MathematicsFoundation20 min read

Ratio, Proportion and Rates

Comparing quantities, and knowing whether more means more or less

This topic appears in:

O Level Mathematics (4024) · ch.1
01

A ratio compares parts; a fraction compares a part with the whole

If a class has 12 boys and 18 girls, the ratio of boys to girls is 12 : 18, which simplifies to 2 : 3. The fraction that are boys is 12/30, which is 2/5. Both describe the same class and they are different numbers, which is why the two are so easily confused.

The rule: a ratio has as many numbers as there are groups, and they need not add to anything in particular. A fraction always compares one part against the total. In 2 : 3 there are 5 parts altogether, and that total is what converts a ratio into a fraction.

Ratio 2 : 3 is not two thirds

It means two parts to three parts, so five parts in all — the first group is two fifths of the total, not two thirds. The number 3 is the size of the other group, never the denominator. This single confusion accounts for more lost marks than anything else in the topic.

02

Sharing in a given ratio

The method never varies: add the parts, divide the total by that number to find one part, then multiply. Writing the "one part" value down explicitly is what makes the rest safe.

Worked example

Rs 4500 is shared between three people in the ratio 2 : 3 : 4. How much does each receive?

  1. Total parts = 2 + 3 + 4 = 9.Always add the parts first; everything else depends on this number.
  2. One part = 4500 ÷ 9 = 500.Write this down as its own line. It is the value you will multiply by three times.
  3. Shares: 2 × 500 = 1000, 3 × 500 = 1500, 4 × 500 = 2000.
  4. Check: 1000 + 1500 + 2000 = 4500 ✓The shares must add back to the original total. This check costs five seconds and catches every arithmetic slip.

Rs 1000, Rs 1500 and Rs 2000

03

Direct and inverse proportion

Two quantities are in direct proportion when doubling one doubles the other — the ratio between them stays fixed and the graph is a straight line through the origin. Five identical books cost five times what one costs.

They are in inverse proportion when doubling one halves the other — their product stays fixed and the graph is a hyperbola. Twice as many workers take half the time; twice the speed takes half as long.

Deciding which applies is the whole question, and the test is simple: ask whether more of one thing means more or less of the other.

direct:y ∝ xy = kxy/x is constantinverse:y ∝ 1/xy = k/xxy is constantsquare:y ∝ x²y = kx²inverse square: y ∝ 1/x²y = k/x²find k from the pair of values you are given, then use it for every other pair
Worked example

Six workers build a wall in 10 days. How long would 15 workers take, assuming they all work at the same rate?

  1. More workers means less time, so this is inverse proportion.Ask the direction first. Treating it as direct would give a longer time, which is obviously wrong.
  2. The product is constant: 6 × 10 = 60 worker-days.This is the total amount of work, and it does not change with the number of workers.
  3. 15 × t = 60, so t = 4 days.
  4. Sanity check: more workers, fewer days — 4 is less than 10 ✓Checking the direction of the answer catches an inverted proportion instantly.

4 days

The assumption these questions hide

Every worker-and-days question assumes all workers are equally fast, that they do not get in each other's way, and that the work divides perfectly. In reality none of those hold — a hundred workers on a small wall would be slower, not faster. A question asking you to "comment on the assumptions" is asking for exactly this.

04

Rates, and the units that give them away

A rate compares two quantities of different kinds: kilometres per hour, rupees per kilogram, litres per 100 km. The word "per" is doing the dividing, and the units tell you which way round.

That is the most useful trick in the topic. If an answer should be in km/h, then whatever you compute must be kilometres divided by hours. Getting the units right forces the arithmetic to be right.

speed = distance / timedistance = speed × timetime = distance / speeddensity = mass / volumeaverage speed = TOTAL distance / TOTAL timeaverage speed is never the average of two speeds unless equal times were spent at each
Worked example

A car travels 60 km at 60 km/h and then 60 km at 120 km/h. Find the average speed for the whole journey.

  1. It is tempting to answer 90 km/h. That is wrong.The average of the two speeds would only be correct if equal times were spent at each, not equal distances.
  2. First stage: 60 ÷ 60 = 1 hour. Second stage: 60 ÷ 120 = 0.5 hours.More time is spent at the slower speed, which is what pulls the average down.
  3. Total distance = 120 km; total time = 1.5 hours.
  4. Average speed = 120 ÷ 1.5 = 80 km/h.Below 90, as expected, because the slower speed occupied twice as much of the journey.

