A recipe for one tray of flapjacks uses oats and syrup in the ratio 5:2. For two trays, how many parts of oats and syrup are needed? Give the answer as parts and then state the simplified ratio for the two trays.
(2)
2
A map uses a scale where 1 cm represents 3 km. Measure the real distance represented by 7 cm on the map.
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3
Write 0.35 as a fraction in its simplest form, then give it as a percentage.
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4
A school orders 84 identical pens. They are packed in boxes that each hold pens in the ratio of red:blue:green = 3:2:1. How many pens of each colour are in the whole order?
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5
A shop reduces the price of a jumper by 20%. The original price was 45 pounds. Work out the sale price.
(2)
6
A runner completes a 10 km route in 50 minutes. What is their average speed in km/h? Give your answer to one decimal place.
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7
Three friends share a takeaway in the ratio 2:3:4. The total bill is 54 pounds. How much does the friend with 3 parts pay?
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8
A student increases the ingredients for a cake by a scale factor of 2.5. The original recipe uses 240 g of flour. How much flour is needed now? Give your answer in grams.
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9
A lemonade recipe uses sugar and water in the ratio 1:7. If 4 litres of lemonade are needed, how many litres of sugar are required? (Assume the ratio parts add to the total volume.)
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10
A garden centre sells potting compost in 3 sizes. The masses are in the ratio 4:6:9. If the smallest bag is 4 kg, what is the mass of the largest bag?
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11
A computer game applies a 15% discount then adds 5% VAT to the discounted price. If the original price is 40 pounds, work out the final price after discount and VAT. Give your answer to the nearest penny.
(2)
12
A mechanic charges labour and parts in the ratio 7:3. If a job costs 280 pounds in total, how much is charged for parts? Show your working.
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13
A photographer prints photos in sizes A and B in the ratio A:B = 5:8. She printed 65 photos in total. How many of each size were printed?
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14
To make a diluted solution, a scientist mixes concentrate and water in the ratio 1:12. If she needs 780 ml of the diluted solution, how much concentrate should she use? Give your answer in ml.
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15
A car uses fuel at a rate of 6.5 litres per 100 km. How many litres are needed for a 275 km journey? Give your answer to one decimal place.
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16
A classroom has a ratio of boys to girls of 7:5. After 3 boys leave and 2 girls join, the ratio becomes 4:3. How many pupils were in the class originally?
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17
A baker mixes flour and water in the ratio 9:4. He needs at least 36 kg of mixture. What is the least whole number of kilograms of flour he must use?
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18
Stretch: A painting mixture uses pigments A, B and C in the ratio A:B = 3:2 and B:C = 5:6. Find the ratio A:B:C. Then, if the painter needs 792 ml of pigment C, how much pigment A is required?
(4)
Mark scheme · KS3.M-R11 Proportional Reasoning Problem Set
Question 1
M1 Scale 5:2 by 2 to give parts 10 and 4
A1 Simplified ratio 5:2 or 10:4 oe (5:2 stated as simplest)
Answer: Oats 10 parts, syrup 4 parts; ratio 5:2
Question 2
M1 Scale method: 7 x 3 km
A1 21 km cao
Answer: 21 km
Question 3
M1 Convert decimal to fraction 35/100 and simplify to 7/20
A1 35% or 35% cao
Answer: 7/20; 35%
Question 4
M1 Find total parts 3+2+1 = 6 and divide 84 by 6 to get one part = 14
M1 Multiply 14 by each part for method: red 42, blue 28, green 14
A1 Correct counts: 42 red, 28 blue, 14 green cao
Answer: 42 red, 28 blue, 14 green
Question 5
M1 Calculate 20% of 45 or find 80%: 45 x 0.2 = 9 or 45 x 0.8 = 36
A1 36 pounds cao
Answer: 36 pounds
Question 6
M1 Convert 50 minutes to hours: 50/60 = 5/6 h or 0.833... h
M1 Compute speed 10 / (50/60) = 10 x 60/50 = 12 km/h
A1 12.0 km/h cao to one decimal place
Answer: 12.0 km/h
Question 7
M1 Total parts 2+3+4 = 9, one part = 54/9 = 6
A1 3 parts = 18 pounds cao
Answer: 18 pounds
Question 8
M1 Multiply 240 by 2.5: 240 x 2.5 = 240 x (5/2) = 120 x 5 = 600
M1 Method shown with correct arithmetic
A1 600 g cao
Answer: 600 g
Question 9
M1 Total parts 1+7 = 8 so one part = 4/8 = 0.5 litres
M1 Sugar is 1 part so 0.5 litres
A1 0.5 litres cao
Answer: 0.5 litres
Question 10
M1 Find scale factor: smallest corresponds to 4 parts = 4 kg so 1 part = 1 kg
M1 Largest 9 parts = 9 x 1 = 9 kg
A1 9 kg cao
Answer: 9 kg
Question 11
M1 Apply 15% discount: 40 x 0.85 = 34.00
A1 Add 5% VAT: 34.00 x 1.05 = 35.70 cao
Answer: 35.70 pounds
Question 12
M1 Total parts 7+3 = 10, one part = 280/10 = 28
M1 Parts cost 3 x 28 = 84
A1 84 pounds cao
Answer: 84 pounds
Question 13
M1 Total parts 5+8 = 13, one part = 65/13 = 5
M1 Calculate A = 5 x 5 = 25 and B = 8 x 5 = 40
A1 25 A-size, 40 B-size cao
Answer: 25 A-size, 40 B-size
Question 14
M1 Total parts 1+12 = 13, one part = 780/13 = 60
M1 Concentrate is 1 part so 60 ml
A1 60 ml cao
Answer: 60 ml
Question 15
M1 Method: fuel = 6.5/100 x 275 or (275/100) x 6.5
M1 Compute 275 x 6.5 = 1787.5 then divide by 100 = 17.875
M1 Round to one decimal place shown method
A1 17.9 litres cao to one decimal place
Answer: 17.9 litres
Question 16
M1 Set up: let multiplier be k so boys = 7k, girls = 5k; after change boys 7k-3, girls 5k+2
M1 Equation (7k-3)/(5k+2) = 4/3 or 3(7k-3) = 4(5k+2)
M1 Solve: 21k -9 = 20k +8 gives k = 17
A1 Total pupils originally = 7k + 5k = 12k = 12 x 17 = 204 cao
Answer: 204
Question 17
M1 Flour:water = 9:4 so water = 4/9 of flour; total = flour + 4/9 flour = 13/9 flour, and 13/9 flour ≥ 36
M1 So flour ≥ 36 x 9/13 = 24.92...; the least whole number of kg of flour meeting this is 25 (24 kg gives total 34.7 kg, below 36 kg)
A1 25 kg cao
Answer: 25 kg
Question 18
M1 Combine ratios: A:B = 3:2 and B:C = 5:6; make B equal by scaling: for B common multiply A:B by 5 gives 15:10, and B:C by 2 gives 10:12
M1 State combined ratio A:B:C = 15:10:12
M1 Find one part value: C corresponds to 12 parts so one part = 792/12 = 66