Two boats on the opposite shores of a river start moving tow

Question

Two boats on the opposite shores of a river start moving towards each other. When they pass each other they are 750 yards from one shoreline. They each continue to the opposite shore, immediately turn around and start back. When they meet again they are 250 yards from the other shoreline. Each boat maintains a constant speed throughout. How wide is the river?

(A) 2400 yards
(B) 3000 yards
(C) 2000 yards
(D) 4000 yards
(E) None of these

(A) 2400 yards 
(B) 3000 yards 
(C) 2000 yards 
(D) 4000 yards 
(E) None of these

chetan2u · Accepted Answer

Untitled111.png Let the boats be B1 and B2 starting form two ends A and B respectively. The movement of B1 is shown by blue lines, while that of B2 is shown by brown lines. Boats meet two times, first at C and then at D, as shown n the sketch. 1) At C B1 has traveled x and B2 has traveled 750. As the time both took is the same, the ratio of the speed of B1:B2=x:750 2) At D B1 has traveled (750)+(x+500) or x+1250, and B2 has traveled x+250. Similarly here ratio of speed of B1:B2=(x+1250):(x+250) TWO APPROACHES from here on (I) LOGICAL but simpler First time they meet, B2 has traveled 750 in a combined total of one width of river. Next time they meet, they have covered three times the width of river within themselves, so B2 will cover 750 for each width and hence 750*3=2250 yards when they meet at D. But the distance B2 travels is also equal to the width + 250 yards. Hence width+250=2250 or width=2250-250=2000 (II) Taking RATIO of speeds Thus x:750=x+1250:x+250 \frac{x}{750}=\frac{x+1250}{x+250} x^2+250x=750x+750*1250 x^2-500x-750*1250=0 x^2-1250x+750x-750*1250=(x-1250)(x+750)=0 So x is 1250 or -750, but x>0, so x=1250. Width of river = x+750=1250+750=2000

TestPrepUnlimited · Answer

There are two variables here we need to watch out for: 
(1) Ratio of their speeds
(2) Width of the river.

The ratio of their speed determines at which point they meet, so their individual speeds are not as important.

Another hint, all the choices are above 2000 so we know 750 marks the distance of the slower boat if we want to find an answer within the choices.

Set the width of the river as W and the ratio of their speeds as r, where r < 1.

The first time they meet:  \frac{r}{1 + r} = \frac{750}{W - 750}

Second time they meet:  \frac{r}{1 + r} = \frac{W + 250}{W + (W - 250)}

\frac{750}{W - 750} = \frac{W + 250}{W + (W - 250)}

750*(2W - 250) = W^2 - 500W - 250*750

W^2 - 2000W= 0

W = 2000

Ans: C

Kinshook · Answer

Given: Two boats on the opposite shores of a river start moving towards each other. When they pass each other they are 750 yards from one shoreline. They each continue to the opposite shore, immediately turn around and start back. When they meet again they are 250 yards from the other shoreline. Each boat maintains a constant speed throughout.

Asked: How wide is the river?

Let the width of the river be w and speeds of the 2 boats be s1 & s2.

When they pass each other they are 750 yards from one shoreline. 
time t = 750/s1
Distance travelled by another boat in time t = 750s2/s1
w = 750 + 750s2/s1 = 750 (1 + s2/s1)

They each continue to the opposite shore, immediately turn around and start back. When they meet again they are 250 yards from the other shoreline.
Distance travelled by boat 1 = w - 750 + 250 = w - 500 = 750 (1 + s2/s1) - 500 = 250 + 750s2/s1
Distance travelled by boat 2 = 750 + (w -250) = w + 500 = 1250 + 750s2/s1
Distances travelled by them are in the ratio of their speeds 
s1/s2 = (250 + 750s2/s1)/(1250 + 750s2/s1)
Let s2/s1 be x
1/x = (250 + 750x)/(1250 + 750x)
1250 + 750x = (250 + 750x)x
5 + 3x = x (1 + 3x)
3xˆ2 - 2x -5 = 0
3xˆ2 - 5x + 3x - 5 = 0
(3x -5)(x+1) = 0
x = s2/s1 = 5/3

w = 750 (1 + x) = 750 (1+5/3) = 750 * 8/3 = 2000 yards

IMO C

saikatdey111 · Answer

Solution: 
Let the width of the river be "d" yards.
1st part:
1st boat crossed 750 yards and 2nd boat crossed (d-750) yards.
2nd part:
1st boat crossed (d+250) yards and 2nd boat crossed (d+(d-250))=2d-250 yards.
A/Q,
 750/(d-750) = (d+250)/(2d-250)
Or, d(d-2000)=0
Or, d=2000 yards (ans)

Posted from my mobile device

ThatDudeKnows · Answer

We can set up the ratios: 
 \frac{750}{w+250} = \frac{w-750}{2w-250}

Rather than solving through, we can Plug In The Answers. Let's assume they're in GMAC order of 2000, 2400, 3000, 4000. I'd want to try one of the two middle values. 3000 looks easier to work with than 2400. Let's try 3000.

