Three rugs have a combined area of $200 \mathrm{~m}^{2}$. By overlapping the rugs to cover a floor area of $140 \mathrm{~m}^{2}$, the area which is covered by exactly two layers of rug is $24 \mathrm{~m}^{2}$. What area of floor is covered by three layers of rug?
(A) $12 \mathrm{~m}^{2}$
(B) $18 \mathrm{~m}^{2}$
(C) $24 \mathrm{~m}^{2}$
(D) $36 \mathrm{~m}^{2}$
(E) $42 \mathrm{~m}^{2}$
Three rugs have a combined area of $200 \mathrm{~m}^{2}$. By overlapping the rugs to cover a floor area of $140 \mathrm{~m}^{2}$, the area which is covered by exactly two layers of rug is $24 \mathrm{~m}^{2}$. What area of floor is covered by three layers of rug?
(A) $12 \mathrm{~m}^{2}$
(B) $18 \mathrm{~m}^{2}$
(C) $24 \mathrm{~m}^{2}$
(D) $36 \mathrm{~m}^{2}$
(E) $42 \mathrm{~m}^{2}$

## Solution

Draw the rugs in the following manner, where $a+b+c$ represents the amount of floor covered by exactly two rugs and $k$ represents the amount of floor covered by exactly three rugs. We are told that $a+b+c=24$ (1).

![](https://cdn.mathpix.com/cropped/2024_04_20_688a3b42f69739f2e244g-285.jpg?height=356&width=355&top_left_y=852&top_left_x=1449)

Since the total amount of floor covered when the rugs do not overlap is $200 \mathrm{~m}^{2}$ and the total covered when they do overlap is $140 \mathrm{~m}^{2}$, then $60 \mathrm{~m}^{2}$ of rug is wasted on double or triple layers. Thus, $a+b+c+2 k=60$ (2). Subtract equation (1) from equation (2) to get $2 k=36$ and solve for $k=18$. Thus, the area of floor covered by exactly three layers of rug is $18 \mathrm{~m}^{2}$.

ANSWER: (B)