A man walks a certain distance and rides bak in `6(1)/(4)`h. he can walk both ways in `7(3)/(4)`h. how long it would take to ride both ways ?

Options

A. A)5 hours
B. B)`4(1)/(2)` hours
C. C)`4(3)/(4)` hours
D. D)6 hours

Answer

To solve the problem step by step, we need to set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( W \) = time taken to walk one way (in hours)
- \( R \) = time taken to ride one way (in hours)

### Step 2: Set Up the Equations
From the problem, we have the following information:
1. The time taken to walk one way and ride back is \( 6 \frac{1}{4} \) hours.
\[ W + R = 6 \frac{1}{4} = \frac{25}{4} \text{ hours} \]

2. The time taken to walk both ways is \( 7 \frac{3}{4} \) hours.
\[ 2W = 7 \frac{3}{4} = \frac{31}{4} \text{ hours} \]

### Step 3: Solve for \( W \)
From the second equation:
\[ 2W = \frac{31}{4} \]
Dividing both sides by 2:
\[ W = \frac{31}{8} \text{ hours} \]

### Step 4: Substitute \( W \) into the First Equation
Now substitute \( W \) into the first equation:
\[ \frac{31}{8} + R = \frac{25}{4} \]
To solve for \( R \), we first convert \( \frac{25}{4} \) to have a common denominator of 8:
\[ \frac{25}{4} = \frac{50}{8} \]
Now, substituting this into the equation:
\[ \frac{31}{8} + R = \frac{50}{8} \]
Subtract \( \frac{31}{8} \) from both sides:
\[ R = \frac{50}{8} - \frac{31}{8} = \frac{19}{8} \text{ hours} \]

### Step 5: Calculate Time for Riding Both Ways
The time taken to ride both ways is:
\[ 2R = 2 \times \frac{19}{8} = \frac{38}{8} = \frac{19}{4} \text{ hours} \]
Converting \( \frac{19}{4} \) to a mixed number:
\[ \frac{19}{4} = 4 \frac{3}{4} \text{ hours} \]

### Final Answer
Thus, the time it would take to ride both ways is \( 4 \frac{3}{4} \) hours.

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