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

Let

Speed of the boat in still water be xkm/hr

Speed Â of the stream be ykm/hr

Speed of boat in downstreamÂ =Â (x+y)km/hr

Speed of boat in upstreamÂ = (x-y)km/hr

According to given problem

Time taken to cover 12km upstreamÂ =12/(xâˆ’y)hrs

Time taken to cover 40km downstreamÂ =40/(x+y)hrs

But, the total time takenÂ =8hr

Â =12/(xâˆ’y)+40/(x+y)=8………(1)

Time taken to cover 16km upstreamÂ =16/(xâˆ’y)â€‹Â hrs

Time taken to cover 32km downstreamÂ =32/(x+y)â€‹hrs

Total time takenÂ =Â 8hr

Â =16/(xâˆ’y)+32/(x+y)â€‹=8…….(2)

Â Put 1/(xâˆ’y)=p and 1/(x+y)â€‹=q

hence we get equation

12p + 40qÂ =Â 8….(3)

16p + 32qÂ =Â 8….(4)

Furthur simplyfying the eq we get

3p + 10qÂ =Â 2……….(3)

2p + 4qÂ =Â 1………(4)

Multiply eq (3) by 2 and eq (4) by 3

6p + 20qÂ =Â 4………..(3)

6p + 12qÂ =Â 3…………(4)

subtracting eq (4) from eq(3) we get

q=1/8â€‹

and we Â getÂ p=1/4â€‹

Hencep=1/(xâˆ’y)â€‹=1/4 â€‹and q=1/(x+y)â€‹=1/8â€‹

x-yÂ =Â 4..(5)

x+y=Â 8….(6)

Solving equation(5) and (6) we get xÂ =Â 6 and y=2

Hence speed of boat in still water =6km/hr and speed of stream 2km/hr.

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