A boat covers a distance of 30 km in `2frac(1)(2)`h running downstream. While returning, it covers the same distance in `3frac(3)(4)` h. What is the speed of the boat in still water?

Options

A. 8 km/h
B. 12 km/h
C. 14 km/h
D. 10 km/h

Answer

To find the speed of the boat in still water, we will first calculate the downstream and upstream speeds based on the given distances and times. Then, we will use these speeds to find the speed of the boat in still water.

### Step 1: Convert Mixed Numbers to Improper Fractions
The time taken downstream is given as \(2 \frac{1}{2}\) hours, which can be converted to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \text{ hours} \]
The time taken upstream is given as \(3 \frac{3}{4}\) hours, which can be converted to an improper fraction:
\[ 3 \frac{3}{4} = \frac{15}{4} \text{ hours} \]

### Step 2: Calculate Downstream Speed
The downstream speed can be calculated using the formula:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
For downstream:
\[ \text{Distance} = 30 \text{ km}, \quad \text{Time} = \frac{5}{2} \text{ hours} \]
Thus, the downstream speed is:
\[ \text{Downstream Speed} = \frac{30}{\frac{5}{2}} = 30 \times \frac{2}{5} = 12 \text{ km/h} \]

### Step 3: Calculate Upstream Speed
Using the same formula for upstream:
\[ \text{Distance} = 30 \text{ km}, \quad \text{Time} = \frac{15}{4} \text{ hours} \]
Thus, the upstream speed is:
\[ \text{Upstream Speed} = \frac{30}{\frac{15}{4}} = 30 \times \frac{4}{15} = 8 \text{ km/h} \]

### Step 4: Calculate Speed of the Boat in Still Water
The speed of the boat in still water can be calculated using the formula:
\[ \text{Speed of Boat in Still Water} = \frac{\text{Downstream Speed} + \text{Upstream Speed}}{2} \]
Substituting the values we found:
\[ \text{Speed of Boat in Still Water} = \frac{12 + 8}{2} = \frac{20}{2} = 10 \text{ km/h} \]

### Final Answer
The speed of the boat in still water is **10 km/h**.

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