Bhanu takes a total of 6 hours 50 minutes in walking a distance and running back to same place where she started. She could walk both ways in 8 hours 30 minutes. The time taken by his to run both ways is :

Options

A. 5 hours 35 min
B. 5 hours 15 min
C. 5 hours 10 min
D. 5 hours 45 min

Answer

To solve the problem step by step, we need to find the time taken by Bhanu to run both ways, given the total time taken for walking and running, and the time taken to walk both ways.

### Step 1: Convert the times into a consistent format
First, we need to convert the total time taken by Bhanu into minutes for easier calculations.

- Total time taken for walking and running: 6 hours 50 minutes
\[ 6 \text{ hours} = 6 \times 60 = 360 \text{ minutes} \]
\[ \text{Total time} = 360 + 50 = 410 \text{ minutes} \]

- Time taken to walk both ways: 8 hours 30 minutes
\[ 8 \text{ hours} = 8 \times 60 = 480 \text{ minutes} \]
\[ \text{Walking time} = 480 + 30 = 510 \text{ minutes} \]

### Step 2: Set up the equations
Let \( W \) be the time taken to walk both ways, and \( R \) be the time taken to run both ways.

From the problem, we have:
- Total time for walking and running:
\[ W + R = 410 \text{ minutes} \]
- Time taken to walk both ways:
\[ W = 510 \text{ minutes} \]

### Step 3: Substitute and solve for \( R \)
Now we can substitute \( W \) into the first equation to find \( R \):
\[ 510 + R = 410 \]
To find \( R \), we rearrange the equation:
\[ R = 410 - 510 \]
\[ R = -100 \text{ minutes} \]

### Step 4: Interpret the result
Since \( R \) cannot be negative, we need to check our calculations. The time taken to run back should be less than the total time taken for walking and running.

### Step 5: Correct the interpretation
The total time for walking both ways is actually longer than the total time taken for walking and running. Thus, we need to find the time taken to run both ways.

### Step 6: Calculate the time taken to run both ways
To find the time taken to run both ways, we can use the difference:
\[ R = 410 - W \]
Substituting \( W = 510 \):
\[ R = 410 - 510 = -100 \text{ minutes} \]

This indicates that the time for running cannot exceed the total time taken for both activities.

### Final Calculation
To find the correct time taken for running both ways, we need to consider:
- Total time for both activities is 6 hours 50 minutes (410 minutes)
- Walking time is 8 hours 30 minutes (510 minutes)

Thus, the time taken by Bhanu to run both ways is:
\[ R = 410 - W \]
\[ R = 410 - 510 = -100 \text{ minutes} \]

This indicates an inconsistency in the problem statement.

### Conclusion
The time taken by Bhanu to run both ways cannot be calculated as it leads to a negative value, suggesting that the walking time exceeds the total time taken for the round trip.