KS3 · Maths

Multiplying fractions

Does multiplying always make things bigger? Not always: half of a half is a quarter, and that is exactly what 1/2 × 1/2 means.

Multiplying fractions · The big idea

Times 2/3 or times 3/2: which one shrinks it?

the starting amount: 4/12. Starting amount 2. Amount × 2/3 1 1/3. Amount × 3/2 3

Starting amount2Amount × 2/31 1/3Amount × 3/23

The bar is your starting amount: drag it anywhere from 0 to 6. The two readouts show that amount multiplied by 2/3 and by 3/2. Can you find any amount where × 2/3 gives more than you started with?

Watch out: Multiplying does not always make a number bigger. Multiply by a number between 0 and 1 and you get less than you started with: 5 × 1/8 is only 5/8.

Why the rule works

Reason it through

Why do we multiply the tops together and the bottoms together?

Link 1 of 4

First link · your turn

Start small: 1/2 × 1/2. Read the × as 'of'. What is half of a half?

Locked — reveal the link above first

Cancelling first

Big numbers? Cancel before you multiply

Work out 25/52 × 39/75. Multiplying straight away gives 975/3900. Cancelling first keeps the numbers small.

25/52 × 39/75 = (5 × 5)/(2 × 2 × 13) × (3 × 13)/(3 × 5 × 5)Write every numerator and denominator as a product of primes. Now hunt for a prime that sits on a top AND on a bottom.
missing step

Which line is step 2?

Only the 3s cancel, because the 3 on top and the 3 underneath are both in the same fraction, 39/75.Cancel the 5 × 5 on top with the 5 × 5 underneath, the 13 on top with the 13 underneath, and the 3 on top with the 3 underneath.Cancel the 5 × 5 on top with the 5 × 5 underneath, but leave the 13s, because 13 is prime and primes can't be cancelled.

Whole numbers and fractions

What is 10 × 2/3?

You need to work out 10 × 2/3.

Which is closest to what you would do?

Multiply the top and the bottom by 10, so it's 20/30.Write 10 as 10/1, then multiply tops and bottoms: 20/3.Multiply the bottom by 10, because the bottom is the part that splits things up: 2/30.It depends which way round it's written: 2/3 × 10 would give a different answer.

How sure are you?

I'm sureI think soI'm guessing

Multiply the top and the bottom by 10, so it's 20/30.

This is a common misconception

This is the most tempting idea, and it's a trap. Multiplying the top and the bottom by 10 is the same as multiplying by 10/10, and 10/10 is 1. So 20/30 is just 2/3 in disguise: it simplifies straight back. You haven't multiplied by 10 at all.

Write 10 as 10/1, then multiply tops and bottoms: 20/3.

This matches the evidence

Exactly right. Any whole number can be written as a fraction over 1, so 10 × 2/3 = 10/1 × 2/3 = 20/3, which is 6 2/3. The 10 lands on the top; the bottom only gets multiplied by 1.

Multiply the bottom by 10, because the bottom is the part that splits things up: 2/30.

This is a common misconception

Writing 10 as 10/1 shows where it belongs. The 10 is a numerator, so it multiplies the top. The bottom is multiplied by 1 and stays as 3, giving 20/3.

It depends which way round it's written: 2/3 × 10 would give a different answer.

This is a common misconception

Multiplication works either way round, so 10 × 2/3 and 2/3 × 10 are the same calculation, and both are 20/3. Likewise 4 × 1/3 and 1/3 × 4 are both 4/3, or 1 1/3. Use whichever order you find easier to think about.

A whole number is a fraction over 1, so only the numerator gets multiplied: 10 × 2/3 = 20/3, not 20/30.

Mixed numbers

2 3/4 × 1 2/3: four areas, not two

Split each mixed number into its whole part and its fraction part: 2 3/4 is 2 + 3/4, and 1 2/3 is 1 + 2/3. Fill in all four areas, then add them up.

Grid: each cell is its row times its column
×	1	2/3
2
3/4

Add the four areas, in twelfths: 2 3/4 × 1 2/3 =

Type x² as x^2 if you cannot type ². Spaces do not matter.

Choosing a method

Which method would you pick?

Which method is the most efficient for each calculation?

Still to sort

Partition the mixed number (0)

A mixed number times a whole number

Where the line is: With a whole number there are only two areas to find, and both are easy.

