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.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.
- 2.An adult cinema ticket costs £15 and a child ticket costs £9. Write the ratio of the adult price to the child price in its simplest form.
- 3.A box contains 12 red counters and 18 blue counters. Write the number of red counters as a fraction of the number of blue counters. Give your answer in its simplest form.
- 4.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.
- 5.x is 2/3 of y. Write the ratio x : y in its simplest form.
- 6.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.
- 7.A metal alloy is made from copper and tin in the ratio 9:1. Write the mass of tin as a fraction of the mass of copper, in its simplest form.
- 8.5 tins of soup cost £8 in total. Work out the multiplier that changes the number of tins into the total cost in pounds.
- 9.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.
- 10.A car uses 6 litres of petrol to travel 90 km. Write the ratio of the distance travelled in km to the amount of petrol used in litres, in its simplest form.
- 11.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.
- 12.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.
- 13.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.
- 14.Water flows into a tank at a constant rate. After 8 minutes the tank holds 200 litres. Write the ratio of time in minutes to volume in litres, in its simplest form.
- 15.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.
Answer key
- (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.)
- (a) 5:3 — Divide both prices by their highest common factor, 3: 15 ÷ 3 = 5 and 9 ÷ 3 = 3, giving the ratio 5:3. Choosing 3:5 comes from writing the ratio the wrong way round, as child price to adult price. Choosing 2:3 comes from using the difference between the two prices (15 − 9 = 6) as the first part of the ratio instead of the adult price, then simplifying 6:9 by dividing by 3. Choosing 5:8 comes from comparing the adult price with the total cost of both tickets (£15 out of £24) instead of comparing it with the child price.
- (c) 2/3 — Method: 'as a fraction of' puts the quantity named after the word 'of' on the bottom, so the red counters go on top and the blue counters go on the bottom. Working: the fraction is 12/18; the highest common factor of 12 and 18 is 6, and 12 ÷ 6 = 2 while 18 ÷ 6 = 3. Answer: 2/3. The distractors: 2/5 comes from comparing the red counters with the total number of counters, 12/30, a part-to-whole fraction when the question asks for one part compared with the other part; 3/2 comes from writing the blue counters over the red counters, 18/12, reversing the order the question gives; 1/2 comes from simplifying 12/18 by subtracting the common factor instead of dividing by it, 12 − 6 = 6 and 18 − 6 = 12, giving 6/12, which then cancels to 1/2.
- (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) 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.
- (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.
- (a) 1/9 — The ratio copper : tin is 9:1, so write tin over copper: 1/9. (9/1 comes from writing the ratio the wrong way round, copper over tin. 1/10 comes from comparing the tin to the total mass of the alloy, 1 part out of 10. 9/10 comes from comparing the copper to the total mass of the alloy, 9 parts out of 10.)
- (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.
- (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.
- (a) 15:1 — Write the ratio distance : petrol using the numbers in the question: 90 : 6. Divide both parts by their highest common factor, 6, to give 15 : 1. (1:15 comes from writing the ratio the wrong way round, petrol : distance. 5:2 comes from dividing only the distance by 6 and leaving the 6 litres unchanged, giving 15:6, which does simplify to 5:2 — the error is the step before, because both parts of a ratio must be divided by the same number. 14:1 comes from subtracting at the first step instead of dividing, 90 − 6 = 84, and then writing 84 : 6 and dividing both parts by 6 to get 14 : 1; subtracting has no part in simplifying a ratio.)
- (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.)
- (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.
- (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) 1:25 — Write the ratio time : volume using the numbers in the question: 8 : 200. Divide both parts by their highest common factor, 8, to give 1 : 25. (25:1 comes from writing the ratio the wrong way round, volume : time. 8:25 comes from dividing only the volume by 8 and leaving the time unchanged. 25:8 is that same mistake written the wrong way round.)
- (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.)
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