There are 28 steps in a temple. In the time A, initially at the 28th s

There are 28 steps in a temple. In the time A, initially at the 28th step, comes down two steps, B, initially at 1st step, goes one step up. If they start simultaneously and keep their speed uniform, then at which step from the bottom will they meet ?

8<sup>th</sup>
9<sup>th</sup>
10<sup>th</sup>
not possible to meet
This question was previously asked in
UPSC CAPF – 2014
Let the steps be numbered from 1 (bottom) to 28 (top). Person A starts at step 28 and moves down 2 steps per unit of time. Person B starts at step 1 and moves up 1 step per unit of time.
Let $t$ be the time elapsed in units of their respective step movements.
A’s position after time $t$ (from the bottom) is $P_A(t) = 28 – 2t$.
B’s position after time $t$ (from the bottom) is $P_B(t) = 1 + 1t$.
They meet when their positions are equal: $P_A(t) = P_B(t)$.
$28 – 2t = 1 + t$
$28 – 1 = t + 2t$
$27 = 3t$
$t = 9$.
So, they meet after 9 units of time. Their position at this time is:
$P_B(9) = 1 + 9 = 10$.
$P_A(9) = 28 – 2(9) = 28 – 18 = 10$.
They meet at the 10th step from the bottom.
This is a problem involving relative speed. The relative speed at which they are closing the distance between them is 2 steps/time (A) + 1 step/time (B) = 3 steps/time. The initial distance between them is $28 – 1 = 27$ steps. The time taken to meet is distance/relative speed = 27 steps / 3 steps/time = 9 units of time. Then calculate the position from the start of either person.