I walk a certain distance and ride back, taking `(13)/(2)` hours in total. I could walk both ways in a total of `(31)/(4)` hours. How long will it take me to ride both ways?

Options

A. 6 h 15 min
B. 5 h 55 min
C. 4 h 15 min
D. 5 h 15 min

Answer

To solve the problem step by step, we will break down the information given and perform the necessary calculations.

### Step 1: Understand the total time taken for walking both ways
We are given that the total time taken to walk both ways is \( \frac{31}{4} \) hours.

### Step 2: Calculate the time taken to walk one way
Since the time taken to walk both ways is \( \frac{31}{4} \) hours, the time taken to walk one way will be:
\[ \text{Time for one way} = \frac{31}{4} \div 2 = \frac{31}{8} \text{ hours} \]

### Step 3: Understand the total time taken for walking one way and riding back
We are also given that the total time taken to walk a certain distance one way and ride back is \( \frac{13}{2} \) hours.

### Step 4: Calculate the time taken to ride back
Let \( t_r \) be the time taken to ride back. The equation for the total time can be expressed as:
\[ \text{Time to walk one way} + \text{Time to ride back} = \frac{13}{2} \]
Substituting the time taken to walk one way:
\[ \frac{31}{8} + t_r = \frac{13}{2} \]

### Step 5: Solve for the time taken to ride back
To isolate \( t_r \), we need to subtract \( \frac{31}{8} \) from \( \frac{13}{2} \):
\[ t_r = \frac{13}{2} - \frac{31}{8} \]
To perform this subtraction, we need a common denominator. The common denominator for 2 and 8 is 8. Thus, we convert \( \frac{13}{2} \):
\[ \frac{13}{2} = \frac{13 \times 4}{2 \times 4} = \frac{52}{8} \]
Now we can subtract:
\[ t_r = \frac{52}{8} - \frac{31}{8} = \frac{21}{8} \text{ hours} \]

### Step 6: Calculate the time taken to ride both ways
To find the time taken to ride both ways, we simply multiply the time taken to ride one way by 2:
\[ \text{Time to ride both ways} = 2 \times t_r = 2 \times \frac{21}{8} = \frac{21 \times 2}{8} = \frac{42}{8} = \frac{21}{4} \text{ hours} \]

### Step 7: Convert to hours and minutes
To convert \( \frac{21}{4} \) hours into hours and minutes:
\[ \frac{21}{4} = 5 \text{ hours and } 15 \text{ minutes} \]

### Final Answer
It will take me **5 hours and 15 minutes** to ride both ways.

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