KS3 · Maths

Dividing fractions

How many halves are there in 3? Not 1 1/2: there are 6. Dividing can make an answer bigger, and by the end of this lesson you'll know why.

How many fit?

Where does the answer land?

Think of the number after the ÷ as a measuring stick, and ask how many of them fit into the first number. Drag each answer to where you think it lands, then check.

A quicker way to count

Problem

Work out 14 ÷ 2/3.

A fraction divided by a whole number

?

Reason it through

Why does dividing a fraction by a whole number only change the numerator?

Link 1 of 4

First link · your turn

9/10 means 9 pieces, each one tenth of a whole. Share those 9 pieces equally into 3 groups. How many pieces does each group get, and how big is each piece?

2
Locked — reveal the link above first
3
Locked — reveal the link above first
4
Locked — reveal the link above first

Reciprocals

What is the reciprocal of 5 1/2?

Every number except 0 has a reciprocal: the number you multiply it by to get 1. The reciprocal of 5 is 1/5, because 5 × 1/5 = 1. The reciprocal of 2/3 is 3/2, because 2/3 × 3/2 = 6/6 = 1.

Which is closest to what you think the reciprocal of 5 1/2 is?
How sure are you?

Predict, then check

Picture the measuring stick from the number line. This time the stick is 3/4 long.

Is 2/3 ÷ 3/4 more than 1 or less than 1?

Your turn

Find the missing length

A rectangle has an area of 4/5 m² and a width of 3/8 m. Find its length.

  1. Area = length × width, so length = area ÷ width.Division undoes multiplication. That's why a missing side comes from dividing.
  2. missing step
Which line is step 2?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Dividing asks how many times one number fits into another, which is why dividing by a fraction can give you more than you started with.

What you need to know

  • Dividing asks how many times the divisor fits into the dividend, and the answer doesn't have to be a whole number.
  • A fraction divided by a whole number only changes the numerator, because the size of the pieces stays the same.
  • Dividing by a fraction gives the same answer as multiplying by its reciprocal, so a/b ÷ c/d = a/b × d/c.

The big picture

Dividing asks how many times the divisor fits into the dividend. Dividing by a fraction between 0 and 1 gives an answer bigger than the number you started with. To divide a fraction by a whole number, divide the numerator, first writing an equivalent fraction if the numerator isn't a multiple. To divide by a fraction, multiply by its reciprocal, turning mixed numbers into improper fractions first. Then simplify, and write improper answers as mixed numbers.

Key points

1Division can mean sharing, grouping or unitising. Unitising treats the divisor as a unit and counts how many of it fit into the dividend.
2For positive numbers, dividing by a number between 0 and 1 gives an answer bigger than the dividend, and dividing by a number bigger than 1 gives a smaller one. Careful: improper fractions and mixed numbers, like 5/2 or 1 1/2, are bigger than 1.
3Fraction ÷ whole number: divide the numerator. If it won't go exactly, write an equivalent fraction first, using the lowest common multiple of the numerator and the whole number.
4Whole number ÷ fraction: write the whole number over the divisor's denominator, then count how many times the numerator fits. A leftover part is written as a fraction of the divisor.
5A number times its reciprocal is 1. To find the reciprocal of a mixed number, turn it into an improper fraction, then flip it.
6Fraction ÷ fraction: keep the first fraction and multiply by the reciprocal of the second. Simplify with the highest common factor and write improper answers as mixed numbers.

Worked example

Problem

Work out 2 1/4 ÷ 1 1/2. Give your answer as a mixed number.

⚠ Watch out

Flipping the wrong fraction. In 3/5 ÷ 2/7, flip only the divisor: 3/5 × 7/2 = 21/10 = 2 1/10. Flipping 3/5 instead gives 10/21. Quick check: 2/7 is less than 1, so the answer must be bigger than 3/5, and 10/21 isn't.

🧠

Memory hook

Ask 'how many fit?' A small stick fits lots of times, so dividing by a fraction less than 1 makes the answer bigger. Then: keep the first, flip the divisor, multiply.

✓

Check yourself

Without calculating, decide whether 5/8 ÷ 1/4 is bigger or smaller than 5/8, and say why. Then work it out by multiplying by the reciprocal, and check your answer matches your prediction.

Flashcards

(14)
What question does a division like 4 ÷ 1/3 ask?
How many thirds fit into 4? Three thirds fit into each whole, so 12 fit.
What are three ways to think about division?
Sharing, grouping and unitising. Unitising treats the divisor as a unit and counts how many times it fits into the dividend.
For positive numbers, when does dividing give an answer bigger than the number you started with?
When the divisor is between 0 and 1. A divisor bigger than 1 gives a smaller answer.
When you divide a fraction by a whole number, which part of the fraction changes?
Only the numerator. The pieces are shared out but stay the same size, so the denominator stays the same: 8/9 ÷ 4 = 2/9.
How do you divide a fraction by a whole number that doesn't go into the numerator?
Write an equivalent fraction whose numerator is a multiple of the whole number, then divide the numerator: 3/4 ÷ 2 = 6/8 ÷ 2 = 3/8.
What is the quick method for a whole number divided by a fraction?
Write the whole number with the divisor's denominator, then count how many times the numerator fits: 10 ÷ 2/5 = 50/5 ÷ 2/5 = 25.
What happens when the divisor doesn't fit a whole number of times?
The leftover becomes a fraction of the divisor. 2 ÷ 3/4 = 8/4 ÷ 3/4: 3 quarters fit into 8 quarters 2 times with 2 quarters left, which is 2/3 of the divisor. So the answer is 2 2/3.
What is a reciprocal?
The number you multiply by to get 1. The reciprocal of 7 is 1/7, and the reciprocal of 4/9 is 9/4.
How do you find the reciprocal of a mixed number?
Turn it into an improper fraction first, then flip it: 3 1/3 = 10/3, so its reciprocal is 3/10.
What is the general rule for dividing by a fraction?
Dividing by a number gives the same result as multiplying by its reciprocal: a/b ÷ c/d = a/b × d/c.
In a fraction division, which fraction do you flip?
Only the divisor, the one after the ÷ sign. The first fraction stays as it is.
What two things do you do to tidy up a fraction answer?
Simplify it by dividing the numerator and denominator by their highest common factor, and write an improper fraction as a mixed number if you're asked to.
How do you find a missing length of a rectangle from its area and width?
Length = area ÷ width, because area = length × width and division undoes multiplication.
How can you check the answer to a division?
Multiply back. If 3/5 ÷ 1/10 = 6, then 6 × 1/10 should give 3/5, and it does, because 6/10 = 3/5.

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

More KS3 Maths topics

See the full KS3 Maths curriculum →

How this lesson was checked. This KS3 Mathslesson was published through Lightbulb Learning's human-designed editorial process — the educational standards, accuracy rules and publication checks it must pass were authored and approved by Philip Halpin. It passed subject-specific assessment, automated educational checks and technical publication verification before going live (publication checks completed 30 September 2026). Published pages are monitored, human spot-checking is ongoing across the lesson library, and anything found wrong is corrected or withdrawn. How our lessons are made and checked. Spotted a mistake? Email hello@lightbulblearning.co and we'll review it.