Define initial and final radius

Let the initial radius be $R_1 = R$. If the radius is reduced by 50%, the new radius is $R_2 = R - 0.5R = 0.5R = \frac{1}{2}R$.

$R_2 = \frac{1}{2}R$

Express volume formula

The volume $V$ of a sphere is given by the formula $V = \frac{4}{3} \pi r^3$.

$V = \frac{4}{3} \pi r^3$

Calculate the new volume

The new volume $V_2$ is calculated by substituting $R_2 = \frac{1}{2}R$ into the formula: $V_2 = \frac{4}{3} \pi (\frac{1}{2}R)^3 = \frac{4}{3} \pi (\frac{1}{8}R^3)$.

$V_2 = \frac{1}{8} \times (\frac{4}{3} \pi R^3)$

Compare with original volume

Since the original volume is $V_1 = \frac{4}{3} \pi R^3$, we can see that $V_2 = \frac{1}{8} V_1$.

$V_2 = \frac{1}{8} V_1$