# Loo roll emergency

When the roll of toilet paper is half as wide, what percentage of the paper is left?

## Problem

New toilet rolls are cylindrical, 12 cm in diameter, and the cardboard tube in the middle is 4 cm in diameter.

Image

My roll has diameter 6 cm, so it looks about half the size.

What percentage of the toilet paper is left?

## Student Solutions

The amount of toilet paper left will be proportional to the area of the paper on the cross-section of the roll. The cross sections of the two rolls are shown below.

On the new roll, this area is $\pi\times6^2-\pi\times2^2=36\pi-4\pi=32\pi$ cm$^2.$

On my roll, this area is $\pi\times3^2-\pi\times2^2=9\pi-4\pi = 5\pi$ cm$^2.$

So $\frac{5}{32}=15.625\%$ of the toilet paper is left.

Image

On the new roll, this area is $\pi\times6^2-\pi\times2^2=36\pi-4\pi=32\pi$ cm$^2.$

On my roll, this area is $\pi\times3^2-\pi\times2^2=9\pi-4\pi = 5\pi$ cm$^2.$

So $\frac{5}{32}=15.625\%$ of the toilet paper is left.