For this sort of thing, the best thing to do is to "convert" the statements into mathematical notation. For example:

You have a rectangle, the sides of which are in a ratio of 4:9.

So the shorter side length is 4x, and the longer side length is 9x, for some value of x.

The shorter side is stretched by 25%

So the new length of the shorter side is 4x * 1.25 = 5x.

while we take away 4cm from the longer side.

So the new length of the longer side is 9x + 4.

The resulting perimeter is therefore increased by 4cm.

So the new perimeter is the old perimeter + 4.

Find the sides in cm.

Okay, the new perimeter is 4 + the old perimeter, so the new perimeter = 4 + 4x + 4x + 9x + 9x = 4 + 26x.

The new perimeter is 5x + 5x + 9x +4 + 9x + 4 = 28x + 8.

So 28x + 8 = 26x + 4, which implies that 2x = 4, so x = 1/2.

Then substitute this into the sides (I think the question means the original sides? It's not too clear), to have a shorter side of 2cm and a longer side of 4.5cm.