The ratio of time taken by Hunny and Bunny to swim a certain distance downstream in a river is 3:4 respectively. The time taken by Bunny to cover a certain distance upstream is 50% more than the time taken by him to cover the same distance downstream. Both of them hired a boat that runs with a speed equal to the sum of their individual speeds. If Hunny can cover a straight path in still water of length 14 km in 60 minutes, then find the time taken by both of them to travel a distance of 48 km to and fro by the hired boat?

Options

A. `5(4)/(143)hr.`
B. `2(4)/(143)hr.`
C. `3(4)/(143)hr.`
D. `4(4)/(143)hr.`

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the Given Information
- The ratio of time taken by Hunny and Bunny to swim downstream is 3:4.
- Bunny takes 50% more time to swim upstream than downstream.
- Hunny can cover 14 km in 60 minutes in still water.

### Step 2: Define Variables
Let:
- \( A \) = speed of Hunny in still water (km/h)
- \( B \) = speed of Bunny in still water (km/h)
- \( R \) = speed of the river (km/h)
- \( D \) = distance they swim downstream (km)

From the information given, we know:
- \( A = 14 \) km/h (since Hunny swims 14 km in 60 minutes).

### Step 3: Set Up the Equations
1. The time taken by Hunny downstream is \( \frac{D}{A + R} \).
2. The time taken by Bunny downstream is \( \frac{D}{B + R} \).

According to the ratio of their times:
\[ \frac{\frac{D}{A + R}}{\frac{D}{B + R}} = \frac{3}{4} \]
This simplifies to:
\[ \frac{B + R}{A + R} = \frac{4}{3} \]
Cross-multiplying gives us:
\[ 3(B + R) = 4(A + R) \]
Expanding this:
\[ 3B + 3R = 4A + 4R \]
Rearranging gives:
\[ 3B = 4A + R \]
(Equation 1)

### Step 4: Analyze Bunny's Upstream Time
Bunny's upstream time is 50% more than his downstream time:
- Downstream time = \( \frac{D}{B + R} \)
- Upstream time = \( \frac{D}{B - R} \)

Setting up the equation:
\[ \frac{D}{B - R} = 1.5 \cdot \frac{D}{B + R} \]
Cancelling \( D \) from both sides gives:
\[ \frac{1}{B - R} = \frac{1.5}{B + R} \]
Cross-multiplying gives:
\[ B + R = 1.5(B - R) \]
Expanding this:
\[ B + R = 1.5B - 1.5R \]
Rearranging gives:
\[ 2.5R = 0.5B \]
Thus:
\[ B = 5R \]
(Equation 2)

### Step 5: Substitute Equation 2 into Equation 1
Substituting \( B = 5R \) into Equation 1:
\[ 3(5R) = 4A + R \]
This simplifies to:
\[ 15R = 4A + R \]
Rearranging gives:
\[ 14R = 4A \]
Thus:
\[ R = \frac{4A}{14} = \frac{2A}{7} \]

### Step 6: Calculate Speed of Bunny
Since \( A = 14 \) km/h:
\[ R = \frac{2 \times 14}{7} = 4 \text{ km/h} \]
Now substituting \( R \) back to find \( B \):
\[ B = 5R = 5 \times 4 = 20 \text{ km/h} \]

### Step 7: Calculate the Speed of the Boat
The speed of the boat is the sum of their individual speeds:
\[ \text{Speed of the boat} = A + B = 14 + 20 = 34 \text{ km/h} \]

### Step 8: Calculate Time Taken to Travel 48 km
The distance to travel is 48 km, and the boat travels downstream and upstream:
\[ \text{Total time} = \frac{48}{\text{Speed downstream}} + \frac{48}{\text{Speed upstream}} \]
- Speed downstream = \( 34 + 4 = 38 \) km/h
- Speed upstream = \( 34 - 4 = 30 \) km/h

Calculating the time:
\[ \text{Total time} = \frac{48}{38} + \frac{48}{30} \]
Calculating each term:
\[ \frac{48}{38} = \frac{24}{19} \text{ hours} \]
\[ \frac{48}{30} = \frac{8}{5} \text{ hours} \]
Finding a common denominator (95):
\[ \frac{24}{19} = \frac{120}{95} \]
\[ \frac{8}{5} = \frac{152}{95} \]
Adding these:
\[ \text{Total time} = \frac{120 + 152}{95} = \frac{272}{95} \text{ hours} \]

### Final Answer
The time taken by both of them to travel a distance of 48 km to and fro by the hired boat is \( \frac{272}{95} \) hours, which can be approximated as \( 2.86 \) hours or \( 2 \) hours and \( 51.6 \) minutes.