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# Takeaway Time

**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.## You may also like

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Age 14 to 16

ShortChallenge Level

- Problem
- Solutions

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.

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.

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.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.