If the radius of the base of a right circular cylinder is halved, then keeping the height same, the ratio of the volume of the reduced cylinder to the volume of the original cylinder is:

Concept -

The formula for the volume of a right circular cylinder is \(V = \pi r^2h\) , where r is the radius of the base and h is the height.

Explanation -

If the radius of the base is halved, the new volume V'  of the cylinder with the halved radius becomes:

\(V' = \pi \left(\frac{r}{2}\right)^2h = \pi \frac{r^2}{4}h \)

Now, let's find the ratio of the volume of the reduced cylinder to the volume of the original cylinder:

Ratio =\( \frac{V'}{V} = \frac{\pi \frac{r^2}{4}h}{\pi r^2h} = \frac{1}{4}\)

Therefore, if the radius of the base of a right circular cylinder is halved while keeping the height constant, the ratio of the volume of the reduced cylinder to the volume of the original cylinder is \( \frac{1}{4}.\)

Hence the ratio of the volume of the reduced cylinder to the volume of the original cylinder is 1:4.