The speed of a boat in still water in 9 kmph and the speed of stream is 6 kmh. If the boat covers a round trip journey in 6 h, then find the time taken by the boat to complete the downstream journey ?

Options

A. `2 (1)/(4) h`
B. `1 h`
C. `1 (3)/(4) h`
D. `1 (1)/(2) h `

Answer

To solve the problem step by step, we will first identify the speeds of the boat in still water and the stream, then calculate the speeds of the boat while going upstream and downstream, and finally determine the time taken for the downstream journey.

### Step 1: Identify the given values
- Speed of the boat in still water (B) = 9 km/h
- Speed of the stream (S) = 6 km/h

### Step 2: Calculate the speeds of the boat upstream and downstream
- Speed of the boat upstream (B_upstream) = B - S = 9 km/h - 6 km/h = 3 km/h
- Speed of the boat downstream (B_downstream) = B + S = 9 km/h + 6 km/h = 15 km/h

### Step 3: Set up the equation for the round trip
Let the distance one way be D km. The total time for the round trip is given as 6 hours. The time taken to go upstream and downstream can be expressed as follows:
- Time taken upstream = Distance / Speed upstream = D / 3
- Time taken downstream = Distance / Speed downstream = D / 15

The total time for the round trip is:
\[ \text{Total time} = \text{Time upstream} + \text{Time downstream} = \frac{D}{3} + \frac{D}{15} \]

### Step 4: Find a common denominator and solve for D
The common denominator for 3 and 15 is 15. Therefore, we can rewrite the equation:
\[ \frac{D}{3} = \frac{5D}{15} \]
Thus, the total time equation becomes:
\[ \frac{5D}{15} + \frac{D}{15} = \frac{6D}{15} \]
Setting this equal to the total time of 6 hours:
\[ \frac{6D}{15} = 6 \]

### Step 5: Solve for D
To eliminate the fraction, multiply both sides by 15:
\[ 6D = 90 \]
Now, divide both sides by 6:
\[ D = 15 \text{ km} \]

### Step 6: Calculate the time taken for the downstream journey
Now that we know the distance D, we can find the time taken for the downstream journey:
\[ \text{Time downstream} = \frac{D}{\text{Speed downstream}} = \frac{15}{15} = 1 \text{ hour} \]

### Final Answer
The time taken by the boat to complete the downstream journey is **1 hour**.