A boat covers 24 km upstream and 36 km downstream in 10 hours, and 36 km upstream and 24 km downstream in 12 hours. The speed of the current is:

Correct option is D

Given:

Case 1: 24 km upstream and 36 km downstream in 10 hours

Case 2: 36 km upstream and 24 km downstream in 12 hours

Formula Used:

Time = $$\frac{\text{Distance}}{\text{Speed}}$$​

Solution:

Let speed of boat in still water = x km/h

Let speed of current = y km/h

Upstream speed = x - y, Downstream speed = x + y
From condition 1:

​$$\frac{24}{x - y} + \frac{36}{x + y} = 10 \quad \tag{1}$$​​

From condition 2:

​$$\frac{36}{x - y} + \frac{24}{x + y} = 12 \quad \tag{2}$$​​

Multiply equation (1) by 3 and equation (2) by 2

and Subtracting:

​$$\left(\frac{72}{x - y} + \frac{108}{x + y}\right) - \left(\frac{72}{x - y} + \frac{48}{x + y}\right) = 6$$

$$\frac{60}{x + y} = 6$$​​​

x + y =$$\frac{60}{6} = 10 \quad\tag{5}$$​

Substitute in (1):

​$$\frac{24}{x - y} + \frac{36}{10} = 10 \\ \ \\\frac{24}{x - y} = 10 - \frac{36}{10} = \frac{64}{10} = \frac{32}{5} \\ \ \\x - y = \frac{24 \times 5}{32} = \frac{120}{32} = \frac{15}{4} \quad \tag{6}$$​​

Add equation 5 & 6 , we have-

​$$2x = \frac{55}{4}$$​​

x = $$\frac{55}{8}$$km/hr

Substitute back:

y = $$10 - \frac{55}{8} = \frac{80 - 55}{8} = \frac{25}{8}$$km/hr