A boat can cover certain distance in upstream in 16 minutes and the same distance in still water in 12 minutes. Find the time taken by boat to cover same distance in downstream

Options

A. 8 minutes
B. 9.6 minutes
C. 10 minutes
D. 8.8 minutes

Answer

To solve the problem step-by-step, we will first define the variables and then use the given information to find the required time taken by the boat to cover the same distance downstream.

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

### Step 2: Write the Equations for Upstream and Still Water
1. The time taken to cover the distance \( d \) upstream is given as 16 minutes:
\[ \text{Upstream speed} = x - y \]
Therefore, the distance can be expressed as:
\[ d = (x - y) \times 16 \]

2. The time taken to cover the same distance \( d \) in still water is given as 12 minutes:
\[ \text{Still water speed} = x \]
Therefore, the distance can also be expressed as:
\[ d = x \times 12 \]

### Step 3: Equate the Two Expressions for Distance
Since both expressions represent the same distance \( d \), we can set them equal to each other:
\[ (x - y) \times 16 = x \times 12 \]

### Step 4: Simplify the Equation
Expanding the left side gives:
\[ 16x - 16y = 12x \]
Now, rearranging the equation:
\[ 16x - 12x = 16y \]
\[ 4x = 16y \]
Dividing both sides by 4:
\[ x = 4y \]

### Step 5: Find the Ratio of Speeds
From the equation \( x = 4y \), we can find the ratio of the speeds:
\[ \frac{x}{y} = \frac{4}{1} \]

### Step 6: Calculate the Distance
Now, we can express the distance \( d \) using the speed in still water:
Using the value of \( x \):
\[ d = 12x = 12(4y) = 48y \]

### Step 7: Find the Downstream Speed
The downstream speed is given by:
\[ \text{Downstream speed} = x + y = 4y + y = 5y \]

### Step 8: Calculate the Time Taken to Cover the Distance Downstream
Using the formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{d}{\text{Downstream speed}} = \frac{48y}{5y} \]
The \( y \) cancels out:
\[ \text{Time} = \frac{48}{5} = 9.6 \text{ minutes} \]

### Conclusion
The time taken by the boat to cover the same distance downstream is **9.6 minutes**.