 = 3 while 35 ÷ 5 = 7. Answer: 3 : 7. The distractors: 7 : 3 comes from writing the two numbers in the order they appear in the question, cars before vans, instead of the order asked for; 3 : 10 comes from comparing the vans with the total number of vehicles, 15 : 50, a part-to-whole ratio in place of the part-to-part ratio asked for; 15 : 35 is the right comparison left unsimplified, and the question asks for the simplest form.
- (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) 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.)
- (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.
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