A man goes downstream with a boat to some destination and returns upstream to his original place in 5 hours. If the speed of the boat in still water and the stream are 10 km/hr and 4 km/ hr respectively, the distance of the destination from the starting place is

Options

A. 16 km
B. 18 km
C. 21 km
D. 25 km

Answer

To solve the problem step by step, we will use the information provided about the speeds of the boat and the stream, as well as the total time taken for the journey.

### Step 1: Define the Variables
Let the distance to the destination be \( x \) km.

### Step 2: Determine the Speeds
- The speed of the boat in still water is given as \( 10 \) km/hr.
- The speed of the stream is given as \( 4 \) km/hr.

**Downstream Speed:**
When going downstream, the speed of the boat increases due to the current of the stream:
\[ \text{Downstream Speed} = \text{Speed of Boat} + \text{Speed of Stream} = 10 + 4 = 14 \text{ km/hr} \]

**Upstream Speed:**
When going upstream, the speed of the boat decreases due to the current of the stream:
\[ \text{Upstream Speed} = \text{Speed of Boat} - \text{Speed of Stream} = 10 - 4 = 6 \text{ km/hr} \]

### Step 3: Set Up the Time Equation
The total time taken for the journey downstream and upstream is given as \( 5 \) hours. We can express the time taken for each leg of the journey using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

**Time Downstream:**
\[ \text{Time Downstream} = \frac{x}{14} \]

**Time Upstream:**
\[ \text{Time Upstream} = \frac{x}{6} \]

### Step 4: Write the Total Time Equation
The total time for the journey is the sum of the time taken downstream and upstream:
\[ \frac{x}{14} + \frac{x}{6} = 5 \]

### Step 5: Solve the Equation
To solve the equation, we need a common denominator. The least common multiple of \( 14 \) and \( 6 \) is \( 42 \).

Rewriting the equation:
\[ \frac{3x}{42} + \frac{7x}{42} = 5 \]

Combining the fractions:
\[ \frac{10x}{42} = 5 \]

Now, multiply both sides by \( 42 \):
\[ 10x = 5 \times 42 \]
\[ 10x = 210 \]

Now, divide both sides by \( 10 \):
\[ x = \frac{210}{10} = 21 \]

### Step 6: Conclusion
The distance of the destination from the starting place is \( 21 \) km.