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
Sure! Let me explain the question and the solution step-by-step.
We are given that P, Q, and R start from the same place X at speeds of a kmph, (a+b) kmph, and (a+2b) kmph respectively.
We are also given that Q starts p hours after P. This means that by the time Q starts, P would have already traveled a distance of a * p kilometers.
Now, our goal is to find out how many hours after Q should R start, so that both Q and R overtake P at the same time.
To solve this, let's consider a scenario when both Q and R overtake P.
When Q overtakes P, they would have traveled the same distance. Let's say they meet after t hours.
In t hours, Q would have traveled at a rate of (a+b) kmph, so the distance traveled by Q would be (a+b) * t kilometers.
On the other hand, P would have traveled for (t + p) hours and at a rate of a kmph, so the distance traveled by P would be a * (t + p) kilometers.
Now, to find out when R should start in order to overtake P at the same time as Q, we need to make sure that R travels the same distance as P in the same amount of time.
Since R starts t hours after Q, by the time R starts, it would have taken R (t - p) hours for Q to overtake P.
In (t - p) hours, Q would have traveled a distance of (a+b) * (t - p) kilometers.
Now, for R to overtake P at the same time as Q, the distance traveled by R in the same amount of time (t - p) hours should be equal to the distance traveled by P, which is a * (t + p) kilometers.
Therefore, we have the equation: (a+2b) * (t - p) = a * (t + p)
Now, we can solve this equation to find the value of t. Simplifying the equation, we get:
at - ap + 2bt - 2bp = at + ap
Cancelling out the common terms, we have:
2bt - 2bp = 2ap
Dividing both sides by 2b, we get:
t - p = ap/b
Finally, to find out how many hours after Q should R start, we need t - p. So, the answer is:
t - p = ap/b
I hope this explanation helps you understand the solution better. Let me know if you have any more questions!
Let's break down the problem step by step.
1. P, Q, and R start from the same place X at speeds of a km/h, (a+b) km/h, and (a+2b) km/h respectively.
2. Q starts p hours after P. This means that when Q starts, P has already been traveling for p hours.
3. We need to find out how many hours after Q should R start, so that both Q and R overtake P at the same time.
To find the solution, let's assume that both Q and R overtake P after t hours.
Distance traveled by P in t hours = Distance traveled by Q in (t - p) hours
Let's calculate the distances traveled by each person:
Distance traveled by P in t hours = Speed of P * Time taken = a * t
Distance traveled by Q in (t - p) hours = Speed of Q * Time taken = (a + b) * (t - p)
Now, since both Q and R overtake P at the same time, their distances traveled should be equal at t hours:
Distance traveled by Q in (t - p) hours = Distance traveled by R in (t - p - x) hours, where x is the additional time R starts after Q.
Distance traveled by Q in (t - p) hours = Speed of Q * Time taken = (a + b) * (t - p)
Distance traveled by R in (t - p - x) hours = Speed of R * Time taken = (a + 2b) * (t - p - x)
Setting these two equal:
(a + b) * (t - p) = (a + 2b) * (t - p - x)
Now, we can solve this equation to find x, the additional time R starts after Q.
(a + b) * (t - p) = (a + 2b) * (t - p - x)
Simplifying the equation:
(a + b) * t - (a + b) * p = (a + 2b) * t - (a + 2b) * (p + x)
(a + b) * t - (a + b) * p = (a + 2b) * t - (a + 2b) * p - (a + 2b) * x
Expanding further:
a * t + b * t - a * p - b * p = a * t + 2b * t - a * p - 2b * p - (a + 2b) * x
Canceling out the common terms:
b * t - b * p = b * t - 2b * p - (a + 2b) * x
Simplifying:
b * p = (a + 2b) * x
Finally, to find x, we can divide both sides by (a + 2b):
x = (b * p) / (a + 2b)
Thus, the answer is x = (b * p) / (a + 2b)
This formula tells you how many hours after Q should R start, so that both Q and R overtake P at the same time.