# Takeaway Time

Pizza, Indian or Chinese takeaway? If everyone liked at least one, how many only liked Indian?

## Problem

$35$ teenagers were asked what takeaways they liked to eat.

$24$ answered Chinese

$16$ answered Indian

$10$ answered pizza

None of the teenagers liked all three.

All who liked pizza also liked Chinese.

$9$ of the Chinese fans didn't like either Indian or pizza.

If all the teenagers liked at least one, how many liked only Indian?

## Student Solutions

**Using a Venn diagram**

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The circles on this Venn diagram represent students who liked Chinese, Indian and pizza. The totals are shown in brackets.

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None of the teenagers liked all three, so there are 0 students in the intersection of all three.

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All who liked pizza also liked Chinese, so all 10 are in this overlap, with none in the other parts of the pizza circle.

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$9$ of the Chinese fans didn't like either Indian or pizza.

That gives enough information to fill in more regions, and see that 11 teenagers liked Indian only.

**Counting students who liked Chinese and pizza**

There were $35$ teenagers altogether.

$24$ liked Chinese

$16$ liked Indian

$10$ liked pizza

Since all the students who liked pizza also liked Chinese, and no student liked all three, we can just think about the remaining $25$ students.

Of these, $14$ liked Chinese and $16$ liked Indian.

Since each student liked at least one of these, and there are a total of $30$ 'likes', $5$ students must like both Indian and Chinese.

This leaves $11$ students who liked Indian only.

**Thinking about the students who liked Chinese**

Let's think about the $24$ students who liked Chinese:

The number who liked Chinese, Indian and pizza: $0$

The number who only liked Chinese: $9$

The number who liked Chinese and pizza: $10$

So the number who liked Chinese and Indian: $24 - 9 - 10 = 5$

Therefore $16 - 5 = 11$ liked Indian only.