In 1000 metres race, P, Q, R scored first, second and third positions respectively. If P beats Q by 100 metres and Q beats R by 200 metres, then the gap between P and R is :

280 m

The correct answer is Option (2) → 280 m

In a 1000 m race, when P finishes (runs 1000 m), Q is 100 m behind, so Q has run 900 m and R has run 800 m when Q finishes the race.

Time taken by T to finish, $T_P=\frac{1000}{V_P}$

and, in the same time, Q covers 900 m

$T_P=\frac{900}{V_Q}$

$⇒\frac{1000}{V_P}=\frac{900}{V_Q}⇒\frac{V_P}{V_Q}=\frac{10}{9}$

Time taken by Q to complete the race, $T_Q=\frac{1000}{V_Q}$

$⇒\frac{1000}{V_Q}=\frac{800}{V_R}⇒V_R=0.8V_Q$

Express $T_P$ in terms of $T_Q$,

$V_P=\frac{10}{9}V_Q⇒T_P=\frac{1000}{(\frac{10}{9})V_Q}=\frac{900}{V_Q}$

∴ Distance covered by R = $V_R×T=(0.8V_Q)×\left(\frac{900}{V_Q}\right)=720m$

The gap between P and R = $1000-720m$

$=280m$