In a river, a motorboat and raft start their journeys in the same

Question

In a river, a motorboat and raft start their journeys in the same direction at the same time, from the same point. The motorboat travels \(\frac{40}{3}\) km downstream and then without stopping, travels \(\frac{28}{3}\) km in the opposite direction and meets the raft. What is the speed of the river to that of the motorboat in still water? (Assume the raft just floats)

A. 1/5

B. 3/17

C. 1/9

D. 3/16

E. 2/5

A. 1/5

B. 3/17

C. 1/9

D. 3/16

E. 2/5

architkap · Answer

Not sure if this kind of question would ever come in the exam. However, wanted to post as I found it difficult to solve as a question.

KarishmaB · Answer

The speed of the raft is the speed of the river since it just floats.

In the same time, the motorboat travels 40/3 + 28/3 = 68/3 km while the raft travels 40/3 - 28/3 = 12/3 km.

Since this distance is covered in the same time, the ratio of this distance is the same as the ratio of speeds.
So ratio of speed of raft (river) : speed of motorboat = 12 : 68 = 3:17

Answer (B)

Time Speed Distance using Ratios:

Tejkulkarni · Answer

Shouldn't we consider effect of speed of river on motorboat?

KarishmaB · Answer

In this case the time taken to go downstream is the same as the time taken to go upstream (since distance upstream is less than what the raft has covered with the river) so the average speed obtained over this distance is simply the speed of the boat in still water.

egmat · Answer

Hi Tejkulkarni,

Great instinct to ask this. The honest answer is that the current's effect on the motorboat  is  in Karishma's solution - it just  cancels out , which is exactly why her shortcut looks like it ignores it.

Here's the piece that makes it cancel. The boat meets the raft, meaning it comes back to where the  floating water  has carried the raft. Relative to the water, the boat moves at its still-water speed  b  both ways, so it spends  equal time  going downstream and coming back upstream. Call that time  t  each leg.

Now look at the actual ground distances with the current included:

- Downstream: speed is  (b + r) , distance =  (b + r)·t 
- Upstream: speed is  (b − r) , distance =  (b − r)·t 
- Total =  (b + r)t + (b − r)t = 2bt

The  +r  and  -r  wipe each other out. So the boat's total path,  68/3 , depends only on  b , not on  r . Meanwhile the raft just drifts at the current's speed the whole time: distance =  r·T = 12/3 .

Since both the  68/3  and the  12/3  happen over the  same total time T , their ratio is the ratio of the relevant speeds:

river : boat =  12 : 68 = 3 : 17 .

So the river definitely speeds the boat up one way and drags it the other - but because the two legs take equal time, those effects neutralize over the round trip, leaving the total distance proportional to  b  alone.

Feel the cancellation with small numbers.  Let still-water speed  b = 10  and equal time  t = 1  each leg:

- If  r = 2 : down =  12 , up =  8 , total =  20 = 2·10 
- If  r = 5 : down =  15 , up =  5 , total =  20  again

The current changes each leg, but the round-trip total never feels it. That's the whole trick.

Answer: B

GurpreetMahli · Answer

R B  = Rate Boat R V  = Rate River

Find R V /R B

The time for the boat moving away from the raft and coming back to raft will be equal in both legs of the journey. T=Time ; Time * Rate = Distance

T*(R B +R V )=40/3 >> (R B +R V ) = (40/3)/T

T*(R B -R V )=28/3 >> (R B -R V ) = (28/3)/T

So then (R B  + R V ) + (R B  - R V ) = 2R B  =   = 68/3T

And (R B  + R V ) - (R B  - R V ) = 2R V  =   = 12/3T

Finally 2R V /2R B  = R V /R B  =   = 3/17

In a river, a motorboat and raft start their journeys in the same

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In a river, a motorboat and raft start their journeys in the same

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

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Perfect 805 Score Debrief by Julia

My Rewards