80 km/h

Before you leave this chapter

  1. Ratio 2 : 3 means five parts in total — the first group is two fifths, not two thirds.
  2. To share in a ratio: add the parts, find one part, multiply, then check the total.
  3. Direct: more means more, y = kx. Inverse: more means less, xy is constant.
  4. Ask which direction before choosing a method — the sanity check afterwards is the same question.
  5. Average speed is total distance over total time, never the average of the speeds.
05

The two shapes proportion makes

Direct and inverse proportion are distinguishable at a glance once you know what each looks like on a graph, and a question giving a table of values is often really asking you to spot which shape it fits.

Direct proportion gives a straight line through the origin — double x and y doubles, and when x is zero so is y. Inverse proportion gives a curve approaching both axes, because as x grows y shrinks towards zero without ever reaching it.

Compare Linear with Reciprocal. Direct proportion is the straight line; inverse is the two-branch curve. The reciprocal never reaches either axis, which is what "as one grows the other shrinks towards nothing" looks like.

Testing a table of values

Given pairs of values, divide each y by its x. If the answers are all the same, it is direct proportion and that constant is k. If instead each product xy is the same, it is inverse proportion. Two quick columns settle it without any graph at all.

Practice questions

6 questions · 20 marks · full working on every one

Try each one on paper first, then open the working. The marks are shown where they are actually awarded, because that is where they are actually lost.

Short questions

3 · 6 marks

Two marks each, in the style of the short-question section of the paper. Answer in two or three lines.

SQ1[2 marks]
Simplify the ratio 45 : 60 : 75.
Model answer

The highest common factor is 15, so dividing each part gives 3 : 4 : 5.

Examiner tip. Find the HCF of all three parts at once. Dividing by 5 first and then by 3 gets there too, but takes an extra line.

SQ2[2 marks]
In a ratio of 3 : 5, what fraction of the total is the first part?
Model answer

There are 3 + 5 = 8 parts altogether, so the first part is 3/8 of the total.

Examiner tip. The denominator is the sum of the parts, never the second number. Answering 3/5 is the classic error.

SQ3[2 marks]
A car covers 150 km in 2 hours 30 minutes. Find its average speed.
Model answer

2 hours 30 minutes is 2.5 hours, so speed = 150 ÷ 2.5 = 60 km/h.

Examiner tip. Convert the time to a decimal number of hours before dividing. Using 2.30 gives 65.2 and is wrong.

Solved numericals

2 · 8 marks

Full working, one step per line, with the marks shown where they are awarded.

N1[4 marks]
A sum of money is divided between Ali, Sara and Bilal in the ratio 5 : 7 : 8. Bilal receives Rs 2400 more than Ali. Find the total sum.
Full working
  1. The difference between Bilal and Ali is 8 − 5 = 3 partswork with the difference in parts, not the shares[1]
  2. 3 parts = 2400, so one part = 800[1]
  3. Total parts = 5 + 7 + 8 = 20[1]
  4. Total sum = 20 × 800 = Rs 16 000check: shares 4000, 5600, 6400 differ by 2400 ✓[1]

Rs 16 000

Examiner tip. When a question gives a difference rather than a total, work out what that difference is in parts first. Everything else follows from the value of one part.

N2[4 marks]
y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10, and x when y = 12.
Full working
  1. Inverse proportion means y = k/x, so xy = k[1]
  2. k = 4 × 15 = 60find k from the given pair[1]
  3. When x = 10: y = 60 ÷ 10 = 6x went up, y went down ✓[1]
  4. When y = 12: x = 60 ÷ 12 = 5[1]

y = 6; x = 5

Examiner tip. Find k once and it serves every later part of the question. Writing k = 60 on its own line makes the rest one division each.

Long questions

1 · 6 marks

Theory and numerical together, as they appear in the long-question section.

LQ1[6 marks]
A recipe for 4 people uses 300 g of rice and takes 25 minutes to cook.
  1. How much rice is needed for 10 people?
  2. Would the cooking time for 10 people be 62.5 minutes? Explain.
  3. Eight workers can paint a house in 6 days. How long would 3 workers take, and what assumption have you made?
Mark scheme
  1. Rice is in direct proportion to the number of people: 300 ÷ 4 = 75 g each[1]
  2. 75 × 10 = 750 g[1]
  3. No — cooking time is not proportional to quantity[1]
  4. Rice cooks by absorbing water at a rate that does not depend on how much is in the pot, so a larger quantity takes only a little longer, not two and a half times as longthe reason is the mark[1]
  5. Painting is inverse: 8 × 6 = 48 worker-days, so 48 ÷ 3 = 16 days[1]
  6. Assumption: all workers paint at the same rate and do not obstruct one another[1]

(a) 750 g (b) no — cooking time is not proportional to quantity (c) 16 days, assuming equal and independent working rates

Examiner tip. Part (b) is the understanding mark. Not everything that increases together is in proportion, and questions plant an example like this deliberately.