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Jewellery Boxes

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

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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