Takeaway Time
Pizza, Indian or Chinese takeaway? If everyone liked at least one, how many only liked Indian?
Age
14 to 16
Challenge level
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.