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Age 14 to 16 Short Challenge Level:

Using a Venn diagram

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


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


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


$9$ 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.





This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.