13. Three rugs have a combined area of $90 \mathrm{~m}^{2}$. When they are laid down to cover completely a floor of area $60 \mathrm{~m}^{2}$, the area which is covered by exactly two layers of rug is $12 \mathrm{~m}^{2}$.
What is the area of floor covered by exactly three layers of rug?
A $2 \mathrm{~m}^{2}$
B $6 \mathrm{~m}^{2}$
C $9 \mathrm{~m}^{2}$
D $10 \mathrm{~m}^{2}$
E $12 \mathrm{~m}^{2}$
Solution
C
The diagram shows the three overlapping rugs.
We let the areas of the different regions, in $\mathrm{m}^{2}$, be $P, Q, R$, $S, T, U$ and $V$, as shown.
The three rugs have areas $P+S+T+V, Q+T+U+V$ and $R+S+U+V$. Therefore, because the three rugs have a combined area of $90 \mathrm{~m}^{2}$,
$$
(P+S+T+V)+(Q+T+U+V)+(R+S+U+V)=90 .
$$

This equation may be rearranged to give
$$
(P+Q+R)+2(S+T+U)+3 V=90 .
$$

Because the floor has an area of $60 \mathrm{~m}^{2}$,
$$
(P+Q+R)+(S+T+U)+V=60 .
$$

By subtracting equation (2) from equation (1), we obtain
$$
(S+T+U)+2 V=30 \text {. }
$$

Because the area covered by exactly two layers of rug is $12 \mathrm{~m}^{2}$,
$$
S+T+U=12 .
$$

By subtracting equation (4) from equation (3), we have $2 V=18$. Therefore $V=9$. Therefore the area of floor covered by exactly three layers of rug is $9 \mathrm{~m}^{2}$.

For INVESTIGATION
13.1 What is the area of the floor that is covered by exactly one rug?