The force, F units, between two objects is inversely proportional to the square of the distance, d units, between them. If the distance decreases by 10%, find the percentage change in the force.

1. Express the inverse-square relationship and set a convenient original distance.

Since $F$ is inversely proportional to the square of $d$,
$F=\frac{k}{d^2}.$
Let the original distance be $d$ units, so the original force is
$F=\frac{k}{d^2}.$

2. Find the new force after the distance decreases by 10%.

A decrease of $10\%$ means the new distance is
$0.9d.$
New force
$=\frac{k}{(0.9d)^2}=\frac{k}{0.81d^2}=\frac{1}{0.81}\cdot \frac{k}{d^2}.$

3. Calculate the percentage change in the force.

Thus
$\frac{\text{new force}}{\text{original force}}=\frac{1}{0.81}=1.234567\ldots$
Percentage change
$=(1.234567\ldots-1)\times 100\%=23.4567\ldots\%.$
So the force increases by $23.5\%$ to 3 significant figures.