GCSE · Maths · AQA · Spec 8300

Ratio notation and simplest form

2 litres of red paint and 3 of white make a pink. 12 of red and 18 of white make exactly the same pink. So what stayed the same?

Ratio · Equivalent ratios

One colour, many ways to write it

Batches mixed: 3/6. Red paint 6 L. White paint 9 L. Red : white 6 : 9. HCF of the parts 3. Simplest form 2 : 3

Red paint6 LWhite paint9 LRed : white6 : 9HCF of the parts3Simplest form2 : 3

Every batch of this pink paint is 2 litres of red to 3 litres of white. Drag to mix more or fewer batches: the two amounts grow and shrink together, and the simplest form never changes.

Watch out: Taking the same amount off both parts is not simplifying. Take 4 off each part of 6 : 9 and you get 2 : 5, which never appears as you drag: it is a different mix, with less red in it.

The method, step by step

Problem

A recipe uses 45 g of butter, 60 g of sugar and 75 g of flour. Write the ratio flour : sugar : butter in its simplest form.

What do you think?

Four ways to simplify one ratio

A club has 18 girls and 12 boys. Kai is asked to write the ratio of girls to boys in its simplest form.

Which of these is closest to what you would do?

Take 6 away from both parts to make them smaller: 12 : 6, which is 2 : 1.Divide both parts by 2 to get 9 : 6. The numbers got smaller, so it's simplified.Put the smaller number first and get 2 : 3. The order doesn't matter once it's simplified.Keep girls first because the question says girls to boys, then divide both parts by 6: 3 : 2.

How sure are you?

I'm sureI think soI'm guessing

Take 6 away from both parts to make them smaller: 12 : 6, which is 2 : 1.

This is a common misconception

Subtracting changes the relationship. 18 : 12 means 3 girls for every 2 boys, but 12 : 6 means 2 girls for every boy, which describes a different club. Only dividing (or multiplying) every part by the same number keeps the ratio the same.

Divide both parts by 2 to get 9 : 6. The numbers got smaller, so it's simplified.

This is a common misconception

Dividing by 2 was allowed, and 9 : 6 is equivalent to 18 : 12. But 9 and 6 still share the factor 3, so it isn't finished. Dividing by the highest common factor, 6, gets to 3 : 2 in one step.

Put the smaller number first and get 2 : 3. The order doesn't matter once it's simplified.

This is a common misconception

2 : 3 would mean 2 girls for every 3 boys, but this club has more girls than boys. The question says girls to boys, so the girls' number goes first: 3 : 2.

Keep girls first because the question says girls to boys, then divide both parts by 6: 3 : 2.

This matches the evidence

Yes. 18 ÷ 6 = 3 and 12 ÷ 6 = 2, and 3 and 2 share no factor except 1. The order matches the wording, so 3 : 2 is the simplest form.

Keep the order the words give you, then divide every part by the same number until no common factor is left.

Spot the slip

Find the line where it goes wrong

Write 40 cm : 2 m in its simplest form.

A student's answer — which line goes wrong?

The ratio is 40 cm : 2 m.
Drop the units: 40 : 2.
40 and 2 have a highest common factor of 2.
Divide both parts by 2: 20 : 1.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same relationship, smallest whole numbers, and always in the order the question asks.

What you need to know

A ratio compares two or more quantities in a set order: red : white = 2 : 3 means 2 parts red for every 3 parts white.
Equivalent ratios describe the same relationship with different numbers, like 2 : 3, 6 : 9 and 12 : 18.
Simplest form is the equivalent ratio written with the smallest possible whole numbers.

The big picture

A ratio compares quantities in a set order, such as 2 : 3 for red paint to white paint. Multiplying or dividing every part by the same number gives an equivalent ratio, which describes the same relationship with different numbers. Simplest form is the equivalent ratio with the smallest whole numbers. To reach it, write the parts in the order the question asks, put any measurements in the same unit, and divide every part by the highest common factor.

Key points

1Write the parts in the order the question names them: 'blue to red' means the blue number comes first.

2Put every measurement in the same unit before you simplify, for example 2 m = 200 cm.

3Divide every part by the highest common factor. In a three-part ratio, divide all three parts.

4Check the answer: the parts should share no factor except 1. Never subtract the same amount from each part, because that changes the ratio.

.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

Write 50 minutes : 2 hours as a ratio in its simplest form.

⚠ Watch out

Stopping too early. Dividing 30 : 45 by 5 gives 6 : 9, which is equivalent but not the simplest form, because 6 and 9 still share the factor 3. Keep dividing (or use the HCF, 15, straight away) to reach 2 : 3.

🧠

Memory hook

Order, units, HCF: say it in the order it's asked, get one unit, then divide every part by the HCF.

✓

Check yourself

Write 1.2 kg : 800 g in its simplest form. Check: change to grams (1200 : 800), then divide both parts by the highest common factor, 400, to get 3 : 2.

Flashcards

(6)

What does 'red : blue = 3 : 5' mean?

For every 3 red there are 5 blue. The numbers follow the order of the words, so red's number comes first.

What are equivalent ratios?

Ratios that describe the same relationship with different numbers, such as 2 : 3, 4 : 6 and 12 : 18. Multiply or divide every part by the same number to move between them.

When is a ratio in its simplest form?

When its parts are whole numbers that share no common factor except 1.

What is the quickest way to simplify any ratio, even a three-part one?

Divide every part by the highest common factor (HCF) of all the parts.

Why can't you simplify a ratio by subtracting the same number from each part?

Subtracting changes the relationship. Take 6 from each part of 8 : 12 and you get 2 : 6, which is 1 : 3, not the 2 : 3 you started with.

What must you check before simplifying a ratio of two measurements?

That both parts are in the same unit. Convert first (for example, metres to centimetres), then simplify.

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

Four ways to simplify one ratio

Find the line where it goes wrong

More AQA GCSE Maths topics