Question

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Answer 1

Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr

Speed upstream = ( x - y ) km/hr
Speed downstream = (x + y) km/hr

Now,

Time is taken to cover 30 km upstream = `(30)/("x"-"y") "hrs"`

Time taken to cover 44 km down stream = `(44)/("x"+"y")"hrs"`

But total time of journey is 10 hours
`(30)/("x"-"y") + (44)/("x"+"y") = 10`   ...(i)

Time taken to cover 40 km upstream= `(40)/("x"-"y")"hrs"`

Time taken to cover 55 km down stream = `(55)/("x"+"y")"hrs"`

In this case total time of journey is given to be 13 hours

Therefore,`(40)/("x"-"y")+(55)/("x"+"y") = 13`  ...(ii)

Putting `(1)/("x"-"y") = "u" and (1)/("x"+"y") = "v"` in equation (i) and (ii) we get

30u + 44v - 10 = 0  ...(iii)
40u + 55v - 13 = 0 ...(iv)

Solving these equations by cross multiplication we get

`("u")/(44xx-13-15 xx-10) = ("-v")/(30 xx-13-14xx -10) = (1)/(30 xx55-40xx44)`

`("u")/(-572+550) = ("-v")/(-390+400) = (1)/(1650-1760)`

`("u")/(-22) = ("-v")/(10) = (1)/(-110)`

`"u" = (-22)/(-110)`

`"v" = (-110)/(-110)`

`"u" = (2)/(10) and "v" =(1)/(11)`

Now,

`"u" = (2)/(10)`

`(1)/("x"-"y") = (2)/(10)`

1 x 10 = 2 (x - y)
10 = 2x - 2y ÷ 2

`"u"= (2)/(10)`

`(1)/("x"-"y") = (2)/(10)`

1 x 10 = 2 (x - y)
10 = 2x - 2y 

5 = x - y ...(v)

`"v" =(1)/(11)`

`(1)/("x"+"y") = (1)/(11)`

1 x 11 = 1 (x + y)

By solving equation (v) and (vi) we get,

x - y = 5

`("x"+"y" = 11)/(2"x" =16)`

`"x" = (16)/(2)`

x = 8

Substituting x = 8 in equation (vi) we get,

x + y = 11
8 + y = 11
y = 11 - 8
y = 3

Hence, speed of the boat in still water is 8km/hr
Speed of the stream is  3km /hr