A person can row a boat d km upstream and the same distance downstream in `5(1)/(4)` hours .Also he can row the boat 2d km upstream in 7 hours. He will row the same distance downstream in

Options

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

Answer

To solve the problem step by step, we will derive the necessary equations based on the information provided in the question.

### Step 1: Define Variables
Let:
- \( d \) = distance in km
- \( SU \) = speed upstream in km/h
- \( SD \) = speed downstream in km/h

### Step 2: Set Up the First Equation
The total time taken to row \( d \) km upstream and \( d \) km downstream is given as \( 5 \frac{1}{4} \) hours, which can be converted to an improper fraction:
\[ 5 \frac{1}{4} = \frac{21}{4} \text{ hours} \]
The time taken to row \( d \) km upstream is \( \frac{d}{SU} \) and downstream is \( \frac{d}{SD} \). Therefore, we can write:
\[ \frac{d}{SU} + \frac{d}{SD} = \frac{21}{4} \]
This is our **Equation 1**.

### Step 3: Set Up the Second Equation
We know that the person can row \( 2d \) km upstream in 7 hours. Therefore, we can set up the second equation:
\[ \frac{2d}{SU} = 7 \]
From this, we can express \( \frac{d}{SU} \):
\[ \frac{d}{SU} = \frac{7}{2} \]
This is our **Equation 2**.

### Step 4: Substitute Equation 2 into Equation 1
Now we will substitute \( \frac{d}{SU} = \frac{7}{2} \) into Equation 1:
\[ \frac{7}{2} + \frac{d}{SD} = \frac{21}{4} \]
To isolate \( \frac{d}{SD} \), we rearrange the equation:
\[ \frac{d}{SD} = \frac{21}{4} - \frac{7}{2} \]
To perform the subtraction, we need a common denominator. The common denominator of 4 and 2 is 4:
\[ \frac{7}{2} = \frac{14}{4} \]
Thus, we have:
\[ \frac{d}{SD} = \frac{21}{4} - \frac{14}{4} = \frac{7}{4} \]

### Step 5: Find the Time to Row \( 2d \) km Downstream
Now we need to find the time taken to row \( 2d \) km downstream. The time is given by:
\[ TD = \frac{2d}{SD} \]
Substituting \( \frac{d}{SD} = \frac{4}{7} \):
\[ TD = 2 \times \frac{d}{SD} = 2 \times \frac{7}{4} = \frac{14}{4} = \frac{7}{2} \text{ hours} \]

### Conclusion
The time taken to row \( 2d \) km downstream is \( \frac{7}{2} \) hours, which is equivalent to \( 3.5 \) hours.