A man can row 50 km upstream and 72 km downstream in 9 hours. He can

Question

A man can row 50 km upstream and 72 km downstream in 9 hours. He can also row 70 km upstream and 90 km downstream in 12 hours. Find the rate of current.

A. 2 kmph
B. 3 kmph
C. 4 kmph
D. 6 kmph
E. 8 kmph

KarishmaB · Accepted Answer

I got the answer in less than 30 secs without writing down a word. Since you are looking for faster approaches, let me write down my thought:

First thing that came to mind was 50/U + 72/D = 9
where U is the upstream speed, D is the downstream speed and D is obviously greater than U.
Now I thought of splitting 9 hrs into 5 hrs and 4 hrs. Why? There was a 50. 5 hrs was screaming out at me and then the remaining would be 4 hrs and 72 is divisible by 4, I know.
If I say that perhaps the time taken to go upstream was 5 hrs, the upstream speed, U, will be 10 km/hr.
If time taken to go downstream is 4 hrs, the downstream speed will be 72/4 = 18 km/hr
Now I need to check if U is actually 10 and D is actually 18. How?
I put it in the other equation.
70/U + 90/D = 12
70/10 + 90/18 = 12
7 + 5 = 12
These values of U and D satisfy.

So difference of speeds of boat and stream is 10 and sum of the speeds is 18.
This would give us two equations: B - S = 10 and B + S = 18
2B = 28
B = 14
So S = 4

GMAT does not require long calculations. If you find yourself making and solving tedious equations, be rest assured that the numbers will fall into place. The 9 hrs given above will not split into 5.2 and 3.8 hrs. It will split into 5 and 4 or 6 and 3 etc. You can pretty much guess the correct answer in the first go by looking at the numerator. 50 is not divisible by 6 but it is divisible by 5. And mind you, all this can be easily done orally. Try to minimize your use of pen and paper; your speed will automatically increase.

sujoykrdatta · Answer

Question)  
A man can row 50 km upstream and 72 km downstream in 9 hours. He can also row 70 km upstream and 90 km downstream in 12 hours. Find the rate of current.

Solution: 
Let the speed of the man and current be  m  and  r  km/hr respectively
Thus, we have: 
Scenario 1: 50/(m-r) + 72/(m+r) = 9 ... (i)
Scenario 1: 70/(m-r) + 90/(m+r) = 12 ... (ii)

Assuming (m-r) = u and (m+r) = v and solving for u and v: 
50/u + 72/v = 9
70/u + 90/v = 12

=> u = 10 and v = 18
=> m - r = 10 and m + r = 18

Solving: r = 4

Alternate method:

We know that:

70 km upstream (U) and 90 km downstream (D) takes 12 hours ... (i)

Also: 50 km upstream and 72 km downstream takes 9 hours 
=> If we increase time by one-third its value, it would become 12 hours (as in first scenario); in that case, we need to increase the distances also by 1/3rd of the respective values
=> 200/3 km upstream (U) and 96 km downstream (D) takes 12 hours ... (ii)

Thus, equating the times from (i) and (ii): 
200/3 (U) + 96 (D) = 70 (U) + 90 (D)
=> 10/3 (U) = 6 (D)
=> 10 (U) = 18 (D)
=> Time taken to cover 10 km upstream is the same as that taken to cover 18 km downstream

Let the speed of the man and current be  m  and  r  km/hr respectively

=> 10/(m-r) = 18/(m+r) 
=> m : r = 7 : 2
=> m = 7k and r = 2k

Thus, from (i): 
50/5k + 72/9k = 9
=> k = 2

=> Speed of the current = 2k = 4 km/hr

Answer C

AabhishekGrover · Answer

How to solve for v and u?

effatara · Answer

AabhishekGrover

This is how you solve for 'u' and 'v' in  sujoykrdatta  solution:
50/u+72/v=9....> (50v+72u)/(uv)=9......(i)
70/u+90/v=12...> (70v+90u)/(uv)=12......(ii)
Dividing (i) by (ii), we get (50v+72u)/(70v+90u)=3/4...>18u=10v...> u=(5/9)v.

Substituting in (i), we get: (50v+40v)/uv=9...> 90v/uv=9...> u=10. BY substituting in either (i) or (ii), we get v=18 from which we can calculate the value of r=4

sujoykrdatta · Answer

Alternative approach: 
Take 1/u = x and 1/v = y
50x + 72y = 9
70x + 90y = 12

Now solve ..

Posted from my mobile device

Geet91 · Answer

Hi can you please explain this part further:

So difference of speeds of boat and stream is 10 and sum of the speeds is 18.
This would give us two equations: B - S = 10 and B + S = 18
2B = 28
B = 14
So S = 4

Kinshook · Answer

Given: A man can row 50 km upstream and 72 km downstream in 9 hours. He can also row 70 km upstream and 90 km downstream in 12 hours.

Asked: Find the rate of current.

Let the rate of rowing be r and rate of current be c.

A man can row 50 km upstream and 72 km downstream in 9 hours.
50/(r-c) + 72/(r+c) = 9 (1)

He can also row 70 km upstream and 90 km downstream in 12 hours. 
70/(r-c) + 90/(r+c) = 12 
7/(r-c) + 9/(r+c) = 1.2 (2)
(2) * 8
56/(r-c) + 72/(r+c) = 9.6 (3)

(3) - (1)
6/(r-c) = .6
r-c = 10
r = c + 10 (4)

Putting value of r in (2)
.7 + 9/(2c+10) = 1.2
9/(2c+10) = .5
18 = 2c + 10
c = 4 kmh

IMO C

pappal · Answer

let boat speed B and stream speed be S
50/(B-S) + 72/(B+S)=9 ----i
70/(B-S) + 90/(B+S)=12 ----ii
since given values of current or stream speed are all integers so most probably should boat speed also be an integer to satisfy above results
also B-S should divide both 50 and 70 ={2,5,10}; 
can absolutely at first instance discard 2 & 5 as result is higher than 9 (first equation). so, (B-S)=10----iii
hence (B+S) can take only 18 as its value----iv
solving iii and iv; S=4
B

A man can row 50 km upstream and 72 km downstream in 9 hours. He can

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A man can row 50 km upstream and 72 km downstream in 9 hours. He can

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