A boat takes 11 hours for travelling downstream from point A to point B and coming back to point C midway between A and B. If the velocity of the stream be 3 km/hr and the speed of the boat in still water be 12 km/hr, what is the distance between A and B?

Options

A. 100 km
B. 90 km
C. 110 km
D. 120 km

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Determine the speeds
The speed of the boat in still water is given as 12 km/hr, and the speed of the stream is 3 km/hr.

- **Downstream speed** (speed of the boat + speed of the stream):
\[ \text{Downstream speed} = 12 + 3 = 15 \text{ km/hr} \]

- **Upstream speed** (speed of the boat - speed of the stream):
\[ \text{Upstream speed} = 12 - 3 = 9 \text{ km/hr} \]

### Step 2: Define the distances
Let the distance between points A and B be \( X \) km. The distance from B to C (midway) will then be \( \frac{X}{2} \) km.

### Step 3: Set up the time equation
The total time taken for the journey downstream from A to B and then upstream from B to C is given as 11 hours.

Using the formula for time, \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can set up the equation:

- **Time taken to travel downstream from A to B**:
\[ \text{Time}_{\text{downstream}} = \frac{X}{15} \]

- **Time taken to travel upstream from B to C**:
\[ \text{Time}_{\text{upstream}} = \frac{\frac{X}{2}}{9} = \frac{X}{18} \]

### Step 4: Combine the times
The total time for the journey is:
\[ \frac{X}{15} + \frac{X}{18} = 11 \]

### Step 5: Find a common denominator
To solve the equation, we need a common denominator. The least common multiple (LCM) of 15 and 18 is 90.

Rewriting the equation:
\[ \frac{6X}{90} + \frac{5X}{90} = 11 \]

Combining the fractions:
\[ \frac{11X}{90} = 11 \]

### Step 6: Solve for \( X \)
To isolate \( X \), multiply both sides by 90:
\[ 11X = 11 \times 90 \]
\[ 11X = 990 \]
Now, divide by 11:
\[ X = \frac{990}{11} = 90 \text{ km} \]

### Conclusion
The distance between points A and B is **90 km**.

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