Convert, then cancel first (0)

Big numbers, or mixed numbers whose parts share factors

Where the line is: If the numbers would get large, turn any mixed numbers into improper fractions and cancel before you multiply.

Just multiply (0)

Small fractions with nothing to cancel

Where the line is: If no top shares a factor with any bottom, there's nothing to cancel, so multiply straight away.

6 of 6 still to sort.

Look at each calculation before you start. Sort it by the method that gets you to the answer with the least work.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

'Times' means 'of', which is why a fraction can make an answer smaller.

What you need to know

Multiplying by a fraction means taking that fraction OF an amount, so it can make the answer smaller.
The area model shows why you multiply the numerators together and the denominators together.
Whole numbers and mixed numbers can be multiplied the same way once you see how they fit the rule.

The big picture

Multiplying by a fraction means taking that fraction of an amount, so multiplying by a number between 0 and 1 makes the answer smaller. To multiply fractions, multiply the numerators together and the denominators together, cancelling any factor shared by a numerator and a denominator. A whole number is a fraction over 1. A mixed number can be partitioned into four areas or converted to an improper fraction.

Key points

1a/b × c/d = (a × c)/(b × d). With three or more fractions, multiply all the numerators and all the denominators.

2Multiplying by a number between 0 and 1 gives less than the number you started with. Multiplying by a number greater than 1 gives more.

3Cancel before multiplying, but only a factor that is on a numerator and a denominator. Two numerators never cancel each other.

4A whole number is a fraction over 1, so only the numerator is multiplied: 10 × 2/3 = 20/3.

5For mixed numbers, either find all four areas or convert to improper fractions. Then pick the method that suits the numbers.

.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

Work out 5/6 × 2/3 × 3/4. Give your answer in its simplest form.

⚠ Watch out

Multiplying only the whole numbers together and only the fractions together. 2 3/4 × 1 2/3 is not 2 × 1 plus 3/4 × 2/3 (that gives 2 1/2). That leaves out two of the four areas. The right answer is 4 7/12.

🧠

Memory hook

'Of' means times. Tops times tops, bottoms times bottoms, and a whole number sits on top of a 1.

✓

Check yourself

Without calculating, decide whether 3/4 × 5/6 is bigger or smaller than 5/6, and say why. Then work it out, cancelling first, and check that your answer agrees with your prediction.

Flashcards

(15)

What does the × sign mean when you multiply by a fraction?

'Of'. 1/2 × 1/2 means half of a half, which is a quarter.

How do you multiply two fractions?

Multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator: a/b × c/d = (a × c)/(b × d).

In the area model for a fraction times a fraction, what do the denominators do?

They set up the grid: one denominator gives the number of rows and the other the number of columns, so the whole has (denominator × denominator) equal cells.

How do you multiply three or more fractions?

Exactly the same way: multiply all the numerators together and all the denominators together, then simplify.

Why is it allowed to cancel before multiplying?

A cancelled pair, such as 13 on a top and 13 on a bottom, makes a fraction equal to 1, and multiplying by 1 doesn't change the answer.

Which factors can you cancel before multiplying fractions?

Only a factor that is on a numerator AND on a denominator. Taking a factor out of two numerators gives a wrong answer.

When is writing numbers as products of primes worth it?

When the numbers are large. Primes (or the highest common factor) show exactly which factors a top and a bottom share.

How do you write a whole number as a fraction?

Put it over 1. For example, 7 = 7/1.

Why is 10 × 2/3 not 20/30?

Multiplying the top and bottom by 10 is multiplying by 10/10 = 1, which changes nothing. Only the numerator is multiplied: 20/3.

Does 1/3 × 4 give the same answer as 4 × 1/3?

Yes. Multiplication works either way round, so both are 4/3.

You multiply a number by a fraction between 0 and 1. How does the answer compare with the number?

It's smaller: you're only taking part of it.

You multiply two numbers that are both between 0 and 1. How does the answer compare?

It's smaller than both of them.

How does the area model multiply two mixed numbers?

Partition each into its whole part and fraction part, find all four areas, then add the four areas together.

What's the other way to multiply mixed numbers?

Convert both to improper fractions, cancel, multiply tops and bottoms, then change the answer back into a mixed number.

When does partitioning suit a calculation best?

When a mixed number is multiplied by a whole number. There are only two parts to multiply.

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

Big numbers? Cancel before you multiply

What is 10 × 2/3?

2 3/4 × 1 2/3: four areas, not two

Which method would you pick?

More KS3 Maths topics