Let the quantity of alloy $P$ to be mixed be $x$ and the quantity of alloy $Q$ to be mixed be $y$.

In alloy $P$, the percentage of copper is $20\%$. In alloy $Q$, the percentage of copper is $60\%$. In the final alloy $R$, the amounts of copper and zinc are equal, which means the percentage of copper in $R$ must be $50\%$.

Method 1: Using Algebra The total amount of copper in the mixture is the sum of the copper from $P$ and $Q$: Total copper = $0.20x + 0.60y$

The total weight of the new alloy $R$ is $(x + y)$. Since copper makes up $50\%$ of alloy $R$, we can set up the following equation: $0.20x + 0.60y = 0.50(x + y)$

Expanding and solving for the ratio of $x$ to $y$: $0.20x + 0.60y = 0.50x + 0.50y$ $0.60y - 0.50y = 0.50x - 0.20x$ $0.10y = 0.30x$ $\frac{x}{y} = \frac{0.10}{0.30} = \frac{1}{3}$

Method 2: Using the Rule of Alligation We can apply the rule of alligation directly to the concentration of copper in the alloys:

• Concentration of copper in $P$ = $20\%$
• Concentration of copper in $Q$ = $60\%$
• Desired mean concentration in $R$ = $50\%$

The ratio of $P$ to $Q$ is calculated by taking the cross-differences: Ratio = $(60 - 50) : (50 - 20)$ Ratio = $10 : 30 = 1 : 3$

Thus, the amounts of $P$ and $Q$ must be mixed in the ratio $1:3$.