Two towns sealand and beachland are on the banks of a river . Amit takes 13 hours to row and back . Sachin , who rows at twice Amit's speed, covers same distance in 6 hours. Find the ratio of Amit's speed to that of flow of river.

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Two towns sealand and beachland are on the banks of a river . Amit takes 13 hours to row and back . Sachin , who rows at twice Amit's speed, covers same distance in 6 hours. Find the ratio of Amit's speed to that of flow of river.

Mathematics
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  • phi
I don't exactly understand Irish's hint. But I would his basic idea, but with distances. Let's say Amit rows at speed v for 13 hours. using rate * time = distance he rowed a distance of 13v (I guess we don't care about the units) I assume he is trying to go straight across to the other side, but the current would carry him downstream, so he rows at an angle, as shown in the picture.

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  • phi
in 13 hours, the current moves him a distance of 13u (u is how fast the current goes) and we will call the distance straight across the river "w" now we can use pythagoras |dw:1443011236342:dw|
  • phi
Sachin, rows for 6 hours, but at a rate of 2v, using rate*time = distance 2v * 6 = 12 v he rows a distance of 12 v the current carries him a distance of 6u (6 hours at a speed of u) |dw:1443011437401:dw|
  • phi
we have two equations \[169 v^2 = 169u^2 + w^2\\ 144 v^2 = 36u^2 + w^2 \] we can subtract the two equations (this will get rid of the w^2) we get \[ 25v^2 = 133 u^2 \]
  • phi
Amit's speed is v and the river's speed is u we have \[ \frac{v^2}{u^2}= \frac{133}{25} \] take the square root of both sides \[ \frac{v}{u}= \frac{\sqrt{133}}{5} \] is the ratio we want
straight from the drawing \[(2w)^2 = (13v)^2 - (13u)^2 = (6\times 2v)^2 - 6u^2\] \[25v^2 = 133u^2\] one more line and you're sone

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