# Prize Pony

## Problem

Helen has two stables, each of which initially housed three ponies.

One day, Helen decided to move her favourite pony from the small stable to the large stable. This increased the mean value of the ponies in the small stable by £10 000. To her surprise, this also increased the mean value of the ponies in the large stable by £10 000.

If her favourite pony is worth £250 000, what is the total value of all six of Helen's ponies?

## Student Solutions

Suppose that initially the ponies in the small stable are worth $(a + b + 250\ 000)$,

and those in the large stable are worth $(c + d + e)$.

The average value of the ponies in the small stable at the start is $\frac {a + b + 250\ 000}3$

After her favourite pony has been moved, the average value is $\frac{a + b}2$.

Since this increases the mean value of the ponies in the small stable by $10\ 000$:

\begin{equation}

\frac {a + b + 250\ 000}3 = \frac{a + b }{2} - 10\ 000

\end{equation}

Multiplying by $6$ and collecting like terms gives $a + b = 560\ 000$

The average value of the ponies in the large stable at the start is $\frac {c + d + e}3$

After her favourite pony has been moved, the average value is $\frac{c + d + e + 250\ 000}4$

Since this increases the mean value of the ponies in the large stable by $10\ 000$:

\begin{equation}

\frac {c + d + e}3 = \frac{c + d + e + 250\ 000}{4} - 10\ 000

\end{equation}

Multiplying by $12$ and collecting like terms gives $c + d + e = 630\ 000$.

Therefore the total value of the ponies is

$$a + b + 250\ 000 + c + d + e = 560\ 000 + 250\ 000 + 630\ 000 = 1\ 440\ 000$$