15. The villages $P, Q, R$ and $S$ are situated, not necessarily in that order, on a long straight road. The distance from $P$ to $R$ is $75 \mathrm{~km}$, the distance from $Q$ to $S$ is $45 \mathrm{~km}$ and the distance from $Q$ to $R$ is $20 \mathrm{~km}$. Which of the following could not be the distance, in $\mathrm{km}$, from $P$ to $S$ ?
A 10
B 50
C 80
D 100
E 140
Solution
C

Since the distance from $P$ to $R$ is $75 \mathrm{~km}$ and the distance of $Q$ from $R$ is $20 \mathrm{~km}$, there are two possible distances of $Q$ from $P,(75+20) \mathrm{km}=95 \mathrm{~km}$ and $(75-20) \mathrm{km}=55 \mathrm{~km}$. For each of the possible positions of $Q$, there are two possible positions of $S$, each $45 \mathrm{~km}$ from $Q$, as shown in the diagrams below.
Therefore the possible distances, in $\mathrm{km}$, of $S$ from $P$ are $95+45=140,95-45=50$, $55+45=100$ and $55-45=10$. Therefore, of the options given, the one which is not a possible distance in km of $S$ from $P$ is 80 .