# Perfect Score

Can you work out how many children answered all the questions correctly?

## Problem

100 children took part in a Mathematics competition consisting of four problems.

90 children solved the first problem

85 solved the second

80 solved the third

75 solved the last problem

What is the minimum number ofÂ children who could have answered all the questions correctly?

This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

Minimising the number of children who solved all four problems correctly is the same as maximising the number of children who got at least one wrong.

10 children got the first problem wrong.

15 got the second problem wrong.

20 got the third wrong.

25 got the fourth wrong.

So if they were all different children, at most 10+15+20+25=70 children got at least one question wrong.

If 70 children got at least one question wrong, 30 got all four questions right.

In the diagram below the children who correctly solved the problems are represented in green and those who didn't in red.

Image

Lined up vertically, the parts of the bars that are green for all four questions represent the contestants who answered all four questions correctly.

To make this as small as possible, the red parts should be set up with as little overlap as possible, i.e. no overlap, as shown above.

So the smallest number of contestants who could have solved all four problems correctly is 30.