TMUA Ratio and Proportion — Questions and Methods

These are SummitPapers original questions, not official past paper questions. Official TMUA questions sorted by topic are on TMUA past papers by topic.

M3Ratio and proportion
18 questions14 on Paper 1 · 4 on Paper 2
8 free solutionsThe rest show the correct letter only

Covers: a mixture in a given ratio, a quantity shared in fractions, two journeys closing a gap, a remainder split into fractions, work rates that add, ratios that change after a transfer, compound percentage change, scale factors for length, area and volume, direct and inverse proportion.

Paper 1 (14)Paper 2 (4)Practise this topic

Back to the syllabus · Ratio and Proportion · 1 of 18

Question 1Ratio and proportion

A box holds red, blue and yellow counters. Two fifths of the counters are red. Of the remaining counters, one third are blue and the other 24 are yellow. How many counters are in the box?

What this topic tests

These questions ask how quantities share a total, or how fast two rates close a gap. You convert a sentence into parts, or into a fraction of what is left, and then you scale.

The questions include a mixture in a given ratio, a fraction of what remains, two speeds that close a gap, combined work rates, and a total split into fractions. Some are inverse proportion, and some are a chain of percentage changes. A percentage inside a mixture is a weight in that ratio. It is not an unweighted average of the percentages.

How it is assessed

Ratio and proportion is Section 1, so the specification allows it on Paper 1 and on Paper 2. This set has questions on both papers.

Each question has five options. A full paper is 20 questions in 75 minutes, and a calculator is not allowed. This note does not guess how many ratio questions a live paper will contain. UAT-UK does not publish a fixed number per topic.

This note is written for SummitPapers. The official list of what can be examined is the specification, together with the Notes on Mathematics and, for Paper 2, the Notes on Logic and Proof.

Key methods

Parts of a ratio

A ratio 5 : 4 : 3 means 5 parts, 4 parts and 3 parts, twelve parts in all. A percentage contributed by one liquid is weighted by its number of parts. The acid in 5 parts at 40% and 4 parts at 15% is 5×40 + 4×15, not (40 + 15) / 2.

A fraction of what remains

If two fifths are red, three fifths are not red. “One third of the remaining” applies to those three fifths, not to the whole box. Write the yellow share as a fraction of the total before you divide the given number by it.

Rates add

Two cyclists riding towards each other close the gap at the sum of their speeds, not at the average. Two workers on the same job add their work rates. The time for the job is 1 divided by the combined rate. Their times do not add.

Inverse proportion is on the specification

Inverse proportion means that doubling one quantity halves the other. A gear train and a quantity inversely proportional to a square root are that pattern. A mixture, and two speeds that close a gap, are not.

Common mistakes

Using the fraction on the wrong whole

Taking “one third of the remainder” as one third of the box, or treating every non-red counter as yellow, changes the total. The blue counters are still there.

Averaging the speeds

The average of 18 km/h and 30 km/h is 24 km/h. The gap closes at 48 km/h. The question is about the gap that remains, so the speeds add.

An unweighted percentage

The mean of 40%, 15% and the unknown percentage is not the acid in the mixture, because the three liquids are not in equal amounts.

Answering with the wrong share

A team total, Cara’s points, and the other players’ points are three different numbers. The last line of the question says which one is required.

Worked example

This question is also in the list below, with the solution folded. It is opened here so the method is on the page before the other questions.

Liquids P, Q and R are mixed in the ratio 5 : 4 : 3. Liquid P is 40% acid and liquid Q is 15% acid. The mixture is 28% acid. What percentage of R is acid?

A. 28
B. 65/3
C. 29
D. 76/3
E. The mixture is impossible

Answer: D. 76/3

Worked solution. Take 5, 4 and 3 units of P, Q and R. With p the acid percentage in R, the acid is 5(40) + 4(15) + 3p = 260 + 3p, and the total volume is 12 units. Setting (260 + 3p)/12 = 28 gives 260 + 3p = 336, so p = 76/3. This lies between 0 and 100. P alone is already above 28% and Q is below it, so the mixture is possible.

