Suppose that initially the total value of the jewels in $P$ is $£A$, and those in $Q$ is $£B$.

The average value in $P$ at the start is $\frac A3$. After the jewel has been moved it is $\frac{A-5000}2$. Therefore:

\begin{equation}

\frac A3 = \frac{A-5000}2 - 1000

\end{equation}

Multiplying by $6$ and collecting like terms gives $A = 21000$.

The average value in $Q$ at the start is $\frac B3$. After the jewel has been moved, the average value is $\frac{B+5000}4$. Therefore:

\begin{equation}

\frac B3 = \frac{B+5000}4 - 1000

\end{equation}

Multiplying by $12$ and collecting like terms gives $B = 3000$.

Therefore the total value is $£24,000$

*This problem is taken from the UKMT Mathematical Challenges.*

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