Question

A boat goes 24 km upstream and 28 km downstream in 6 hours. It goes 30 km upstream and 21 km downstream in 6.5 hours. Find the speed of the boat in still water and also the speed of the stream

Answer 1

$\huge{\underline{\purple{\rm{Solution:}}}}$

$\large{\underline{\red{\rm{Let:}}}}$

• $\rm{Speed\:of\:boat=x\:km/h}$
• $\rm{Speed\:of\:stream=y\:km/h}$
• $\rm{Speed\:of\: upstream=x-y}$
• $\rm{Speed\:of\:downstream=x+y}$

$\small{\underline{\red{\rm{To\: Find:}}}}$

• $\rm{The\:speed\:of\: boat\:in\: still\:in\: water}$
• $\rm{The\:speed\:of\: stream}$

$\small{\underline{\green{\rm{As\:per\:the\: above\: information:}}}}$

$\small{\underline{\red{\rm{Given\:in\:1st\:case:}}}}$

• $\sf{A\:boat\:goes\:24\:km\: upstream\:and\:28\:km\: downstream\:in\:6\: hours}$

$\rm{\implies Time=\dfrac{Distance}{speed}}$

$\rm{\implies \dfrac{28}{x+y}+\dfrac{24}{x-y}=6}$

$\small{\underline{\red{\rm{Given\:in\:2nd\:case:}}}}$

• $\sf{It\:goes\:30\:km\: upstream\:and\:21\: downstream\:in\:6.5\: hours}$

$\rm{\implies \dfrac{21}{x+y}+\dfrac{30}{x-y}=6.5}$

• $\sf{Let,\:x+y=u\:and\:x-y=v}$

$\rm{\implies \dfrac{28}{u}+\dfrac{24}{v}=6}$

$\rm\orange{\implies 28v+24u=6uv------(i)}$

$\rm{\implies \dfrac{21}{u}+\dfrac{30}{v}=6.5}$

$\rm\orange{\implies 21v+30u=6.5uv-----(ii)}$

• $\sf{In\:eq\:1st\: multiplying\:by\:21\:and\:in\:eq\:2nd\:by\:28}$

$\rm{\implies 588v+504u=126uv}$

$\rm{\implies 588v+840u=182uv}$

• $\sf{by\: solving\: equation\:we\:get}$

$\rm{\implies -336v=-56uv}$

$\rm\red{\implies v=6}$

• $\sf{putting\:the\: value\:v=1/6\:in\:eq\:1}$

$\rm{\implies 28v+24u=6uv}$

$\rm{\implies 28*6+24u=36u}$

$\rm{\implies 28*6=36u-24u}$

$\rm{\implies 28*6=12u}$

$\rm\red{\implies u=14}$

• [Now, solving equation]

$\rm{\implies x+y=14}$

$\rm{\implies x-y=6}$

• [adding equation here]

$\rm{\implies 2x=20}$

$\rm\red{\implies x=10}$

• [putting the value of X=10 in 1]

$\rm{\implies x+y=14}$

$\rm{\implies 10+y=14}$

$\rm{\implies y=14-10}$

$\rm\red{\implies y=4}$

Hence,

The speed of boat=10km/h

The speed of stream=4km/h