Why the other options look right. A assumes R must have the same acid percentage as the final mixture, 28%. B divides the acid from P and Q alone by the total volume, (200 + 60)/12 = 65/3, and reports that as the percentage for R instead of solving 260 + 3p = 336. C is 29, from the unweighted condition (40 + 15 + p)/3 = 28. E says the result is impossible, but 76/3 lies between 0 and 100.

Paper 1 questions

Paper 1 is Applications of Mathematical Knowledge. Calculators are not allowed.

Q1. A box holds red, blue and yellow counters. Two fifths of the counters are red. Of the remaining counters, one third are blue and the other 24 are yellow. How many counters are in the box?
Free · worked solution included
Answer and worked solution
Answer: C. 60
Worked solution. The counters that are not red form 1 − 2/5 = 3/5 of the box. Two thirds of these are yellow, so the yellow counters are (2/3) × (3/5) = 2/5 of the box. Since 2/5 of the total is 24, the total is 24 × 5/2 = 60. Check: 24 red, leaving 36, of which 12 are blue and 24 are yellow.
Why the other options look right. A takes the 24 yellow counters to be two thirds of the whole box, forgetting that the thirds apply only to the counters left after the red ones: 24 ÷ (2/3) = 36. B ignores the blue counters and treats all the non-red counters as yellow: 24 ÷ (3/5) = 40. D takes one third of the whole box to be blue, so the yellow fraction becomes 1 − 2/5 − 1/3 = 4/15 and the total becomes 24 × 15/4 = 90. E swaps the two shares of the remainder, taking the yellow counters to be one third of the 3/5, which is 1/5 of the box, so the total becomes 24 × 5 = 120.
Q2. Liquids P, Q and R are mixed in the ratio 5 : 4 : 3. Liquid P is 40% acid and liquid Q is 15% acid. The mixture is 28% acid. What percentage of R is acid?
Free · worked solution included
Answer and worked solution
Answer: D. 76/3
Worked solution. Take 5, 4 and 3 units of P, Q and R. With p the acid percentage in R, the acid is 5(40) + 4(15) + 3p = 260 + 3p, and the total volume is 12 units. Setting (260 + 3p)/12 = 28 gives 260 + 3p = 336, so p = 76/3. This lies between 0 and 100. P alone is already above 28% and Q is below it, so the mixture is possible.
Why the other options look right. A assumes R must have the same acid percentage as the final mixture, 28%. B divides the acid from P and Q alone by the total volume, (200 + 60)/12 = 65/3, and reports that as the percentage for R instead of solving 260 + 3p = 336. C is 29, from the unweighted condition (40 + 15 + p)/3 = 28. E says the result is impossible, but 76/3 lies between 0 and 100.
Q3. Two cyclists start 240 km apart and ride towards each other at constant speeds of 18 km/h and 30 km/h. How many kilometres apart are they 40 minutes before they meet?
Free · worked solution included
Answer and worked solution
Answer: C. 32
Worked solution. Together they close the gap at 48 km/h, so they meet after 240/48 = 5 hours. Forty minutes is 2/3 of an hour, and in that time the gap they still have to close is 48 × 2/3 = 32 km. That moment is well after the start, so the answer does not depend on any further detail of the 240 km.
Why the other options look right. A uses only the slower speed for 2/3 of an hour. B uses the average of the two speeds, 24 km/h, instead of their sum. D is the combined speed in km/h, with the 40 minutes left unused. E is the distance already closed by that moment, 240 − 32, rather than the gap that remains.
Q4. Two bags hold marbles in the ratio 5 : 3. After 12 marbles are moved from the first bag to the second, the ratio of the number of marbles in the first bag to the number in the second becomes 7 : 9. How many marbles were in the first bag originally?
Free · worked solution included
Answer and worked solution
Answer: C. 40
Worked solution. Let the bags hold 5k and 3k marbles. After the move they hold 5k − 12 and 3k + 12, so 9(5k − 12) = 7(3k + 12). This gives 45k − 108 = 21k + 84, so 24k = 192 and k = 8. The first bag originally held 5 × 8 = 40 marbles. Check: after the move the bags hold 28 and 36, and 28 : 36 = 7 : 9.
Why the other options look right. A gives the original number in the second bag, 3k = 24, instead of the first. B gives the number left in the first bag after the move, 40 − 12 = 28, instead of the original number. D gives the total number of marbles, 8k = 64, instead of the number in the first bag. E writes the new ratio the wrong way round as 9 : 7, solving 7(5k − 12) = 9(3k + 12) to get k = 24 and 5k = 120.
Q5. A tank contains 40 litres of a solution that is 30% acid by volume. Some of the solution is drained off and replaced by the same volume of pure acid. The tank then contains 40 litres of solution that is 50% acid by volume. How many litres of solution were drained off?
Free · worked solution included
Answer and worked solution
Answer: B. 80/7
Worked solution. At the start the tank holds 0.3 × 40 = 12 litres of acid, and at the end it must hold 0.5 × 40 = 20 litres. Draining x litres removes 0.3x litres of acid, and adding x litres of pure acid adds x litres. So 12 − 0.3x + x = 20, which gives 0.7x = 8 and x = 80/7. This is less than 40, so it is possible.
Why the other options look right. A forgets that the drained solution carries acid away, solving 12 + x = 20 to get 8. C adds pure acid without draining any solution, solving (12 + x)/(40 + x) = 1/2 to get 16. D gives the 20 litres of acid needed in the final solution instead of the volume drained. E divides the 8 extra litres of acid by 0.3 instead of by 1 − 0.3 = 0.7, giving 80/3.
