Orange Drink

Age

7 to 11

Challenge level

A 750 ml bottle of concentrated orange squash is enough to make fifteen 250 ml glasses of diluted orange drink. How much water is needed to make 10 litres of this drink?

Thinking mathematically tags

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Explaining, convincing and proving

Mathematical mindsets tags

Being curious Being collaborative Being resourceful Being resilient

Problem

This is a 750ml bottle of concentrated orange squash.

Undiluted 750ml bottle of orange squash.

It is enough to make fifteen 250ml glasses of diluted orange drink.

How much water is needed to make 10 litres of this drink?

Student Solutions

You might have had to read this problem over several times before you understood it clearly - it wasn't easy! However, we have received many correct solutions and the majority of you tackled it in the same way. Esther explains how she went about solving the problem very well:

The answer is 8 litres.

First I found out how much juice there is in each cup of drink. The 750mls in the bottle makes 15 cups so each cup contains 750/15=50mls.

This means that each cup contains 250-50=200mls of water.

10 litres=10,000mls. We need to know how many cups this is. As each cup holds 250mls we need 10,000/250=40 cups.

Each cup holds 200mls of water so 40 cups will need 200 times 40=8,000mls or 8 litres of water.

Esther also wrote:

There is another way using fractions.

The fraction of water in each cup is 200/250 or 4/5. This means that only 4/5 of the 10 litres is water. 1/5 of 10 is 2, so 4/5 is 8, and the answer is 8 litres.

Excellent reasoning Esther, thank you. Pupils from Queens College Junior School used a similar method to the first way Esther described. Greg, Liam, George and Joe from Moorfield Junior School found a slightly different way:

First we found out how much orange squash was in each 250 ml glass. We did this by sharing the 750 ml into the fifteen glasses. This gave us 50 ml in each glass. We then knew that there must be 200 ml water in each glass (250-50).

We then worked out how much water was in one litre of drink. This was 800 ml (200ml times 4).

Finally to find out how much water was in 10 litres we multiplied 800 ml by 10.

So the solution is that in 10 litres of drink there is 8000 ml (8 litres) of water. So we also worked out that there must be 2 litres of squash in the drink.

Alex from Columbia Independent School, MO, USA worked at the problem in a similar way to Esther's second method:

Fifteen glasses contain 3750 ml because each glass has 250 ml and 250 * 15 = 3750 ml.

Since we have only 750 ml of Orange drink, then we have 3000 ml, or 3 litres, of water.

So then each glass contains 200 ml of water which means that 80% of the diluted drink is water, because 200 is 80% of 250.

Then I found out that 4 glasses are equal to 1000 ml then I knew that 200 * 4 = 800. I knew that to prepare 1000 ml of orange drink I need 800 ml of water.

So 10 litres of orange drink need 8000 ml, or 8 litres, of water.

Finally, pupils from Wilby School tackled the problem in a slightly different way again. Here is Georgina's solution:

If you have a 750ml bottle of concentrated orange juice you can make 15 cups of juice. Each cup contains 250ml of juice.

I am trying to make 40 cups of juice, I need to know how much water I need.

If I make 15 cups of juice I need 750ml of juice and 3000ml of water.

If Imake 30 cups of water Ineed 1500ml of juice and 6000ml of water.

I cannot make 40 by adding 15 and 30 together so Ihave to make 10 cups. I only need to find out how much water Ineed to make 10 cups because if Iadd how much water Ihave in 30 cups and how much water Ihave in 10 cups together I will get 40 cups worth of water.

In 10 cups there is 2000ml of water. 6000ml + 2000ml = 8000ml.

There are 8000ml of water in 40 cups of juice.

Thank you very much to all those who sent solutions - even if there isn't room here to mention you personally.

Teachers' Resources

Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.

Why do this problem?

This question tackles proportion in a real context. It also needs systematic thinking to sort out the information and take a step-by-step route to the solution.

Possible approach

You could introduce this problem to learners simply as it stands and then, without saying anything more, give them a few moments to think completely on their own about what they might do. (They might like to jot some ideas down on paper or a mini-whiteboard.) Next, invite children to talk to a partner about a possible approach and suggest that they come to an agreement about how the problem might be tackled. At this stage, you could ask for some suggestions, or you might want to leave them to begin. However, after some time of working together, it would be good to draw attention to a range of different approaches by asking a few pairs to explain what they are doing. Emphasise that there is not just one way to go about this problem - you are looking for clear descriptions of a possible start. You could also invite pupils to share different ways of recording or jotting.

In a plenary session you could use this as an opportunity for some children to model a logical approach. In order to reach a solution to this problem, it is a matter of thinking about what we can work out from the information and then using this to answer the question.

Key questions

How much juice is there in each glass of drink?

How much water is there in each glass of drink?

How many glasses of drink are there in a litre? In 10 litres?

What fraction of the made-up drink is water?

Possible extension

You could extend this problem into a school-based context, for example, if every child in your school had a 250ml drink of this drink on sports day, how many 750ml bottles of concentrated orange squash would be needed? You might like to encourage some children to look at Mixing Lemonade.

Possible support

Before trying this problem, some children might find it helpful to look at Blackcurrantiest which looks at the concept of proportion.

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