KS3 · Maths

Ratio notation and equivalent ratios

30 : 10 and 15 : 5 look like completely different ratios. Are they? Divide each by its common factor and see what you get — you might be surprised.

Maths · Ratio

Same proportion, different numbers

Pick a ratio, then pick the family it belongs in. Simplify it first: divide both parts by a common factor and see what is left.

Still to sort

2 : 3 (0)

Simplifies to 2 : 3.

Where the line is: Dividing both parts by the same common factor must land exactly on 2 : 3.

3 : 1 (0)

Simplifies to 3 : 1.

Where the line is: Dividing both parts by the same common factor must land exactly on 3 : 1.

4 : 5 (0)

Simplifies to 4 : 5 (or is 4 : 5 multiplied or divided by one number, decimals included).

Where the line is: A ratio belongs here if 4 : 5 multiplied or divided by one number gives it.

None of these (0)

Simplifies to something that isn't 2 : 3, 3 : 1 or 4 : 5.

Where the line is: Nothing you do to both parts will turn these into one of the three families.

8 of 8 still to sort.

Here's a pile of ratios. They look different, but they're hiding in just a few families.

Watch out: Whatever you divide one part by, divide every part by the same number.

Maths · Ratio

What do the numbers in a ratio mean?

Which of these is closest to what you think right now?

The numbers can go either way round. For circles to squares, 4 : 3 means the same as 3 : 4.One of the numbers in a ratio is the total, usually the biggest one.Red to blue is 2 : 1, so 10 chocolates in total would work fine.The numbers follow the words, each number is a part, and the parts add up to the whole.

How sure are you?

I'm sureI think soI'm guessing

The numbers can go either way round. For circles to squares, 4 : 3 means the same as 3 : 4.

This is a common misconception

Easy slip, but the order matters. 'For every three circles there are four squares' is circles to squares 3 : 4, because the numbers follow the order of the words. Write 4 : 3 and you're now saying four circles for every three squares.

One of the numbers in a ratio is the total, usually the biggest one.

This is a common misconception

Each number in a ratio is a part, never the total. Three parts raisins and four parts chocolate is 3 : 4, and it's the two numbers added together, 3 + 4 = 7, that tell you how many equal parts the whole is split into.

Red to blue is 2 : 1, so 10 chocolates in total would work fine.

This is a common misconception

Not quite. For red to blue 2 : 1 the chocolates come in groups of 3 (2 red and 1 blue). A total of 10 isn't a multiple of 3, so 10 is not possible. 10 red means 5 blue, and 10 blue means 20 red.

The numbers follow the words, each number is a part, and the parts add up to the whole.

This matches the evidence

Exactly. Order follows the words, every number is a part, and adding the parts tells you how many equal parts the whole is split into. That one sentence stops most of the classic ratio slips.

Order follows the words, each number is a part, and the parts add up to the whole.

Maths · Ratio

Find the slip

A student finds two ratios equivalent to the three-part ratio 2 : 3 : 1. One line of the working goes wrong. Which one?

A student's working — which line goes wrong?

Start with 2 : 3 : 1.
Multiply every part by 4: 2 × 4 = 8, 3 × 4 = 12 and 1 × 4 = 4, so 8 : 12 : 4 is equivalent.
Go back to 2 : 3 : 1 and divide by 2: 2 ÷ 2 = 1 and 3 ÷ 2 = 1.5. Leave the last part as 1.
So 1 : 1.5 : 1 is equivalent to 2 : 3 : 1.

Using a ratio with a total

Problem

A 35 g chocolate bar has raisins to chocolate in the ratio 3 : 4. How many grams of raisins and how many grams of chocolate are in the bar?

Maths · Ratio

Four ways to show a ratio

Tick the box where each way of showing a ratio has that property, then check your grid.