Q6. Pipe P alone fills an empty tank in 6 hours, and pipe Q alone fills it in 10 hours. Drain R alone empties a full tank in 15 hours. Starting with the tank empty, P, Q and R are all opened at the same time. As soon as the tank is half full, R is closed, while P and Q stay open. How many hours does it take to fill the tank, starting from empty?
Free · worked solution included
Answer and worked solution
Answer: D. 35/8
Worked solution. With all three open the tank fills at 1/6 + 1/10 − 1/15 = (5 + 3 − 2)/30 = 1/5 of a tank per hour, so the first half takes (1/2) ÷ (1/5) = 5/2 hours. With R closed the rate is 1/6 + 1/10 = 8/30 = 4/15 of a tank per hour, so the second half takes (1/2) ÷ (4/15) = 15/8 hours. The total is 5/2 + 15/8 = 35/8 hours.
Why the other options look right. A adds the drain's rate instead of subtracting it in the first half, using 1/3 of a tank per hour for 3/2 hours, then 15/8 hours, giving 27/8. B ignores the drain completely, so the whole tank fills at 4/15 per hour in 15/4 hours. C averages the two rates, 1/5 and 4/15, to get 7/30 and uses it for the whole tank, giving 30/7, instead of timing each half separately. E keeps the drain open for the whole time, so the tank fills at 1/5 per hour in 5 hours.
Q7. Sam walks up an escalator that is moving upwards and reaches the top in 24 seconds. Walking at the same speed relative to the steps, Sam walks down the same escalator, still moving upwards, from the top to the bottom in 120 seconds. How many seconds would Sam take to travel from the bottom to the top by standing still on the moving escalator?
Free · worked solution included
Answer and worked solution
Answer: C. 60
Worked solution. Measure speed in escalator lengths per second. Let Sam's walking speed be w and the escalator's speed be e. Then w + e = 1/24 and w − e = 1/120. Subtracting gives 2e = 1/24 − 1/120 = 4/120, so e = 1/60. Standing still, Sam takes 1 ÷ (1/60) = 60 seconds.
Why the other options look right. A forgets to halve after subtracting, taking e = 1/24 − 1/120 = 1/30 and getting 30 seconds. B finds Sam's walking speed w = 1/40 instead of the escalator's speed, giving the time to walk up a stationary escalator. D averages the two times, (24 + 120)/2 = 72, instead of working with speeds. E subtracts the times, 120 − 24 = 96, instead of the speeds.
Q8. The price of a coat is increased by 25%. The new price is then reduced by x%, and the reduced price is then increased by 20%. The final price equals the original price. What is the value of x?
Free · worked solution included
Answer and worked solution
Answer: B. 100/3
Worked solution. The three changes multiply the price by 1.25, by (1 − x/100) and by 1.2. So 1.25 × 1.2 × (1 − x/100) = 1, that is 1.5(1 − x/100) = 1. Hence 1 − x/100 = 2/3 and x = 100/3.
Why the other options look right. A adds the two rises to make a single 45% rise and then finds the reduction that undoes it: 1 − 1/1.45 = 9/29, so x = 900/29, instead of multiplying the factors 1.25 and 1.2. C undoes each rise separately and adds the reductions, 20% + 50/3% = 110/3%, instead of combining the multipliers. D simply adds 25% and 20%. E finds the combined factor 1.5 but takes a 50% reduction to undo a 50% rise, instead of dividing by 1.5.
Q9. Two solid cylinders, P and Q, are mathematically similar and are made of the same material. The total surface area of Q is 44% greater than the total surface area of P. By what percentage is the mass of Q greater than the mass of P?
Free · correct letter only
Answer
Answer: D. 72.8
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Q10. A sum of money was going to be shared between Pat, Quinn and Ravi in the ratio 2 : 3 : 5. Instead, it is shared in the ratio 3 : 4 : 5, and as a result Ravi receives £40 less. How much more does Pat receive than under the original ratio?
Free · correct letter only
Answer
Answer: C. £24
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Q11. Positive quantities a, b and c are related as follows: a is directly proportional to b³, and b is inversely proportional to √c. If c is multiplied by 9, by what factor is a multiplied?
Free · correct letter only
Answer
Answer: B. 1/27
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Q12. Gear P has 24 teeth and drives gear Q, which has 36 teeth. Gear Q is fixed on the same axle as gear R, so Q and R turn at the same rate. Gear R has 16 teeth and drives gear S, which has 40 teeth. Gear P turns at 150 revolutions per minute. At how many revolutions per minute does gear S turn?
Free · correct letter only
Answer
Answer: A. 40
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Q13. Worker n, for each integer n ≥ 0, finishes a job alone in 2ⁿ days. Let f(k) be the number of days needed when workers 0, 1, ..., k work together and their work rates add. What is f⁻¹(128/255)?
Free · correct letter only
Answer
Answer: B. 7
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Q14. One quarter of a team's points were scored by Amir and one sixth by Bea. Cara scored 11 points. None of the other 4 players scored more than 2 points. How many points did those other 4 players score altogether?
Free · correct letter only
Answer
Answer: A. 3
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Paper 2 questions

