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Everyday arithmetic

Ratio & proportion: compare and share

Use equal-sized ratio units to compare quantities, divide a total, and recognise direct and inverse relationships.

BeginnerAbout 13 min3 worked examples
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01 / THE STARTING POINT

Before you begin

What you need: Fractions and basic multiplication.

  • Simplify ratios in matching units
  • Divide a total in a given ratio
  • Distinguish direct from inverse proportion
02 / BUILD THE CONCEPT

Understand the idea

A ratio compares quantities

The ratio 3:5 means three units of one quantity for every five equal units of another. Before simplifying, convert both quantities to the same measurement unit.

Parts are not the whole

If two shares are in the ratio 3:5, there are 8 parts altogether. The first share is 3/8 of the total, not 3/5.

What stays constant?

In direct proportion, y/x is constant: twice the quantity costs twice as much at a fixed unit price. In inverse proportion, xy is constant: twice as many equally efficient workers need half the time for the same job.

03 / A DIFFERENT WAY TO SEE IT

Share 8 equal parts

First share: 3 parts
₹90₹90₹90
Second share: 5 parts
₹90₹90₹90₹90₹90
In a 3:5 split of ₹720, each of the 8 parts is worth ₹90.
04 / FROM IDEA TO ANSWER

A method you can reuse

KEEP THIS HANDY

First share = total × a/(a + b)

For a positive total divided in the positive ratio a:b.
  1. Match units and simplify by dividing both ratio terms by their HCF.
  2. Add ratio terms to find the total number of equal parts.
  3. Divide the total by the number of parts, then multiply for each share.
A useful insight

For a:b = c:d, cross products satisfy ad = bc, provided the denominators in the corresponding fractions are nonzero.

05 / WATCH THE METHOD WORK

Worked examples

Read the question first. Try a step yourself, then compare your reasoning.

EXAMPLE 01

Simplify a comparison

Simplify 42:56.

  1. The HCF of 42 and 56 is 14.
  2. 42 ÷ 14 = 3; 56 ÷ 14 = 4.
  3. Both terms were scaled by the same amount.
Answer3:4
EXAMPLE 02

Divide a total

Share ₹720 in the ratio 3:5.

  1. Total parts = 3 + 5 = 8.
  2. One part = ₹720 ÷ 8 = ₹90.
  3. The shares are 3 × ₹90 and 5 × ₹90.
Answer₹270 and ₹450
EXAMPLE 03

Inverse proportion

6 equally efficient workers finish a job in 12 days. How long would 8 take?

  1. The job requires 6 × 12 = 72 worker-days.
  2. With 8 workers, time = 72 ÷ 8.
  3. This assumes identical rates and no interference.
Answer9 days
06 / YOUR TURN

Check your understanding

Quick questions, with explanations. These are for self-study and do not affect your account score.

0 of 3 questions checked

QUESTION 1Simplify 2 metres : 50 centimetres.

See the explanation

4:1. Convert 2 m to 200 cm; 200:50 = 4:1.

QUESTION 2Split 60 in the ratio 2:3. What is the smaller share?

See the explanation

24. There are 5 parts, each worth 12. The smaller share is 2 × 12 = 24.

QUESTION 34 notebooks cost ₹60. At the same rate, what do 10 cost?

See the explanation

₹150. The price per notebook is ₹15, so 10 cost ₹150.

07 / TAKE THE NEXT STEP

Put your understanding to work

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EXPLORE THE NUMBERS

Ratio Simplifier

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Ratio and Proportion Practice Quiz 1

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Ratio and Proportion Practice Quiz 2

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Ratio and Proportion Practice Quiz 3

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Ratio and proportion practice

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