\frac{750}{3250} = \frac{2250}{5750} ? Nope, the first fraction is much smaller.

Do we need something larger or smaller? I don't know, but we get more information by trying smaller, so let's do that. 2000 looks easier to work with than 2400. Let's try 2000.

\frac{750}{2250} = \frac{1250}{3750} ? Yep, both are 1/3.

2000 is correct.

Answer choice C.

Lipun · Answer

Let the width of the river be W yards. Let B1 cover 750 yards & B2 cover 'x' yards when they meet for the first time.

Given, W = 750 + x.

Next time both the boats meet, the total distance covered by them will be 2W, since they will move away from each other at first and then turn around to meet again.

So B1 covers 2*750 yards = 1500 yards, which is equal to 'x+250' yards.

This implies, x = 1250 yards and W = x +750 = 2000 yards.

Abhinandan Dey · Answer

Solution:

3*750 = D + 250

D = 2000 yards. (Ans)

Explanation:

Let, width of the river is D and distance covered by Boat A and B are a and b respectively.

|--------D--------|

Then, SUM of the distance covered by A and B will be width of the river when they first meet.
 A B
|-------><-------|

So,
D = a + b ----------------------------------------- (1)

Given, when they pass each other they are 750 yards from one shoreline, it means one of the boat has covered 750 yards. Let's say that boat is A.

a = 750 yards ------------------------------------------------------ (2)

By the second passing, each boat has covered the width of the river, and turned around. It means both of the boats have individually travelled the width of the river D.

When they MEET AGAIN then the boats have covered the width of the river ONCE MORE, so the sum of the distances they've traveled is 3 times the width of the river.

Since they travel at a constant rate, and together they've gone three times as far as when they first passed, it follows that one of them has traveled a distance of 3a and the other has traveled 3b.

When the boats passed a second time, 250 yards from the "OTHER" shoreline, it means that the same boat that had traveled 750 yards by their first passing has traveled D+250 yards by the second passing  .

So,
3a = D + 250 ------------------------------------ (3)

Now, Putting the value of equation (2) in equation (1).

3*750 = D + 250

D = 2000 yards. (Ans)

Vibhatu · Answer

THIS IS VERY EASY!!!! 
Just use the formula of distance= 3x-y
x=730
y=250
Distance=2000
 Ans:C

jjannyyys · Answer

Assumed that total wide of river = Z

If two boats meet each other first time when 10 min passed 
- First boat move 750 (means speed of first boat = 75)
- Second boat move Z-750 (means speed of second boat = Z-750/10)

Then, two boats meet each other second time when x time
- First boat move Z+250 (mean speed of first 75 = z+250/x)
- Second boat move Z+(Z-250) (means speed of second boat Z-750/10 = 2Z-250/x)

75x = Z+250
X = Z+250/75 - (1)

Z-750/10 = 2Z-250/(Z+250/75)
Z-750(Z+250/75) = (2Z-250)10
Z^2 - 750Z + 250Z - 187500 = 1500Z - 187500
Z^2 -2000Z = 0
Z(Z-2000) = 0
Z = 2000

shihab7944 · Answer

How did you do this? Could you please explain this?

manish8242 · Answer

I got two equation first is Va/Vb = x/750 same as you but the second equ is diff my equ is x+500/250 = va/vb. I dont know how you deduce the second equ i used the same diagram as yours. Have you taken care that after meeting they reached their target and then they turned back so their direc will be opposite to what was earlier.

paragw · Answer

Your 1st equation is absolutely correct but in second equation, I know where it goes wrong. On writing second equation as x+500/250 = va/vb, you assumed that after crossing each other, both B1, who starts from A & B2, who starts from B, reached their respective destination B & A at the same time, Therefore on this assumption, we wrote our ratio as x+500/250, assuming that they both now started at the same time from opposite side after reaching their destination and met at a point which is 250 m away from A.

Instead of this, use the ratio of total distance in which they covered the distance from the beginning till they meet for the second time i.e. (750+x+250)/(750+x+500+x), where (750+x) is the total width of the river.

I'll also attach my solution in few minutes. I hope you understand this and if not, go through my solution for once.

paragw · Answer

https://gmatclub.com/forum/download/file.php?id=86168 For first meeting point, Vx/Vy = 750/d For second meeting point, Vx/Vy = 1000+d/1250+2d here we can't take the ratio as (250/500+d) coz we don't know if they reached at their destination point at the same time or not, so we've to take whole distance that they covered from the beginning till the second meeting point in our ratio Solving further, 750 / d = 1000+d / 1250+2d you'll get d = 1250 hence, total width of the river will be 750+d = 750+1250 = 2000 GMAT-Club-Forum-umd6t4rr.png

Two boats on the opposite shores of a river start moving tow

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