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# A boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time. Find the speed of the boat in still water and the speed of the stream.

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One of the most important and conceptual question from linear equations in two variables in which we have to find the speed of the stream and the speed of the boat if the boat goes 12 km upstream and 40 km downstream in 8 hours. It can go 16 km upstream and 32 km downstream in the same time.

Kindly solve the above problem

RS Aggarwal, class 10, chapter 3E, question no 36

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1. Let
Speed of the boat in still water be xkm/hr
Speed  of the stream be ykm/hr
Speed of boat in downstream = (x+y)km/hr
Speed of boat in upstream = (x-y)km/hr
According to given problem
Time taken to cover 12km upstream =12/(xy)hrs
Time taken to cover 40km downstream =40/(x+y)hrs
But, the total time taken =8hr
=12/(xy)+40/(x+y)=8………(1)
Time taken to cover 16km upstream =16/(xy) hrs
Time taken to cover 32km downstream =32/(x+y)hrs
Total time taken = 8hr
=16/(xy)+32/(x+y)=8…….(2)
Put 1/(xy)=p and 1/(x+y)=q
hence we get equation
12p + 40q = 8….(3)
16p + 32q = 8….(4)
Furthur simplyfying the eq we get
3p + 10q = 2……….(3)
2p + 4q = 1………(4)
Multiply eq (3) by 2 and eq (4) by 3
6p + 20q = 4………..(3)
6p + 12q = 3…………(4)
subtracting eq (4) from eq(3) we get
q=1/8
and we  get p=1/4
Hencep=1/(xy)=1/4 and q=1/(x+y)=1/8
x-y = 4..(5)
x+y= 8….(6)
Solving equation(5) and (6) we get x = 6 and y=2
Hence speed of boat in still water =6km/hr and speed of stream 2km/hr.

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