Bar model

Concise: uses symbols and takes up little spaceMakes the relationships easy to see

Ratio table

Concise: uses symbols and takes up little spaceMakes the relationships easy to see

Double number line

Concise: uses symbols and takes up little spaceMakes the relationships easy to see

Ratio notation

Concise: uses symbols and takes up little spaceMakes the relationships easy to see

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Different numbers, same proportion

What you need to know

A ratio shows the relative sizes of two or more values. It lets you compare one part with another part of a whole.
Write a ratio with a colon. 'For every three circles there are four squares' is circles to squares, 3 : 4.
Say it as '3 to 4'. The numbers follow the order of the words, so swap the words and you swap the numbers.
Have a goFor every 2 apples there are 5 pears. Write the ratio of pears to apples.
5 : 2
Pears are mentioned first in 'pears to apples', so the 5 goes first. The words set the order, not whichever number you spotted first.
Add the numbers to get the total parts. Three parts raisins and four parts chocolate make 7 parts in all.
A ratio can compare more than two values. Stars to hearts to circles 1 : 1 : 2 is a three-part ratio.
Multiply or divide every part by the same number to make an equivalent ratio. 4 : 5 times 2 is 8 : 10.
Have a goMake an equivalent ratio by multiplying 3 : 5 by 4. What do you get?
12 : 20
You multiply both parts by 4: 3 × 4 and 5 × 4. Multiplying only one part would give a different ratio.
There are infinitely many equivalent ratios. They use different numbers but share one multiplicative relationship, so they are the same proportion.
To simplify, divide every part by a common factor. 4 : 6 divided by 2 gives 2 : 3.
Have a goSimplify 12 : 18 by dividing by the highest common factor.
2 : 3 (divide both parts by 6)
6 is the highest number that goes into both 12 and 18, so dividing both parts by 6 reaches simplest form in one step.
With one known amount, find the multiplier, then multiply every part by it to find the other amounts.
With a total, add the parts to count the equal parts, then use the total to find the multiplier.
Ratio notation is quick and concise, but it is harder to see relationships than in a bar model or double number line.

The big picture

A ratio compares the sizes of two or more values, written with colons. Multiply or divide every part by the same number and you get an equivalent ratio, so simplifying shows which ratios are really the same proportion. A ratio plus one known amount, or a total, gives you all the other amounts.

Key points

1A ratio compares the sizes of two or more values, and the order of the numbers follows the order of the words.

2The parts of a ratio add up to the whole, so a total must fit the number of parts.

3Equivalent ratios come from multiplying or dividing every part by the same number.

4Simplify a ratio by dividing every part by a common factor; ratios with the same simplest form are the same proportion.

5A ratio plus one known amount or a total gives the other amounts: find the multiplier, then multiply every part.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

Orange juice to milk is 2 : 3 in a drink. You use 120 ml of milk. How much orange juice do you need?

⚠ Watch out

Treating the total as one of the parts. The total is shared out across all the parts of the ratio, so it never sits in the ratio itself.

🧠

Memory hook

Words set the order. Every number is a part. Whatever you do to one part, do to all of them.

✓

Check yourself

PVA to water is 5 : 2 and you have 20 ml of PVA. How much water? Find the multiplier first. (Answer: 8 ml, from 5 × 4 = 20.)

Flashcards

(13)

What does a ratio show?

The relative sizes of two or more values. It lets you compare one part with another part in a whole.

How do you write 'for every three circles there are four squares' as a ratio?

Circles to squares is 3 : 4, said '3 to 4'. The numbers follow the order of the words.

What do the numbers in a ratio add up to?

The total number of equal parts the whole is split into.

Can a ratio compare more than two values?

Yes. Stars to hearts to circles 1 : 1 : 2 compares three values.

How do you find an equivalent ratio?

Multiply or divide every part by the same number.

How many ratios are equivalent to 4 : 5?

Infinitely many. They use different numbers but the same multiplicative relationship, so they are the same proportion.

How do you simplify a ratio?

Divide every part by a common factor. Use the highest common factor to reach simplest form in one step.

Write 4 : 6 in simplest form.

2 : 3, dividing both parts by the highest common factor, 2.

You know one amount in a ratio. How do you find the others?

Find the multiplier that turns that part into the known amount, then multiply every part by it.

You know the total for a ratio. What is your first step?

Add the parts of the ratio to find how many equal parts there are, then find the multiplier.

Triangles to squares is 4 : 1 with 40 shapes in total. Why isn't the answer 40 and 10?

40 is the total, not a part. There are 5 parts, so it is 32 triangles and 8 squares.

Red to blue is 2 : 1. Why can't the total be 10?

The chocolates come in groups of 3, and 10 is not a multiple of 3.

What is the trade-off with ratio notation?

It is efficient and concise, but it is harder to see relationships.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Same proportion, different numbers

What do the numbers in a ratio mean?

Find the slip

Four ways to show a ratio

More KS3 Maths topics