A Bowl of Fruit
Can you work out how many apples there are in this fruit bowl if you know what fraction there are?
Here is a bowl of fruit.
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Half of the pieces of fruit in the bowl are apples. There are also $3$ oranges, $2$ pears and a banana.
How many apples are there in the bowl?
If, instead, one quarter were apples and one quarter were oranges and there were also $4$ bananas, $3$ pears and $3$ plums how many would be apples?
In total, what fraction of the fruit in the bowl are oranges, pears and bananas?
How many pieces of fruit which are not apples are there altogether?
In the second part of the problem, what fraction of the fruit in the bowl are bananas, pears and plums?
How many bananas, pears and plums are there altogether?
We had a good selection of solutions submitted. Children from Mef School in Turkey explained it like this:
1+2+3= 6 Half of the total fruits is equal to the number of apples. They showed this in a picture:
Image3 oranges, 2 pears and a banana apples
Year 1 from St. Giles' Primary School in Shrewsbury wrote;
At first, we decided to draw the fruits that we knew were in the basket.
Delilah noticed that 2 (pears) + 1 (banana) =3, and 3 more (oranges) makes 6 fruits.
Some children suggested that there might be 12 apples but not everyone was convinced.
We read the clues again thinking about what we understood about 'half'.
We decided to draw the bowl and draw a line through the middle to show two halves.
Some then drew the 6 fruit on one side, others wrote 3+2+1 or simply 6.
We then thought again about what it means if half are apples.
Now many more children could 'see' that there must be 6 apples... "because the two halves have to be equal".
We didn't all get it but we thought that drawing a picture was helpful.
Danny from Swarland Primary School sent this in
Hi, My Mum changed the fruit in the problem, hope you don't mind. It's because we take it in turns in our class to take Stuart (a Minion) home to have maths adventures and (as everyone knows) Minions LOVE bananas. I solved the problem using real fruit and by creating pictures. It was fun! I've sent you a picture of what I wrote in Stuart's diary. From Danny.
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That is absolutely wonderful, one of the most lovely solutions to come in in the last 20 years since NRICH started.
Scott from Hagley Primary School sent in the following;
First, I wrote the question and drew the 3 oranges, 2 pears and the banana.
Then, I drew my bar and coloured 1 for banana, 2 for pears and 3 for oranges.
After that, I added 1,2 and 3. My answer was 6. I know that double 6 is 12 and doubling is the opposite of a half. So 6 more cubes makes 12, that is how many apples there will be.
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Emily from Burrough Green Primary sent in a picture of her work.
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Well done everyone. It was good to see different ways of expressing the solution.
Why do this problem?
This problem requires a sound understanding of the relationship between part and whole. It could be used as part of a lesson on finding fractions of numbers and quantities.
Possible approach
A good introduction to this problem could be to have the image of a bowl of fruit for all the group to see and to invite them to talk about it. This could be the one in the problem on an interactive whiteboard or another picture. A real bowl of fruit could also be used if that were possible.
You could steer the conversation towards fractions if the children do not naturally bring it up. Asking general questions about the fractions of different fruits in the bowl and referring also to the fraction of "other fruit" will give children the confidence to tackle this problem.
Children should be encouraged to record in any way they find useful while working on this problem. Many may find it helpful to use practical equipment to represent the fruit, for example blocks or counters, perhaps with different colours standing for different types.
Key questions
What fraction of fruit in the bowl is apples?
What fraction of fruit in the bowl is not apples?
Possible extension
Children could make up similar problems for each other to do. Some learners could be encouraged to use cards or symbols and move into a kind of algebra.