Ratio between speed of boat in still water to speed of stream is 5: 1. If boat travels 48 km upstream in 3 hours less than 144 km downstream, then find the speed of boat in still water?

Options

A. 12 kmph
B. 24 kmph
C. 20 kmph
D. 16 kmph

Answer

To solve the problem step by step, let's break it down as follows:

### Step 1: Define the Variables
Given the ratio of the speed of the boat in still water to the speed of the stream is 5:1, let:
- Speed of the stream = \( x \) km/h
- Speed of the boat in still water = \( 5x \) km/h

**Hint:** Start by defining the variables based on the given ratio.

### Step 2: Determine the Speeds Upstream and Downstream
- Upstream speed (against the stream) = Speed of boat - Speed of stream = \( 5x - x = 4x \) km/h
- Downstream speed (with the stream) = Speed of boat + Speed of stream = \( 5x + x = 6x \) km/h

**Hint:** Remember that upstream speed is reduced by the speed of the stream, while downstream speed is increased by it.

### Step 3: Set Up the Time Equations
We know that the boat travels 48 km upstream and 144 km downstream. The time taken for each journey can be expressed as:
- Time taken to travel upstream = \( \frac{48}{4x} \) hours
- Time taken to travel downstream = \( \frac{144}{6x} \) hours

According to the problem, the time taken upstream is 3 hours less than the time taken downstream:
\[ \frac{48}{4x} = \frac{144}{6x} - 3 \]

**Hint:** Use the formula for time, which is distance divided by speed.

### Step 4: Simplify the Equation
Now, simplify the equation:
1. Multiply both sides by \( 12x \) (the least common multiple of the denominators) to eliminate the fractions:
\[ 12x \cdot \frac{48}{4x} = 12x \cdot \left(\frac{144}{6x} - 3\right) \]
2. This simplifies to:
\[ 144 = 288 - 36x \]

**Hint:** Multiplying through by the least common multiple helps to eliminate fractions.

### Step 5: Solve for \( x \)
Rearranging the equation gives:
\[ 36x = 288 - 144 \]
\[ 36x = 144 \]
\[ x = 4 \]

**Hint:** Isolate the variable to find its value.

### Step 6: Find the Speed of the Boat in Still Water
Now that we have \( x \), we can find the speed of the boat in still water:
\[ \text{Speed of the boat} = 5x = 5 \times 4 = 20 \text{ km/h} \]

**Hint:** Substitute the value of \( x \) back into the equation for the speed of the boat.

### Final Answer
The speed of the boat in still water is **20 km/h**.