​ A boat covers 24 km upstream and 36 km downstream in 6 hours, while it covers 36 km upstream and 24 km downstream in hours. ​

Question

A boat covers 24 km upstream and 36 km downstream in 6 hours, while it covers 36 km upstream and 24 km downstream in \(6\frac12\) hours. The speed of the current is :

(a) 1 km/hr

(b) 2 km/hr

(c) 1.5 km/hr

(d) 2.5 km/hr

Answer 1

Correct option is (b) 2 km/hr

We take the speed of the boat = x km/hr. and that of the water current = y km/hr.

In the first case, their relative speed = x − y upstream & x + y downstream..

UPSTREAM-

The distance = 24km

So the time taken = \(\frac{24}{x -y}\) 

DOWNSTREAM-

The distance = 36km.

So the time taken = \(\frac{36}{x+y}\)

∴ The total time taken = \(\frac{24}{x -y}\) + \(\frac{36}{x+y}\) = 6 (given)

Then the above equation becomes,

24a + 36b = 6

Or 4a + 6b = 1   .........(i).

In the second case, their relative speed = x − y upstream & + y downstream..

UPSTREAM-

The distance = 36km.

So the time taken = \(\frac{36}{x-y}\) 

DOWNSTREAM-

The distance = 24km.

So the time taken = \(\frac{24}{x +y}\)

∴ The total time taken = \(\frac{36}{x-y}\) + \(\frac{24}{x +y}\) = \(6\frac12\) hr

\(= \frac{13}2\) (given)

Now we have assumed that  x − y = a & x + y = b.

Then the above equation becomes

36a + 24b = \( \frac{13}2\)

Or 72a + 48b = 13   .........(ii)

(ii) − (iii) ⟶ 60b = 5

Or b = \(\frac 1{12}\)

Or x + y = 12   .........(iv)

Putting b = \(\frac 1{12}\)​ in (i),

4a + 6 × \(\frac 1{12}\) =  1

Or a = \(\frac 18\)

Or x − y = 8  ........(v)

Subtracting (iv)  from (v),

2y = 4

Or y = 2km/hr.

So, the speed of the current = 2km/hr.