A boat takes 19 h for travelling downstream from point A to point B and coming back to a point C midway between A and B. If the velocity of the stream is 4 km/h and the speed of the boat in still water is 14 km/h, what is the distance between A and B?

Options

A. 200 km
B. 180 km
C. 160 km
D. 220 km

Answer

To solve the problem, we need to find the distance between points A and B based on the information given about the boat's travel time, speed in still water, and the speed of the stream.

### Step 1: Determine the effective speeds of the boat
1. **Speed of the boat in still water (B)** = 14 km/h
2. **Speed of the stream (S)** = 4 km/h

**Downstream speed (D)** = Speed of the boat + Speed of the stream = B + S = 14 km/h + 4 km/h = 18 km/h

**Upstream speed (U)** = Speed of the boat - Speed of the stream = B - S = 14 km/h - 4 km/h = 10 km/h

**Hint for Step 1:** Remember that downstream speed is the sum of the boat's speed and the stream's speed, while upstream speed is the difference.

### Step 2: Define the distance
Let the distance between points A and B be **D** km. Since point C is midway between A and B, the distance from A to C is **D/2** km.

### Step 3: Calculate the time taken for each part of the journey
1. **Time taken to travel downstream from A to B** = Distance / Speed = D / 18 hours
2. **Time taken to travel upstream from B to C** = Distance / Speed = (D/2) / 10 hours = D / 20 hours

### Step 4: Set up the equation for total time
The total time for the journey is the sum of the time taken to go downstream and the time taken to go upstream:

Total time = Time downstream + Time upstream
19 hours = (D / 18) + (D / 20)

### Step 5: Solve the equation
To solve for D, we need a common denominator for the fractions. The least common multiple of 18 and 20 is 180.

Rewriting the equation:
19 = (10D / 180) + (9D / 180)

Combine the fractions:
19 = (10D + 9D) / 180
19 = 19D / 180

Now, multiply both sides by 180 to eliminate the fraction:
19 * 180 = 19D
3420 = 19D

Now, divide both sides by 19:
D = 3420 / 19
D = 180 km

### Conclusion
The distance between points A and B is **180 km**.