Myriora Maths › 11+ maths › Chapter 2: Classic word problems

Fractions of what's left

11+Chapter 2 · Lesson 7

In this lesson

Draw a bar for the whole amount.
After each step, look only at what's left.
Cut what's left into equal pieces.
Then find one piece, and scale up.

Your turn

Mia spends half of her money, then a third of what's left. She has twelve pounds left. How much did she start with?

Show the answer

Let's draw a bar for all her money. She spends half, so we split it into two equal parts, and cross one out. Now look only at what's left: one part. A third means three equal pieces, so we cut it into three. She spends one piece, so we cross it out. Two pieces are left, and they're worth twelve pounds. So one piece is worth six pounds. The part we cut is three pieces, so it's worth eighteen pounds. The half she spent is the same size, so it's worth eighteen pounds too. Two parts of eighteen pounds make thirty-six pounds!

The common mistake

A common mistake is taking the quarter of the whole bar. But the question says a quarter of what's left! By then he only has forty pounds, so a quarter of it is ten pounds, not fifteen. After each step, look only at what's left.

Try it at home

Share a bag of grapes: half to one person, a third of what's left to another. How many were there at the start?

Read the whole lesson

Hello, and welcome to Myriora Maths! Sam spends a third of his money. Then he spends a quarter of what's left. He has thirty pounds left. How much did he start with?

Let's draw a bar for all his money. He spends a third, so we split it into three equal parts. He spends one of those parts, so we cross it out. Two parts are left.

Next, he spends a quarter of what's left. So forget the part he has already spent, and look only at these two parts. A quarter means four equal pieces. How can we make four pieces from two parts? We cut each part in half. Now what's left is four small pieces. He spends one of the four, so we cross it out.

Three small pieces are left. The question says he has thirty pounds left. So three pieces are worth thirty pounds. That means one piece is worth ten pounds.

Now, how much did he start with? Each part we cut is two small pieces, so it's worth twenty pounds. The part he spent first is exactly the same size. So it's worth twenty pounds too, and we never needed to cut it. Three parts of twenty pounds make sixty pounds!

Let's check. A third of sixty is twenty, which leaves forty. A quarter of forty is ten, which leaves thirty. It works!

Your turn! Mia spends half of her money, then a third of what's left. She has twelve pounds left. How much did she start with? You've got twenty seconds, or pause the video if you need more time. Go!

Time's up! Let's draw a bar for all her money. She spends half, so we split it into two equal parts, and cross one out. Now look only at what's left: one part. A third means three equal pieces, so we cut it into three. She spends one piece, so we cross it out. Two pieces are left, and they're worth twelve pounds. So one piece is worth six pounds. The part we cut is three pieces, so it's worth eighteen pounds. The half she spent is the same size, so it's worth eighteen pounds too. Two parts of eighteen pounds make thirty-six pounds!

Quick review. Draw a bar for the whole amount. After each step, look only at what's left. Cut what's left into equal pieces. Then find one piece, and scale up. See you next time!