# Average Surroundings

Can you work out Ali's age based on the diagram?

## Problem

Image

Ali (A) and Baba (B) are shown surrounded by six thieves. The thieves' ages are given. Ali's age is the average of his four nearest neighbours', and so is Baba's. How old is Ali?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.

## Student Solutions

**Answer**: Ali is $38$ years old

$$A=\frac {B+31+64+20}{4}=\frac {B+115}{4}\\

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B=\frac {A+18+54+38}{4}=\frac{A+110}{4}$$

Substitute $B=\dfrac{A+110}{4}$ into $A=\dfrac {B+115}{4}$: $$\begin{align}A&=\dfrac{\frac{A+110}{4}+115}4\\

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\Rightarrow 4A &= \dfrac{A+110}{4} +115\\

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\Rightarrow 16A &= A + 110 + 460\\

\Rightarrow 15A &= 570\\

\Rightarrow A &= 38\end{align}$$