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Maths explained

Ratio and proportion explained simply, with worked examples

Ratio and proportion explained simply: simplify ratios, share an amount in a ratio, and solve direct and inverse proportion problems, with full workings.

For students. By the Qlive team. . 5 min read.

This guide is for you if ratio questions seem to need a different trick every time. In CBSE you first meet ratio and proportion in Class 6, and it returns in Classes 7 and 8 and in later chapters. It is a major topic in Key Stage 3, GCSE and IGCSE maths as well. The good news is that almost every question uses the same small set of ideas.

The idea in plain words

A ratio compares two or more quantities. If a recipe uses 2 cups of rice for every 3 cups of water, the ratio of rice to water is 2:3 (said “2 to 3”).

Ratios work like fractions in one important way: you can multiply or divide all the parts by the same number and the ratio stays the same. So 2:3 = 4:6 = 10:15.

A proportion says two ratios are equal. If 2:3 = 4:6, the quantities are in proportion.

There are two kinds of proportion between changing quantities:

Type What happens Example
Direct proportion Both go up together at the same rate More notebooks, more cost
Inverse proportion One goes up as the other goes down More workers, fewer days to finish

Worked examples

Example 1: simplifying a ratio

Simplify 12:18.

  1. Find the highest common factor of 12 and 18. It is 6.
  2. Divide both parts by 6: 12 ÷ 6 = 2 and 18 ÷ 6 = 3

Answer: 2:3

If the units are different, make them the same first. For 50 cm : 2 m, change 2 m into 200 cm. Then 50:200, and dividing by 50 gives 1:4.

Example 2: sharing an amount in a given ratio

Share ₹600 between Asha and Ben in the ratio 2:3.

  1. Add the parts: 2 + 3 = 5 parts in total
  2. Find the value of one part: 600 ÷ 5 = 120
  3. Asha gets 2 parts: 2 × 120 = 240
  4. Ben gets 3 parts: 3 × 120 = 360

Answer: Asha ₹240, Ben ₹360

Check: 240 + 360 = 600, and 240:360 simplifies to 2:3 (divide both by 120).

Example 3: when you know one part

In a club, the ratio of boys to girls is 4:5. There are 20 boys. How many girls are there?

  1. The boys are 4 parts, and 4 parts = 20
  2. One part = 20 ÷ 4 = 5
  3. The girls are 5 parts: 5 × 5 = 25

Answer: 25 girls

Notice that here you did not add the parts, because you were given one side of the ratio, not the total.

Example 4: direct proportion (the unitary method)

5 pens cost ₹60. How much do 8 pens cost?

  1. Find the cost of one pen: 60 ÷ 5 = 12
  2. Multiply by the number you want: 12 × 8 = 96

Answer: ₹96

Example 5: inverse proportion

6 workers take 10 days to paint a building. How long would 4 workers take, working at the same rate?

Fewer workers means more days, so this is inverse proportion.

  1. Find the total amount of work in “worker-days”: 6 × 10 = 60
  2. Share it among 4 workers: 60 ÷ 4 = 15

Answer: 15 days

Example 6: finding a missing term

Find x if 3:x = 9:12.

In a proportion a:b = c:d, the product of the outer terms equals the product of the inner terms: a × d = b × c.

  1. 3 × 12 = x × 9
  2. 36 = 9x
  3. x = 4

Check: 3:4 and 9:12 both simplify to 3:4.

Common mistakes and how to avoid them

Dividing by one part instead of the total. In Example 2, some students do 600 ÷ 2 and 600 ÷ 3. Always add the parts first when you are sharing a total.

Adding the parts when you should not. If you are told the size of one share, as in Example 3, that share is the number of parts it represents. Do not add the parts.

Writing the ratio in the wrong order. “Girls to boys” is not the same as “boys to girls”. Write the labels above the numbers if it helps.

Mixing units. 50 cm : 2 m is not 50:2. Convert to the same unit before simplifying.

Treating inverse proportion as direct. If the answer to “4 workers” comes out smaller than the answer for 6 workers, stop and think. Fewer people should take longer. Ask “should the answer go up or down?” before you calculate.

Not simplifying fully. 4:6 is correct but not simplest. Keep dividing until the parts share no common factor.

Try these

  1. Simplify 45:60.
  2. Share 840 in the ratio 3:4.
  3. Two numbers are in the ratio 2:7. The smaller number is 14. Find the larger number.
  4. 3 kg of apples cost ₹135. How much do 7 kg cost?
  5. 8 taps fill a tank in 15 minutes. How long would 12 identical taps take?

Answers

  1. The HCF is 15, so 45:60 = 3:4.
  2. 7 parts, one part = 840 ÷ 7 = 120. Shares are 360 and 480.
  3. 2 parts = 14, so one part = 7. The larger is 7 × 7 = 49.
  4. One kg costs 135 ÷ 3 = 45. Seven kg cost 45 × 7 = ₹315.
  5. Inverse proportion. Total tap-minutes = 8 × 15 = 120. 120 ÷ 12 = 10 minutes.

Still unsure?

Ratio questions are often really about fractions and division, so a small gap there can make the whole topic feel harder than it is. A Qlive teacher can work through it with you one-to-one, starting with a free assessment that finds the exact gap. You can read more on our maths page, and our guide to percentages made simple is a natural next step.

Common questions

What is the difference between a ratio and a fraction?

A ratio compares parts with each other, while a fraction compares a part with the whole. If a class has boys and girls in the ratio 2:3, there are 5 parts in total, so the fraction of the class that are boys is 2/5, not 2/3.

Does the order in a ratio matter?

Yes. The ratio 2:3 for boys to girls means something different from 3:2. Always write the ratio in the same order as the words in the question.

How do I tell direct proportion from inverse proportion?

Ask what happens when one quantity doubles. If the other also doubles, such as more pens costing more money, it is direct. If the other halves, such as more workers taking less time, it is inverse.

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