# Combined overlap area question

I'm having a bit of trouble with this question:

Sue works in a rug shop. She as three rugs with a combined area of $200 m^2$.

She arranged the rugs so they overlap and cover a floor area of $140 m^2$.

The total floor area that is covered by exactly two layers of rugs is $24 m^2$.

What is the floor area that is covered by exactly three layers of rugs, in $m^2$?

The answer I get is $36$, and searching online that's what other people are getting too. However, the given answer is 18. How is that so? Thanks.

• Use the inclusion exclusion principle. – Rogelio Molina Jul 10 '15 at 3:58

The $200 m^2$ of rug have to be "used up".

A single layer will use up $140 m^2$, $60 m^2$ carpet area left.

The 2-layer part will use up 60 - 24 = $36 m^2$

In the 3-layer part, two carpets are overlapping with the single layer,

thus part needing 3 layers = $\frac{36}{2} = 18 m^2$

Or if you must have algebra, let x, 24, z be $m^2$ of single, double, triple layered portions, then

x + 2*24 + 3z = 200 ... (i )

x + 24 + z = 140 ... (ii)

Subtracting, 24 + 2z = 60

z = 18