Paper 2 is Mathematical Reasoning. It can test this same topic. Argument, proof, and identifying errors are the topics that appear on Paper 2 only.

Q1. The positive quantities y and z depend on the positive quantity x. The quantity y is directly proportional to x², and z is inversely proportional to √x. Which one of the following must be constant for all x > 0?
Free · correct letter only
Answer
Answer: A. yz⁴
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Q2. A town's population increases by a% in one year and then by b% in the next year, where a and b are non-negative numbers with a + b = 20. Which one of the following must be true about the overall percentage increase in the population over the two years?
Free · correct letter only
Answer
Answer: E. It is at least 20% and at most 21%.
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Q3. The positive quantity y is inversely proportional to the positive quantity x. Which of the following statements must be true? I: If x increases by 50%, then y decreases by 50%. II: If x decreases by 50%, then y increases by 100%. III: For every k > 0, if x increases by k%, then y decreases by less than k%.
Free · correct letter only
Answer
Answer: E. II and III only
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Q4. Alex and Bea each make the same journey. Alex drives for half of the time at a constant speed u and for the other half of the time at a constant speed v. Bea drives half of the distance at speed u and the other half of the distance at speed v. The speeds u and v are positive and u ≠ v. Which one of the following must be true?
Free · correct letter only
Answer
Answer: A. Alex takes less time than Bea.
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A box holds red, blue and yellow counters. Two fifths of the counters are red. Of the remaining counters, one third are blue and the other 24 are yellow. How many counters are in the box?

