$P$ is inversely proportional to $Q^3$. Find the percentage change in $P$ when $Q$ is reduced by 20%. Answer ...%

1. Express the new value of $Q$ after a 20% reduction.

Let the original value of $Q$ be $Q$.

Reduced by $20\%$ means the new value is

$80\% \times Q = 0.8Q.$

2. Use inverse proportionality to find the percentage change in $P$.

Since $P$ is inversely proportional to $Q^3$,

$P \propto \frac{1}{Q^3}.$

So

$\frac{P_{\text{new}}}{P_{\text{old}}}=\frac{\frac{1}{(0.8Q)^3}}{\frac{1}{Q^3}}=\frac{Q^3}{(0.8Q)^3}=\frac{1}{0.8^3}.$

$\frac{1}{0.8^3}=\frac{1}{0.512}=1.953125.$

Hence $P$ becomes $195.3125\%$ of its original value.

Percentage increase:

$195.3125\%-100\%=95.3125\%.$