Two towns P and Q are 400 km apar. A bus left for P and Q. It stopped at Q for one hour and then started the return to P. one hour after the departure of the buse from P, a trail also heading for Q left P. the trailer meth the returning bus ¾ of the way from P to Q. they met t hours after the departure of the buse from from P.

(a) Epress the average speed of the trailer in terms of t
(b) Find the ration of the speed of the bus O that of the trailer

To solve this question, we need to analyze the situation and set up equations based on the given information.

Let's start with the distance between towns P and Q, which is 400 km. We'll use this as a reference point for all calculations.

(a) To express the average speed of the trailer in terms of t, we need to find the distance traveled by the trailer during time t.

The bus traveled for t + 1 hours before meeting the trailer. Since the trailer met the returning bus ¾ of the way from P to Q, it traveled ¾ * 400 km = 300 km during t + 1 hours.

To find the average speed of the trailer, we divide the distance traveled by the time taken:
Average speed of the trailer = 300 km / (t + 1) hours
So the average speed of the trailer in terms of t is (300 / (t + 1)) km/h.

(b) To find the ratio of the speed of the bus to that of the trailer, we need to compare their speeds.

Let's assume the speed of the bus is B km/h, and the speed of the trailer is T km/h.

Since the trailer met the returning bus ¾ of the way from P to Q, the distance traveled by the bus is ¾ * 400 km = 300 km during t hours.

To find the ratio of their speeds:
Speed of the bus / Speed of the trailer = B / T

We know that the time taken by the bus to cover 300 km is t hours, so:
Speed of the bus = Distance / Time = 300 km / t hours
Speed of the trailer = Distance / Time = 300 km / (t + 1) hours

Therefore, the ratio of their speeds is:
B / T = (300 km / t hours) / (300 km / (t + 1) hours)
= (300 km / t) * ((t + 1) / 300)
= (t + 1) / t

Hence, the ratio of the speed of the bus to that of the trailer is (t + 1) / t.