If the radius of a cylinder is doubled and the height is reduced by 50%, then by how much percent does the volume increase/decrease?

Let's assume the radius and height of the cylinder initially are 'r' and 'h' respectively.

the volume of cylinder = $$\pi\times r^2\times\ h$$

= $$\pi r^2h$$   Eq.(i)

If the radius of a cylinder is doubled and the height is reduced by 50%.

New radius of a cylinder = 2r

The new height of a cylinder = h of (100-50)% = h of 50% = 0.5h

the new volume of cylinder = $$\pi\times (2r)^2\times\ 0.5h$$

= $$\pi\times4r^2\times\ 0.5h$$

= $$2\times \pi r^2h$$    Eq.(ii)

Here we can see that there is an increase in the volume from Eq.(i) to Eq.(ii).

Percentage increase in the volume = $$\frac{\left(2\pi r^2h\ -\ \pi r^2h\right)}{\pi r^2h}\times\ 100$$

= $$\frac{\ \pi r^2h}{\pi r^2h}\times\ 100$$

= 100%