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August 27th, 2018, 07:25 AM  #1 
Senior Member Joined: Aug 2014 From: India Posts: 310 Thanks: 1  How to solve this problem in simple way?
A boat goes 24 km upstream and 36 km upstream in 8 hours and 30 mins while if it goes 36 km upstream and 24 km downstream, the boat takes 9 hours. What is the speed of the boat in still water? Let c be the current and s be the speed of the boat. t1, t2, t3, and t4 be the times. (cs)t1 = 24 [takes t1 hours to go upstream 24 km] (c+s)t2 = 36 [takes t2 hours to go downstream 36 km] t1 + t2 = 8.5 [total time is 8.5 hours] (cs)t3 = 36 (c+s)t4 = 24 t3 + t4 = 9 T1 + T2 = 8.5 $\frac {24}{C  S}$ + $\frac {36}{C + S}$ = 8.5 (24)(C + S) + 36(C  S) = (8.5)($C^2$  $S^2$) 60 C  12 S = 8.5 ($C^2$  $S^2$) It is getting complicated. Do you have any shortcuts to proceed this problem? Last edited by skipjack; August 27th, 2018 at 10:50 AM. 
August 27th, 2018, 07:55 AM  #2 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,129 Thanks: 914  
August 27th, 2018, 08:45 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,542 Thanks: 1750  Accepting the amendment Denis suggested, it's straightforward. Let p = 1/(c  s) and q = 1/(c + s). 24p + 36q = 8.5 and (similarly) 36p + 24q = 9, so 48p + 72q = 17 and 108p + 72q = 27. By subtraction, 60p = 10 and so 72q = 9. Hence c  s = 1/p = 6 and c + s = 1/q = 8. Solving is now easy and gives c = 7 and s = 1. I omitted the units in the above, so the answer is 1 km/h. 
August 27th, 2018, 09:23 AM  #4 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,129 Thanks: 914 
If you're not too busy or tired, you can always make one up to see how it works, like making boat speed = 25 and current speed = 5. then creating this scenario (as example): A boat goes 120 km downstream and 60 km upstream in 7 hours. Same boat goes 60 km downstream and 120 km upstream in 8 hours. What is the speed of the boat in still water? v = boat speed, c = current speed @v+c................120..................>@vc........60...........>7 hr @v+c.......60........>@vc..................120....................>8 hr Now solve that for v: you should get v=25. 
August 27th, 2018, 07:38 PM  #5 
Senior Member Joined: Aug 2014 From: India Posts: 310 Thanks: 1  
August 28th, 2018, 03:52 AM  #6 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,129 Thanks: 914 
Represents a path, like river or road... @v+c................120..................>@vc........60...........>7 hr Here, represents the river, 180 total length, travelled from left to right, as indicated with ">". 

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