A boat traveled upstream a distance of 90 miles at an average speed of
(v-3) miles per hour and then traveled the same distance downstream at an
average speed of (v+3) miles per hour. If the trip upstream took half an
hour longer than the trip downstream, how many hours did it take the boat
to travel downstream?

a) 2.5
b) 2.4
c) 2.3
d) 2.2
e) 2.1

The speed downstream (v+3) is 6 miles per hour greater than the speed upstream (v-3).
We can PLUG IN THE ANSWERS, which represent the time to travel downstream.
When the correct answer choice is plugged in, the difference between the time upstream and the time downstream will be 0.5.
The most likely answer choice is A -- the only option that divides evenly into 90.

Answer choice A: 2.5 hours to travel downstream.
Rate downstream = d/t = 90/(2.5) = 36 miles per hour.
Rate upstream = 36-6 = 30 miles per hour.
Time upstream = d/r = 90/30 = 3 hours.
Time upstream - time downstream = 3-2.5 = 0.5.
Success!

The correct answer is A.

prachi18oct wrote: The difference in speed upstream and downstream is 6 miles/h and time taken is less by 0.5 hour for downstream.
So, 6 * 0.5 = 3 miles will be difference in the distance travelled in the same time i.e if the downstream

The portion in red is not quite correct.
The following line of reasoning is correct:
Since the rate upstream is 6 miles per hour less than the rate downstream, 3 fewer miles will be traveled upstream than downstream FOR EVERY 1/2 HOUR OF TRAVEL TIME.

Case 1: Time to travel 90 miles downstream = 1/2 hour
Since the distance traveled upstream is 3 miles less for every 1/2 hour of travel time, the boat travels 3 fewer miles upstream.
Thus, the distance traveled upstream = 90-3 = 87 miles.
In this case, the boat upstream must travel the remaining 3 miles in 1/2 hour.

Case 2: Time to travel 90 miles downstream = 1 hour
Since the distance traveled upstream is 3 miles less for every 1/2 hour of travel time, the boat travels 6 fewer miles upstream.
Thus, the distance traveled upstream = 90-6 = 84 miles.
In this case, the boat upstream must travel the remaining 6 miles in 1/2 hour.

Case 3: Time to travel 90 miles downstream = 1.5 hours
Since the distance traveled upstream is 3 miles less for every 1/2 hour of travel time, the boat travels 9 fewer miles upstream.
Thus, the distance traveled upstream = 90-9 = 81 miles.
In this case, the boat upstream must travel the remaining 9 miles in 1/2 hour.

So, (v+3)t1 = 90 & (v-3)t1 = 87

This equation is valid only for Case 1, in which the time to travel downstream = 1/2 hour.
Since the time to travel downstream is unknown, this equation is not valid.

You could organize your info as follows:

Upstream:
D = 90
R = (v - 3)
T = t + .5

Downstream
D = 90
R = (v + 3)
T = t

Since the distances are the same, we have

(v - 3)(t + .5) = (v + 3)t
vt - 3t + .5v - 1.5 = vt + 3t
.5v - 1.5 = 6t
v - 3 = 12t

We know from the upstream trip that
90 = (v - 3) * (t + .5)

Since (v - 3) = 12t, we really have
90 = 12t * (t + .5)
90 = 12t² + 6t
15 = 2t² + t
0 = 2t² + t - 15
0 = (2t - 5)(t + 3)

So 2t = 5, or t = 2.5.