A steamer goes downstream and covers the distance between two ports in `5` hours while it covers the same distance upstream in `6` hours. If the speed of the stream is `1` km/hr, find the speed of the steamer in still water.

Options

A. `10` km/h
B. `9` km/h
C. `11` km/h
D. `12` km/h

Answer

To solve the problem, we need to find the speed of the steamer in still water (let's denote it as \( U \)). We are given the following information:

1. The speed of the stream is \( 1 \) km/hr.
2. The time taken to cover the distance downstream is \( 5 \) hours.
3. The time taken to cover the same distance upstream is \( 6 \) hours.

### Step 1: Define the distance
Let the distance between the two ports be \( D \) km.

### Step 2: Write the speed equations
When the steamer is going downstream, its effective speed is the sum of its speed in still water and the speed of the stream:
\[ \text{Downstream speed} = U + 1 \text{ km/hr} \]
The distance covered downstream can be expressed using the formula:
\[ D = \text{Speed} \times \text{Time} \]
So, for downstream:
\[ D = (U + 1) \times 5 \]
This simplifies to:
\[ D = 5U + 5 \quad \text{(Equation 1)} \]

When the steamer is going upstream, its effective speed is the difference between its speed in still water and the speed of the stream:
\[ \text{Upstream speed} = U - 1 \text{ km/hr} \]
Using the same distance formula for upstream:
\[ D = (U - 1) \times 6 \]
This simplifies to:
\[ D = 6U - 6 \quad \text{(Equation 2)} \]

### Step 3: Set the two equations for distance equal to each other
Since both expressions represent the same distance \( D \), we can set them equal to each other:
\[ 5U + 5 = 6U - 6 \]

### Step 4: Solve for \( U \)
Rearranging the equation:
\[ 5 + 6 = 6U - 5U \]
This simplifies to:
\[ 11 = U \]
Thus, the speed of the steamer in still water is:
\[ U = 11 \text{ km/hr} \]

### Final Answer
The speed of the steamer in still water is \( 11 \) km/hr.

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