Let $O, P$ and $Z$ represent the numbers $0, 3$ and $- 5$ respectively on the number line. Points $Q, R$ and $S$ are between $O$ and $P$ such that $O Q = Q R = R S = S P$ . What are the rational numbers represented by the points $Q, R$ and $S$ ?

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Solution

As $O Q = Q R = R S = S P$ and $O Q + Q R + R S + S P = O P$
therefore $Q, R$ and $S$ divide $O P$ into four equal parts.
So, $R$ is the midpoint of $O P$ , i.e. $R = \frac{1}{2} (0 + 3) = \frac{3}{2}$
$Q$ is the midpoint of $O R$ , i.e. $Q = \frac{1}{2} (0 + \frac{3}{2}) = \frac{3}{4}$
and, $S$ is the midpoint of $R P$ , i.e. $S = \frac{1}{2} (\frac{3}{2} + 3) = \frac{9}{4}$
therefore, $Q = \frac{3}{4}, R = \frac{3}{2},$ and $S = \frac{9}{4}$

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