Jewellery boxes
Problem
Two boxes, P and Q, each contain 3 jewels.
When a jewel worth £5000 is transferred from P to Q, the average value of the jewels in each box increases by £1000.
What is the total value of all six jewels?
Student Solutions
Suppose that initially the jewels in $P$ are worth $(a + b + 5000)$,
and those in $Q$ are worth $(c + d + e)$.
The average value in $P$ at the start is $\frac {a + b + 5000}3$
After the jewel has been moved it is $\frac{a + b}2$.
Therefore:
\begin{equation}
\frac {a + b + 5000}3 = \frac{a + b }{2} - 1000
\end{equation}
Multiplying by $6$ and collecting like terms gives $a + b = 16000$
The average value in $Q$ at the start is $\frac {c + d + e}3$
After the jewel has been moved, the average value is $\frac{c + d + e + 5000}4$
Therefore:
\begin{equation}
\frac {c + d + e}3 = \frac{c + d + e + 5000}{4} - 1000
\end{equation}
Multiplying by $12$ and collecting like terms gives $c + d + e = 3000$.
Therefore the total value is
$$a + b + 5000 + c + d + e = 16000 + 5000 + 3000 = 24000$$