A motor boat can travel at 15km/hr in still water. It traveled 63 km downstream in a river and then returned taking altogether 10 hr. What is the rate of flow of the river?

6 km/hr

The correct answer is Option (2) → 6 km/hr

Step 1: Define the variables

• Speed of the boat in still water ($u$) = 15 km/hr
• Distance traveled ($d$) = 63 km
• Total time taken ($T$) = 10 hours
• Let the rate of flow of the river be $v$ km/hr.

Step 2: Determine Relative Speeds

• Downstream Speed: When the boat moves with the flow, its speeds add up: $(15 + v)$ km/hr.
• Upstream Speed: When the boat moves against the flow, its speed is reduced: $(15 - v)$ km/hr.

Step 3: Set up the Equation

The total time is the sum of the time taken to go downstream and the time taken to return upstream.

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

$\frac{63}{15 + v} + \frac{63}{15 - v} = 10$

Step 4: Solve for $v$

Factor out 63 and find a common denominator:

$63 \left[ \frac{(15 - v) + (15 + v)}{(15 + v)(15 - v)} \right] = 10$

$63 \left[ \frac{30}{225 - v^2} \right] = 10$

$63 \times 3 = 225 - v^2$

(After dividing both sides by 10)

$189 = 225 - v^2$

$v^2 = 225 - 189$

$v^2 = 36$

$v = 6 \text{ km/hr}$

Answer: The rate of flow of the river is 6 km/hr.