Question

A motor boat can travel 30 km upstream and 28 km downstream in 7 hours. It can travel 21 km upstream and return in 5 hours. Find the speed of the boat in still water and the speed of the stream.

Answer 1

Let the speed of the motorboat in still water and the speed of the stream are u km/h and v km/h, respectively

Then, downstream speed of motorboat = (u + v) km/h

And upstream speed of motorboat = (u – v) km/h

Time taken to travel 30 km upstream,

t₁ = `30/(u - v)` hours  ...`[because  "Speed" = "Distance"/"Time"]`

And time taken to travel 28 km downstream,

t₂ = `28/(u + v)` hours

By first condition,

t₁ + t₂ = 7 hours

⇒ `30/(u - v) + 2/(u + v)` = 7

Now, time taken to travel 21 km upstream,

t₃ = `21/(u - v)` hours

And time taken to travel 21 km downstream,

t₄ = `21/(u + v)` hours

By second condition,

t₄ + t₃ = 5 hours

⇒ `21/(u + v) + 21/(u - v)` = 5  ....(ii)

Let x = `1/(u + v)` and y = `1/(u - v)`

Equation (i) and equation (ii) becomes,

30x + 28y = 7 ......(iii)

and 21x + 21y = 5

⇒ x + y = `5/21`   .....(iv)

Now, multiplying in equation (iv) by 28 and then subtracting from equation (iii), we get

(30x – 28y) – (28x + 28y) = `7 - 140/21`

⇒ 2x = `7 - 20/3`

⇒ 2x = `1/3`

⇒ x = `1/6`

On putting the value of x in equation (iv), we get

`1/6 + y = 5/21`

⇒ y = `5/21 - 1/6`

= `(10 - 7)/42`

= `3/42`

⇒ y = `1/14`

Now, x = `1/(u + v) = 1/6`

⇒ u + v = 6  ....(v)

And y = `1/(u - v) = 1/14`

⇒ u – v = 14  ....(vi)

Now, adding equation (v) and equation (vi), we get

2u = 20

⇒ u = 10

On putting the value of u in equation (v), we get

10 + v = 6

⇒ v = – 4

Hence, the speed of the motorboat in still water is 10 km/h and the speed of the stream 4 km/h.