A man take a 5 hours 45 mins in walking to a certain place and riding back. He would have gained 2 hours by riding both ways . The time he would take to walk both ways, is ______.

Options

A. `3` hrs `45` mins
B. `7` hrs `30` mins
C. `7` hrs `45` mins
D. `11` hrs `45` mins

Answer

To solve the problem, we need to find the time a man would take to walk both ways to a certain place, given that he takes 5 hours and 45 minutes to walk to the place and ride back, and he would have gained 2 hours by riding both ways.

Let's break it down step by step:

### Step 1: Convert the total time into hours
The total time taken by the man is 5 hours and 45 minutes.
- Convert 45 minutes into hours:
\[ 45 \text{ minutes} = \frac{45}{60} = 0.75 \text{ hours} \]
- Therefore, the total time in hours is:
\[ 5 + 0.75 = 5.75 \text{ hours} \]

**Hint:** Always convert mixed time formats (hours and minutes) into a single unit (hours) for easier calculations.

### Step 2: Set up the equation for walking and riding
Let \( t_w \) be the time taken to walk one way, and \( t_r \) be the time taken to ride one way. The total time for walking to the place and riding back can be expressed as:
\[ t_w + t_r = 5.75 \text{ hours} \]

### Step 3: Set up the equation for the time saved by riding both ways
If the man rides both ways, he would save 2 hours. Therefore, the time taken to ride both ways is:
\[ 2t_r = 5.75 - 2 = 3.75 \text{ hours} \]

**Hint:** When dealing with time savings, subtract the time saved from the total time to find the new time taken.

### Step 4: Solve for the time taken to ride one way
From the equation \( 2t_r = 3.75 \):
\[ t_r = \frac{3.75}{2} = 1.875 \text{ hours} \]

### Step 5: Substitute \( t_r \) back into the first equation
Now substitute \( t_r \) back into the first equation:
\[ t_w + 1.875 = 5.75 \]
Solving for \( t_w \):
\[ t_w = 5.75 - 1.875 = 3.875 \text{ hours} \]

### Step 6: Calculate the total time for walking both ways
The total time to walk both ways is:
\[ 2t_w = 2 \times 3.875 = 7.75 \text{ hours} \]

### Step 7: Convert back to hours and minutes
Convert 7.75 hours into hours and minutes:
- The whole number part is 7 hours.
- The decimal part (0.75) represents 45 minutes (since \( 0.75 \times 60 = 45 \)).

Thus, the total time taken to walk both ways is:
\[ \text{Total time} = 7 \text{ hours } 45 \text{ minutes} \]

### Final Answer
The time he would take to walk both ways is **7 hours 45 minutes**.

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