A boat takes a total time of twelve hours to travel 105 km upstream and the same distance downstream. The speed of the boat in still water is six times that of the current. What is the speed of the boat in still water?( in km`//`hr)

Options

A. 12
B. 30
C. 18
D. 24

Answer

To solve the problem step by step, let's define the variables and use the information given in the question.

### Step 1: Define the Variables
Let:
- \( x \) = speed of the boat in still water (in km/hr)
- \( y \) = speed of the current (in km/hr)

### Step 2: Set Up the Relationship
According to the problem, the speed of the boat in still water is six times that of the current. This gives us our first equation:
\[ x = 6y \]

### Step 3: Calculate Upstream and Downstream Speeds
When the boat is going upstream, its effective speed is:
\[ \text{Upstream speed} = x - y \]
When the boat is going downstream, its effective speed is:
\[ \text{Downstream speed} = x + y \]

### Step 4: Set Up the Time Equation
The time taken to travel a distance is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
The total time taken for both upstream and downstream is 12 hours. Therefore, we can write:
\[ \frac{105}{x - y} + \frac{105}{x + y} = 12 \]

### Step 5: Substitute \( x \) in Terms of \( y \)
Substituting \( x = 6y \) into the time equation:
\[ \frac{105}{(6y - y)} + \frac{105}{(6y + y)} = 12 \]
This simplifies to:
\[ \frac{105}{5y} + \frac{105}{7y} = 12 \]

### Step 6: Find a Common Denominator
The common denominator for \( 5y \) and \( 7y \) is \( 35y \). Rewriting the equation gives:
\[ \frac{105 \cdot 7}{35y} + \frac{105 \cdot 5}{35y} = 12 \]
This simplifies to:
\[ \frac{735 + 525}{35y} = 12 \]
\[ \frac{1260}{35y} = 12 \]

### Step 7: Solve for \( y \)
Cross-multiplying gives:
\[ 1260 = 12 \cdot 35y \]
\[ 1260 = 420y \]
Dividing both sides by 420:
\[ y = \frac{1260}{420} = 3 \text{ km/hr} \]

### Step 8: Find \( x \)
Now substitute \( y \) back into the first equation to find \( x \):
\[ x = 6y = 6 \cdot 3 = 18 \text{ km/hr} \]

### Final Answer
The speed of the boat in still water is:
\[ \boxed{18 \text{ km/hr}} \]