Launching your learning experience…
Ratio compares amounts, and proportion keeps that comparison fixed as the amounts grow or shrink. This guide shows you how to write and simplify ratios, share an amount in a given ratio, and solve direct proportion problems with the unitary method.
Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.
A ratio compares amounts side by side, and the order of the numbers matters.
A ratio is a way of comparing two or more amounts. If a fruit bowl holds 3 apples for every 2 oranges, we write the ratio of apples to oranges as , said "three to two".
The colon is read as "to". The order matters a great deal — tells you apples come first, so swapping it to would describe a completely different bowl.
Divide every part of a ratio by a common factor to write it as simply as possible.
A ratio like can be written more simply. Just as with fractions, you divide every part by a common factor.
Both 12 and 8 divide by 4, so becomes . The comparison is unchanged — for every 3 of the first thing there are still 2 of the second — but the numbers are now as small as they can be.
Equivalent ratios use different numbers but describe the same comparison.
Just as and are equivalent fractions, ratios have versions too.
Add the parts to find how many shares there are, then size one share.
A very common task is splitting an amount into a given ratio. The method has three clear steps.
Suppose $40 is shared between Amy and Ben in the ratio .
In direct proportion two amounts grow together at the same rate.
Two quantities are in direct proportion when they grow or shrink together at the same rate. Double one and the other doubles too; halve one and the other halves.
If 4 pens cost $6, then 8 pens cost $12 and 2 pens cost $3 — the cost stays in step with the number of pens.
The most reliable way to solve proportion questions is the unitary method. The word "unitary" comes from "unit" — meaning one. The idea is to find the value of one item first, then scale up.
For the pens: 4 pens cost $6, so one pen costs 6 \div 4 = \1.5010 \times 1.50 = $15$.
Ratio and proportion reach into geometry, science and everyday decisions.
Ratio and proportion are some of the most practical skills in maths:
If a later topic about scaling or comparing feels tricky, it is often a ratio or proportion step underneath. The "find one, then scale" habit is worth practising until it feels natural.
Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 7). Last reviewed 2026-05-19.
Step-by-step solutions for ratio and proportion, written exactly the way a tutor would explain them at the board.
Question
A box holds 5 blue counters and 8 red counters. Write the ratio of blue counters to red counters.
Step-by-step solution
Step 1
The question asks for blue to red, so the blue amount comes first.
Step 2
Write the two amounts with a colon between them.
Answer
The ratio of blue to red is .
Question
Write the ratio in its simplest form.
Step-by-step solution
Step 1
Find a number that divides both parts — 5 works.
Step 2
Divide every part of the ratio by 5.
Question
6 identical notebooks cost $9. How much would 10 notebooks cost?
Step-by-step solution
Step 1
Find the cost of one notebook by dividing.
Question
A recipe for 4 people uses 320 g of rice and 600 ml of stock. How much rice and stock are needed for 7 people?
Step-by-step solution
Step 1
Find the amounts for one person by dividing each by 4.
The important words for ratio and proportion and what they mean — learn these so you can explain your thinking clearly.
A way of comparing two or more amounts, written with a colon, such as .
The symbol used to separate the parts of a ratio. It is read as 'to'.
Example
is read 'four to one'.
One of the numbers in a ratio, showing one share of the comparison.
Example
In the parts are 3 and 2.
A ratio whose parts have no common factor other than 1.
Example
in simplest form is .
A ratio that describes the same comparison as another, written with different numbers.
Example
and are equivalent ratios.
Dividing an amount into parts so that the parts match a given ratio.
The sum of all the parts of a ratio, used to work out the value of one share.
Example
For the total number of shares is 5.
A statement that two ratios are equal, or a relationship where amounts keep the same comparison.
A relationship where two quantities grow or shrink together at the same rate.
Example
If 2 tickets cost $10, then 4 tickets cost $20.
A method for solving proportion problems by first finding the value of one unit, then scaling up.
A comparison of two quantities measured in different units, such as price per item or distance per hour.
Example
$1.50 per pen is a rate.
A ratio that compares a size on a drawing or map with the matching real size.
Example
A map scale of means 1 cm stands for 1000 cm.
The slip-ups students most often make with ratio and proportion — and simple ways to avoid them.
Why it happens
The numbers are written in the order they appear in the question, not the order named.
How to avoid it
Match each number to the word order in the question — name first, number first.
Why it happens
Only one number is divided, like simplifying a single value.
How to avoid it
Divide every part of the ratio by the same number: becomes .
Why it happens
It feels faster to guess a multiplier than to work through the steps.
How to avoid it
Use the unitary method: find the value of one item first, then multiply up to the amount you need.
All rights reserved
©2026 Tutopiya
All resources on this platform are independently created by Tutopiya and have no endorsement from the International Baccalaureate Organization.
Ratios turn up everywhere — mixing paint colours, following a recipe, sharing a prize, or setting the scale on a map. They are useful because they describe a relationship rather than a fixed amount, so the same ratio still works whether you have a small bowl or a giant crate.
A ratio compares part to part. This is slightly different from a fraction, which compares a part to the whole — you have already met fractions, and ratios sit neatly alongside them.
A ratio is in its simplest form when the only number that divides every part is 1. Using the highest common factor gets you there in one step, but dividing in stages works too — to to .
One thing to watch: every part must be divided by the same number, and the parts should be in the same units first. A ratio of 1 hour to 30 minutes should be written as , which simplifies to .
, and are all equivalent ratios. Each describes the very same comparison — for every 1 of the first thing there are 3 of the second.
To build an equivalent ratio, multiply every part by the same number — that is exactly the reverse of simplifying. From , multiplying both parts by 4 gives .
Equivalent ratios are how you scale recipes and mixtures. If a smoothie needs juice and yoghurt in the ratio and you want to use 6 cups of juice, scale up by 3 to get — so 3 cups of yoghurt.
So Amy gets 3 \times 8 = \242 \times 8 = $16$.
A brilliant check: the two answers should add back to the original amount. , so it works. If your shares do not add up, a step has gone wrong.
This "find one, then scale" approach handles recipes, speeds, currency exchange and best-buy comparisons. Whenever a question gives you a quantity and asks about a different quantity, reach for the unitary method.
Step 3
3 and 2 share no common factor except 1, so it is simplest.
Answer
Step 2
Divide the amount by the total to size one share.
Step 3
Multiply each part by the value of one share.
Step 4
Check the shares add back to $48.
Answer
The friends receive $30 and $18.
Step 2
One notebook costs $1.50. Multiply to find the cost of 10.
Answer
10 notebooks would cost $15.
Step 2
One person needs 80 g of rice and 150 ml of stock. Multiply by 7.
Step 3
State both scaled amounts.
Answer
For 7 people you need 560 g of rice and 1050 ml of stock.
Why it happens
Ratio and fraction both describe parts, so they get muddled.
How to avoid it
A ratio compares part to part. In the first share is of the whole, not .