Two towns P and Q are 400 km apart. 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 bus from P, a trail also heading for Q left P. the trailer met the returning bus ¾ of the way from P to Q. they met t hours after the departure of the bus from P.

(a) Express the average speed of the trailer in terms of t

(b) Find the ratio of the speed of the bus so that of the trailer.

The Jiskha tutor Steve posted this answer to the exact same question in 2015...

Perhaps it will start you in the right direction :)

The trailer went 300 km in t-1 hours, so its avg speed is 300/(t-1)

Let the bus's speed be x, and the trailer's speed be y.

The bus went 500 km.
Its time spent on the trip before meeting the trailer was t = 400/x + 1 + 100/x = 1 + 500/x hours

The trailer was on the road for 300/y hours

so, 300/y = 500/x
(b) x/y = 5/3

check:
x=50, y=30

400/50 + 1 + 100/50 = 11 hrs
300/30 = 10, or 1 hours less

am not understad where 400/50 come from

(a) Well, the average speed of the trailer can be expressed as the distance traveled divided by the time taken. In this case, the trailer traveled 3/4 of the distance between P and Q in t hours. So we can say the average speed of the trailer is (3/4) * (400/t) km/h.

(b) To find the ratio of the speed of the bus to that of the trailer, we need to compare their average speeds. Let's assume the speed of the bus is x km/h. Therefore, the average speed of the bus is 400/(t + 1) km/h (since it spent one additional hour at point Q).
Now, we divide the average speed of the bus by the average speed of the trailer:
x / ((3/4) * (400 / t)) = (4x * t) / (3 * 400)
Simplifying this, we get:
x / ((3/4) * (400 / t)) = (4x * t) / (3 * 400)
4xt = x * (3/4) * (400 / t)
4xt = 3x
4t = 3
t = 3/4

So, the ratio of the speed of the bus to that of the trailer is 3/4.

Let's analyze the given information step-by-step to find the answers.

Step 1: Calculate the distance from P to the meeting point of the trailer and the returning bus.
Given that the distance between towns P and Q is 400 km, the meeting point is located (3/4) * 400 km = 300 km away from P.

Step 2: Calculate the time it takes for the bus to travel from P to the meeting point.
Since the trailer left P one hour after the bus, and they met t hours after the departure of the bus from P, the bus traveled for t + 1 hours.

Step 3: Calculate the average speed of the bus.
Average speed = Total distance / Total time
Total distance = 400 km
Total time = t + 1 hours
Average speed of the bus = 400 km / (t + 1) hours

(a) Express the average speed of the trailer in terms of t:
The average speed of the trailer can be calculated by dividing the distance traveled by the trailer by the time it took.
Given that the trailer met the bus at the meeting point, which is 300 km away from P:
Average speed of the trailer = Distance traveled by the trailer / Time taken by the trailer
The trailer traveled 300 km in t hours (since the bus traveled for t + 1 hours).
Average speed of the trailer = 300 km / t hours

(b) Find the ratio of the speed of the bus to that of the trailer:
To find the ratio between the speeds, divide the average speed of the bus by the average speed of the trailer:
Ratio of the speed of the bus to that of the trailer = (Average speed of the bus) / (Average speed of the trailer)
Ratio = [(400 km) / (t + 1) hours] / [(300 km) / t hours]
Ratio = [(400 km) * t] / [(t + 1) * (300 km)]

So, the ratio of the speed of the bus to that of the trailer is [(400 km) * t] / [(t + 1) * (300 km)].

To solve this problem, we need to analyze the distances and times involved. Let's break it down step by step.

Step 1: Calculate the distance traveled by the bus and the trailer during their respective journeys.

Given that the two towns, P and Q, are 400 km apart, the bus travels the full 400 km distance between the towns. The trailer starts its journey 1 hour after the departure of the bus from town P and meets the returning bus ¾ of the way from P to Q.

Let's denote the distance traveled by the bus as D, which is 400 km.

The trailer meets the returning bus at a point ¾ * D from town P. So, the distance traveled by the trailer is ¾ * D.

Step 2: Calculate the time taken by the bus and the trailer.

Let's denote the time taken by the bus as T (in hours).

The trailer starts 1 hour after the bus and meets the returning bus t hours after the departure of the bus. So, the time taken by the trailer is T + 1 + t.

Step 3: Calculate the average speed of the trailer.

The average speed of an object can be calculated by dividing the distance traveled by the time taken. Therefore, the average speed of the trailer (Vt) can be expressed as:

Vt = (¾ * D) / (T + 1 + t)

This equation gives us the average speed of the trailer in terms of t (the time taken by the trailer).

(a) Expressing the average speed of the trailer in terms of t:

Vt = (¾ * 400) / (T + 1 + t)

Simplifying the equation further, we get:

Vt = 300 / (T + t + 1)

So, the average speed of the trailer in terms of t is 300 divided by the sum of the time taken by the bus, the time taken by the trailer, and 1.

(b) Finding the ratio of the speed of the bus to that of the trailer:

To find the ratio of the speed of the bus (Vb) to that of the trailer (Vt), we need to divide the speed of the bus by the average speed of the trailer:

Vb / Vt = (D / T) / (300 / (T + t + 1))

Simplifying the equation further, we get:

Vb / Vt = (D * (T + t + 1)) / (300 * T)

So, the ratio of the speed of the bus to that of the trailer is given by the expression (D * (T + t + 1)) / (300 * T).