Maths Week Scotland 2022 - Problem 2 - Fruit Cocktail

CHRIS This challenge is all about figuring out the mystery prices of some fruit.

A mathematical greengrocer puts a poster in his window to advertise the fruit that he’s selling.

The poster shows how the cost of apples, bananas, pears and cherries are related and we need to work out how much each individual fruit costs. Here's what we know so far.

We know that an apple, banana and bunch of cherries cost ninety pence.

Two bananas cost the same as two apples and two pears.

Buying an apple, pear and cherries would cost the same as buying one banana and four pears.

Two bunches of cherries cost the same as four apples.

How much does each kind of fruit cost?

There’s a mathematical greengrocer and he tells us about the prices of cherries and bananas and pears and apples and how they are connected to each other.

PUPIL It's a good idea to think about what information we can work out right away.

PUPIL Can you simplify any information you have?

CHRIS No, no, no that’s fine, keep going.

PUPIL We can't work out all the prices at once so can you work out how the prices of different fruits relate to each other?

CHRIS Say you think apple’s ten, right, instead. If apple was ten, what would that mean for cherries?

PUPIL You might want to use algebra to work this one out.

PUPIL Give it a go.

In this tasty challenge, we had to work out the price of different kinds of fruit. Let’s figure this one out together.

First let’s simplify some of the equations. From picture four, we know two bunches of cherries cost the same amount as four apples.

We can simplify this by dividing both sides equally to work out that one bunch of cherries cost the same as two apples.

Now let’s look at picture two. Two bananas costs the same as two apples and two pears.

We can simplify this by dividing both sides by two.

A banana costs the same as an apple and a pear.

Now we’ve simplified these two equations, we can put this information into equation three.

An apple, a pear and a bunch of cherries costs the same as a banana and four pears.

We can swap the bunch of cherries for two apples.

And we can swap the banana for an apple and a pear. So, three apples and a pear costs the same as one apple and five pears.

Because both sides now have a pear, we can remove a pear from each side.

And both sides have an apple, so we can take an apple away from each side. So, two apples costs the same as four pears.

We can now divide both sides by two. One apple costs the same as two pears.

Now we know how all the different fruits relate to each other, we can look at equation one.

This equation is the only one with the price in it. If we can change all the different types of fruit in this equation to just one type, we can start to work out the individual costs.

We know that a bunch of cherries costs the same as two apples so we can swap that.

And we know that a banana costs the same as an apple and a pear, so we can swap that.

Now we’ve got four apples and a pear.

An apple costs the same as two pears, so we can swap all of these.

So, nine pears costs 90 pence.

That means one pear costs 10 pence.

We can use the price of a pear to work out all the other prices.

An apple costs the same as two pears, which is two times 10 pence, 20 pence.

A banana costs the same as an apple and a pear. So that’s 20 pence plus 10 pence, which is 30 pence.

And a bunch of cherries costs the same as two apples, which is two times 20, which is 40 pence.

You could’ve used algebra to work this out too, replacing each fruit with a letter and turning those pictures into equations.

Let apples be ‘a’, bananas be ‘b’, cherries be ‘c’ and pears be ‘p’.

So you get the equations

a + b + c = 90

2b = 2a + 2p

a + c + p = b = 4p

2c = 4a

You can then simplify these equations and combine them just like we did with the pictures.

It’s just a slightly different method but it gets us to the exact same answer.