What this topic tests

How it is assessed

Key methods

Parts of a ratio

A fraction of what remains

Rates add

Inverse proportion is on the specification

Common mistakes

Using the fraction on the wrong whole

Averaging the speeds

An unweighted percentage

Answering with the wrong share

Worked example

Liquids P, Q and R are mixed in the ratio 5 : 4 : 3. Liquid P is 40% acid and liquid Q is 15% acid. The mixture is 28% acid. What percentage of R is acid?

Paper 1 questions

Q1. A box holds red, blue and yellow counters. Two fifths of the counters are red. Of the remaining counters, one third are blue and the other 24 are yellow. How many counters are in the box?

Q2. Liquids P, Q and R are mixed in the ratio 5 : 4 : 3. Liquid P is 40% acid and liquid Q is 15% acid. The mixture is 28% acid. What percentage of R is acid?

Q3. Two cyclists start 240 km apart and ride towards each other at constant speeds of 18 km/h and 30 km/h. How many kilometres apart are they 40 minutes before they meet?

Q4. Two bags hold marbles in the ratio 5 : 3. After 12 marbles are moved from the first bag to the second, the ratio of the number of marbles in the first bag to the number in the second becomes 7 : 9. How many marbles were in the first bag originally?

Q5. A tank contains 40 litres of a solution that is 30% acid by volume. Some of the solution is drained off and replaced by the same volume of pure acid. The tank then contains 40 litres of solution that is 50% acid by volume. How many litres of solution were drained off?

Q8. The price of a coat is increased by 25%. The new price is then reduced by x%, and the reduced price is then increased by 20%. The final price equals the original price. What is the value of x?

Q9. Two solid cylinders, P and Q, are mathematically similar and are made of the same material. The total surface area of Q is 44% greater than the total surface area of P. By what percentage is the mass of Q greater than the mass of P?

Q10. A sum of money was going to be shared between Pat, Quinn and Ravi in the ratio 2 : 3 : 5. Instead, it is shared in the ratio 3 : 4 : 5, and as a result Ravi receives £40 less. How much more does Pat receive than under the original ratio?

Q11. Positive quantities a, b and c are related as follows: a is directly proportional to b³, and b is inversely proportional to √c. If c is multiplied by 9, by what factor is a multiplied?

Q12. Gear P has 24 teeth and drives gear Q, which has 36 teeth. Gear Q is fixed on the same axle as gear R, so Q and R turn at the same rate. Gear R has 16 teeth and drives gear S, which has 40 teeth. Gear P turns at 150 revolutions per minute. At how many revolutions per minute does gear S turn?

Q13. Worker n, for each integer n ≥ 0, finishes a job alone in 2n days. Let f(k) be the number of days needed when workers 0, 1, ..., k work together and their work rates add. What is f⁻¹(128/255)?

Q14. One quarter of a team's points were scored by Amir and one sixth by Bea. Cara scored 11 points. None of the other 4 players scored more than 2 points. How many points did those other 4 players score altogether?

Paper 2 questions

Q1. The positive quantities y and z depend on the positive quantity x. The quantity y is directly proportional to x², and z is inversely proportional to √x. Which one of the following must be constant for all x > 0?

Q2. A town's population increases by a% in one year and then by b% in the next year, where a and b are non-negative numbers with a + b = 20. Which one of the following must be true about the overall percentage increase in the population over the two years?

Related topics

Q13. Worker n, for each integer n ≥ 0, finishes a job alone in 2ⁿ days. Let f(k) be the number of days needed when workers 0, 1, ..., k work together and their work rates add. What is f⁻¹(128/255)?