A and B cross the lake in a straight line with the help of a one-seat

Question

A and B cross the lake in a straight line with the help of a one-seat boat. Each can row the boat at a speed of 7 km/hour and swim at a speed of 2 km/hour. They start from the same point, where A starts by rowing while B swims. After some time A got out of his boat and started swimming by leaving the boat in the position he started swimming. Once B reaches the position of the boat, he will begin rowing. If they arrived on the other side of the lake at the same time which is 90 minutes after they started, the length of time the boat was empty would be?
A. 35 minutes
B. 40 minutes
C. 45 minutes
D. 50 minutes
E. 55 minutes

Kinshook · Answer

Given: A and B cross the lake in a straight line with the help of a one-seat boat. Each can row the boat at a speed of 7 km/hour and swim at a speed of 2 km/hour. They start from the same point, where A starts by rowing while B swims. After some time A got out of his boat and started swimming by leaving the boat in the position he started swimming. Once B reaches the position of the boat, he will begin rowing.

Asked: If they arrived on the other side of the lake at the same time which is 90 minutes (1.5 hours) after they started, the length of time the boat was empty would be?

Let us assume that A rows for   t_1  hours and then swims for remaining  t_2 = 1.5-t_1  hours
Distance covered by A =  7t_1 + 2t_2 = 7t_1 + 2(1.5-t_1) = 5t_1 +3  km

B takes time =  \frac{7t_1}{2} = 3.5t_1  hours to reach the boat, then rows for remaining (1.5 -  3.5t_1 ) hours.
Distance covered by B =  7t_1 + 7(1.5- 3.5t_1) = 10.5 - 17.5t_1  km

Since both distance are same
 5t_1 + 3 = 10.5 - 17.5t_1

22.5t_1  = 7.5

t_1  =  \frac{7.5}{22.5}  =  \frac{1}{3}  hours = 20 minutes

Idle time of boat =  \frac{(7t_1 - 2t_1)}{2}  =  \frac{5t_1}{2}  = 5* \frac{20}{2}  = 50 minutes.

IMO D

Shhh103 · Answer

let the distance when A stopped = x 
remaining distance = y

A,B_________X__________P ( STOP) ________Y_________F (FINAL stop)

so x/7( TIME TAKEN BY A TO REACH P) + y/2 (TIME TAKEN BY A TO REACH FINAL POINT AFTER P)
 = 90 mins = x/2 ( TIME TAKEN BY B TO REACH P) + y/7 (TIME TAKEN BY B TO REACH FINAL POINT AFTER P)
solving this we get

x = y = 140

time taken by A to reach point P = x/7 = 140/7 = 20 min

time taken by B to reach point P = x/2 = 140/2 = 70 min

time when boat was empty = 70-20 = 50 mins
Correct answer is D

ThatDudeKnows · Answer

Another good question by a Newcomer! We've had a few of these lately...good stuff!

This is a nice example of a question where "Quantitative" and "Reasoning" collide. If you get good at reasoning through some of these more complex QR questions, the actual math is really simple.

A and B each moves the entire time. When they are both swimming, they progress at the same rate. When they are both rowing, they progress at the same rate. They start at the same time and finish at the same time. Given all that, we know the amount of time that A swims while B rows must be equal to the amount of time that B swims while A rows.

Let s be the amount of time that each swims and r be the amount of time that each rows.

s+r = 90--> 2s+2r = 180
7r = 2s --> -2s+7r = 0

Add those together:
9r = 180
r = 20

Each rows for 20 minutes, so the boat gets used for 40 minutes. That means it sits vacant for 50 minutes.

ScottTargetTestPrep · Answer

Let A row for t minutes and let B row for s minutes. Since each person swims for the rest of the time, A swims for 90 - t minutes and B swims for 90 - s minutes.

Notice that the total distance that A travels is equal to the total distance that B travels. Since A rows for at 7 km/h for t minutes and swims at 2 km/h for 90 - t minutes, the total distance that A travels is 7 * (t/60) + 2 * ((90 - t)/60). Similarly, the total distance that B travels is 7 * (s/60) + 2 * ((90 - s)/60). Let's set these two expression equal and simplify:

\Rightarrow  7 * (t/60) + 2 * ((90 - t)/60) = 7 * (s/60) + 2 * ((90 - s)/60)

\Rightarrow  7t + 2(90 - t) = 7s + 2(90 - s)

\Rightarrow  7t + 180 - 2t = 7s + 180 - 2s

\Rightarrow  5t = 5s

\Rightarrow  t = s

This shows that A and B row and swim for equal amounts of time.

Next, notice that the distance that A swims is equal to the distance that B rows. This is because B starts rowing from the same point A starts swimming, and they finish at the same point as well. The distance that A swims is 2 * ((90 - t)/60) and the distance that B rows is 7 * s/60 = 7s/60 = 7t/60. Let's set these two expression equal to each other:

\Rightarrow  2 * ((90 - t)/60) = 7t/60

\Rightarrow  2 * (90 - t) = 7t

\Rightarrow  180 - 2t = 7t

\Rightarrow  180 = 9t

\Rightarrow  t = 20

We see that both A and B row for 20 minutes. Since the total time is 90 minutes and since the boat was occupied for 20 + 20 = 40 minutes, it follows that the boat was empty for 90 - 40 = 50 minutes.

Answer: D

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A and B cross the lake in a straight line with the help of a one-seat

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