I started on my bicycle at 7 a.m. to reach a certain place. After going a certain distance by bicycle went out of order. Consequently, I rested for 35 minutes and came back to my house walking all the way. I reached my house at 1 p.m. If my cycling speed is 10 kmph and my walking speed is 1 kmph, then on my bicycle I covered a distance of :

Options

A. A) `4 (61)/(66)` km
B. B) `13 4/9` km
C. C) `14 3/8` km
D. D) `15 (10)/(21)` km

Answer

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

### Step 1: Determine the total time spent
The total time from 7 a.m. to 1 p.m. is 6 hours.

**Hint:** To find the total time, subtract the start time from the end time.

### Step 2: Convert the rest time into hours
The rest time is given as 35 minutes. To convert this into hours, we divide by 60:
\[ \text{Rest time in hours} = \frac{35}{60} = \frac{7}{12} \text{ hours} \]

**Hint:** Remember that there are 60 minutes in an hour.

### Step 3: Calculate the time spent cycling
Let \( t \) be the time spent cycling in hours. The time spent walking will then be:
\[ \text{Time spent walking} = \text{Total time} - \text{Rest time} - \text{Cycling time} \]
This can be expressed as:
\[ \text{Time spent walking} = 6 - \frac{7}{12} - t \]

**Hint:** Make sure to express all times in the same unit (hours).

### Step 4: Calculate the distance covered
The distance covered while cycling can be expressed as:
\[ \text{Distance by bicycle} = \text{Speed} \times \text{Time} = 10t \text{ km} \]
The distance covered while walking can be expressed as:
\[ \text{Distance by walking} = \text{Speed} \times \text{Time} = 1 \left( 6 - \frac{7}{12} - t \right) \text{ km} \]

**Hint:** Use the formula for distance, which is speed multiplied by time.

### Step 5: Set up the equation
Since the distance covered while cycling and the distance covered while walking back to the house must be equal, we can set up the equation:
\[ 10t = 1 \left( 6 - \frac{7}{12} - t \right) \]

**Hint:** This equation represents the fact that the distance to the destination is the same regardless of the mode of transport.

### Step 6: Solve the equation
First, simplify the right side:
\[ 10t = 6 - \frac{7}{12} - t \]
Convert 6 into a fraction with a denominator of 12:
\[ 10t = \frac{72}{12} - \frac{7}{12} - t \]
Combine the fractions:
\[ 10t = \frac{65}{12} - t \]
Add \( t \) to both sides:
\[ 10t + t = \frac{65}{12} \]
This simplifies to:
\[ 11t = \frac{65}{12} \]
Now, solve for \( t \):
\[ t = \frac{65}{12 \times 11} = \frac{65}{132} \text{ hours} \]

**Hint:** When solving for \( t \), make sure to isolate \( t \) on one side of the equation.

### Step 7: Calculate the distance covered by bicycle
Now, substitute \( t \) back into the distance formula:
\[ \text{Distance by bicycle} = 10t = 10 \times \frac{65}{132} = \frac{650}{132} \text{ km} \]
This can be simplified:
\[ \frac{650}{132} = \frac{325}{66} \text{ km} \]

**Hint:** Always simplify your final answer if possible.

### Final Answer
The distance covered by bicycle is \( \frac{325}{66} \) km.