The ratio of speed of a boat in still water to speed of the current is 10 : 1. If ratio of the time taken by the boat to travel D km downstream to the time taken to travel (D – 45) km upstream is 3 : 2. Find the value of D.

Options

A. 88
B. 54
C. 110
D. 99

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define Variables
Let:
- Speed of the boat in still water = \( 10x \) km/h
- Speed of the current = \( x \) km/h

### Step 2: Calculate Effective Speeds
- Downstream speed of the boat = Speed of the boat + Speed of the current = \( 10x + x = 11x \) km/h
- Upstream speed of the boat = Speed of the boat - Speed of the current = \( 10x - x = 9x \) km/h

### Step 3: Write the Time Formulas
- Time taken to travel \( D \) km downstream = \( \frac{D}{11x} \)
- Time taken to travel \( D - 45 \) km upstream = \( \frac{D - 45}{9x} \)

### Step 4: Set Up the Ratio
According to the problem, the ratio of the time taken downstream to the time taken upstream is given as \( 3:2 \):
\[ \frac{\frac{D}{11x}}{\frac{D - 45}{9x}} = \frac{3}{2} \]

### Step 5: Simplify the Ratio
Cross-multiply to eliminate the fractions:
\[ 2 \cdot \frac{D}{11x} = 3 \cdot \frac{D - 45}{9x} \]
This simplifies to:
\[ 2D \cdot 9x = 3(D - 45) \cdot 11x \]

### Step 6: Cancel \( x \) and Simplify
Since \( x \) is common in both sides, we can cancel it out:
\[ 2D \cdot 9 = 3(D - 45) \cdot 11 \]
This simplifies to:
\[ 18D = 33(D - 45) \]

### Step 7: Expand and Rearrange
Expanding the right side:
\[ 18D = 33D - 1485 \]
Rearranging gives:
\[ 1485 = 33D - 18D \]
\[ 1485 = 15D \]

### Step 8: Solve for \( D \)
Dividing both sides by 15:
\[ D = \frac{1485}{15} = 99 \]

### Final Answer
The value of \( D \) is \( 99 \) km.

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