Proportional Reasoning Problem Set
A proportional reasoning problem set mixes together several KS3 ratio and proportion skills in one sitting, without labelling which method each question needs, the way these questions appear on a real GCSE-style paper: sharing an amount in a given ratio, scaling a recipe using the unitary method, working out inverse proportion, comparing prices to find the best buy, applying a percentage change, or using a compound measure such as speed. The skill being tested is recognising, from the wording alone, which method a question needs before calculating.
Before you start
Make sure you're comfortable with these topics first:
Method
- Before calculating anything, read the whole question and look for key words that signal the type of proportion: 'in the ratio' or 'share' signals ratio sharing; 'the same rate' or 'each' signals the unitary method (direct proportion); phrasing linking more of one thing to less of another (such as workers and the time a job takes) signals inverse proportion; 'better value' or 'per' signals a best-buy comparison.
- Check whether the relationship is DIRECT (both quantities increase together, such as more hours worked meaning more pay) or INVERSE (one increases as the other decreases, such as more workers needing less time) before choosing a method - using the wrong one is the single most common error in a mixed problem set.
- When working through a full set under time pressure, do the ratio-sharing and unitary-method questions first, since the method is usually quickest to spot and calculate; leave inverse proportion and best-buy questions, which need an extra check, until the more straightforward marks are safely banked.
- For a direct proportion or scaling question, find the value of ONE unit first by dividing, then multiply up or down to the amount needed - this is the unitary method.
- For an inverse proportion question, multiply the two paired values together to find a constant total (such as workers x days), then divide this constant by the new value of the other quantity.
- For a best-buy question, convert every option to the same common unit, such as price per 100 g or price per litre, before comparing - never compare two totals, or two unit prices measured in different units.
- Show every step of working, including any intermediate unit value found along the way (such as 'cost per kg ='), since method marks in a real exam are awarded for a correct process even when the final answer has an arithmetic slip.
- At the end, sense-check the answer against the context: a shared amount should add back up to the original total, and a 'days to finish a job' answer should get SMALLER as workers are ADDED, never bigger.
Worked example
A school office photocopier prints 480 exam papers in 15 minutes, working at a constant rate. At this same rate, how many minutes would it take a SLOWER photocopier, which prints only half as many pages per minute, to print 600 exam papers?
- Identify the method: since each copier works at its own constant rate, this is direct proportion, so first find the fast copier's unit rate using the unitary method.
- Fast copier's rate: 480 pages / 15 minutes = 32 pages per minute.
- The slower copier prints half as many pages per minute, so its rate is 32 / 2 = 16 pages per minute.
- To find how long the slow copier takes to print 600 pages, divide the total by its rate: 600 / 16 = 37.5 minutes.
- Sense-check: the slow copier is half as fast, so for the same number of pages it should take about twice as long as the fast copier would. The fast copier would take 600 / 32 = 18.75 minutes, and 37.5 is exactly double 18.75, which confirms the answer.
Practice questions
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Q18 identical candles burn for a total of 20 hours if lit one after another. How many hours would 5 of these candles burn for, in total?Show answer
Answer: 12.5 hours (one candle burns for 20 / 8 = 2.5 hours, so 5 candles burn for 2.5 x 5 = 12.5 hours)
Q2It takes 6 machines 9 hours to fill a warehouse order, all working at the same rate. How many hours would it take 4 machines, working at the same rate, to fill an identical order?Show answer
Answer: 13.5 hours (this is inverse proportion: 6 x 9 = 54 machine-hours in total, so 4 machines take 54 / 4 = 13.5 hours)
Q3A 750 g box of cereal costs 3.30 pounds. A 500 g box of the same cereal costs 2.40 pounds. Which box is better value for money?Show answer
Answer: The 750 g box (3.30 / 750 x 100 = 0.44 pounds per 100 g, against 2.40 / 500 x 100 = 0.48 pounds per 100 g for the 500 g box)
Q4A charity bake sale raises 216 pounds, to be split between two classes in the ratio 5 : 4 according to how many cakes each class contributed. Work out how much more money the class with the larger share receives than the class with the smaller share.Show answer
Answer: 24 pounds (one part is 216 / 9 = 24 pounds, so the shares are 120 pounds and 96 pounds, a difference of 24 pounds)
Q5A recipe for 6 people uses 450 g of rice. Neha is cooking the same recipe for 15 people. How much rice does she need?Show answer
Answer: 1125 g, or 1.125 kg (one person needs 450 / 6 = 75 g, so 15 people need 75 x 15 = 1125 g)
Q6A laptop's original price was 480 pounds. In a sale, the price is reduced by 15 percent. Work out the sale price.Show answer
Answer: 408 pounds (480 x 0.85 = 408, or 480 minus 15 percent of 480, which is 480 - 72)
Q7A delivery van travels 175 miles in 3.5 hours at a constant speed. Work out the van's average speed, in miles per hour.Show answer
Answer: 50 mph (175 / 3.5)
Q8It takes 3 painters 8 days to paint a large fence, working at the same rate. A student says: '6 painters would take 16 days, because you double the painters so you double the days.' Explain why the student is wrong, and find the correct number of days 6 painters would take.Show answer
Answer: The student is wrong because this is inverse proportion, not direct: more painters means the job is finished FASTER, so doubling the painters should HALVE the time, not double it. The correct total is 3 x 8 = 24 painter-days, so 6 painters take 24 / 6 = 4 days
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A tank is filled with water by 5 identical pumps in 12 hours, all working at the same rate. (a) How long would it take 3 of these pumps to fill the same tank? (b) The water company then fills a SECOND tank, exactly twice the volume of the first, using all 5 original pumps. How long would this take?
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A market stall sells apples in two bag sizes: a 3 kg bag for 5.40 pounds, and an 8 kg bag for 13.60 pounds. (a) Work out the price per kg for each bag size, and state which is better value for money. (b) Farah buys the better-value bag and shares all of it with her two neighbours in the ratio 4 : 2 : 2, with Farah getting the largest share. How many kg does each neighbour with the smaller share receive?
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A recipe that serves 8 people uses 320 g of butter. A chef is scaling the recipe down to serve 3 people. (a) Work out how much butter is needed for 3 people. (b) The chef only has a 100 g block of butter. Explain whether this is enough.
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Free printable worksheet
Want more practice on paper? Download the proportional reasoning problem set worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 44 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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