A boat goes a certain distance downstream and then returns and covers 40% of distance covered in downstream. Ratio of time taken in covering downstream and upstream distances is 3:2. If speed of boat in still water is reduced by 50% then it covers 60 km downstream in 10 hours. Find the speed of boat in still water.

Options

A. A)9 km/hr
B. B)8 km /hr
C. C)6 km/hr
D. D)10 km/hr

Answer

To solve the problem step by step, we will break it down into manageable parts.

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

### Step 2: Establish the Distance and Time Relationships
The boat travels a distance \( D \) downstream and returns, covering 40% of the distance \( D \) upstream. The time taken for downstream and upstream is in the ratio of 3:2.

- Downstream distance = \( D \)
- Upstream distance = \( 0.4D \)

### Step 3: Write the Time Equations
The time taken to travel downstream and upstream can be expressed as:
- Time downstream = \( \frac{D}{B + W} \)
- Time upstream = \( \frac{0.4D}{B - W} \)

Given the ratio of time taken:
\[ \frac{\frac{D}{B + W}}{\frac{0.4D}{B - W}} = \frac{3}{2} \]

### Step 4: Simplify the Equation
Cancelling \( D \) from both sides:
\[ \frac{1}{B + W} \cdot \frac{B - W}{0.4} = \frac{3}{2} \]

Cross-multiplying gives:
\[ 2(B - W) = 3 \cdot 0.4(B + W) \]
\[ 2B - 2W = 1.2B + 1.2W \]

### Step 5: Rearranging the Equation
Rearranging the equation:
\[ 2B - 1.2B = 1.2W + 2W \]
\[ 0.8B = 3.2W \]
\[ \frac{B}{W} = \frac{3.2}{0.8} = 4 \]

Thus, the ratio of the speed of the boat to the speed of the water is:
\[ B : W = 4 : 1 \]

### Step 6: Use the Given Condition
If the speed of the boat is reduced by 50%, the new speed of the boat becomes \( \frac{B}{2} \).

Given that the boat covers 60 km downstream in 10 hours:
\[ \text{Speed downstream} = \frac{60 \text{ km}}{10 \text{ hours}} = 6 \text{ km/h} \]

### Step 7: Set Up the Equation
The speed downstream can be expressed as:
\[ \frac{B}{2} + W = 6 \]

### Step 8: Substitute for \( W \)
From the ratio \( B = 4W \):
\[ W = \frac{B}{4} \]

Substituting \( W \) into the downstream speed equation:
\[ \frac{B}{2} + \frac{B}{4} = 6 \]

### Step 9: Solve for \( B \)
Finding a common denominator (4):
\[ \frac{2B}{4} + \frac{B}{4} = 6 \]
\[ \frac{3B}{4} = 6 \]
Multiplying both sides by 4:
\[ 3B = 24 \]
Dividing by 3:
\[ B = 8 \text{ km/h} \]

### Final Answer
The speed of the boat in still water is **8 km/h**.

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