# Mean Median

Can you work out the median weight of these children from the given means?

## Problem

The mean weight of five children is 45 kg.

The mean weight of the lightest three children is 42 kg.

The mean weight of the heaviest three children is 49 kg.

What is the median weight of the children?

This problem is taken from the UKMT Mathematical Challenges.

## Student Solutions

We are looking for the weight of the third child, as they will have the median weight.

The total weight of all five of the children is $45 \times 5 = 225$kg.

The weight of the lightest three children is $42 \times 3 = 126$kg

The weight of the heaviest three children is $49 \times 3 = 147$kg

Adding the weights of the three heaviest children to the three lightest children ($126 + 147 = 273$kg) is equal to the weight of all 5 children plus the weight of the third child, since we have counted their weight twice.

Subtracting the weight of all five children from this, we are left with the weight of the third child, the median: $273 - 225 = 48$kg.