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To solve this problem, let's break down the given information and use a logical approach to determine the distance traveled by bicycle.

### Information provided:
1. **Start time**: 7 a.m.
2. **Breakdown event**: After traveling a certain distance, the bicycle goes out of order.
3. **Rest time**: 35 minutes.
4. **Walking back**: The person walks back home from the point of breakdown.
5. **Arrival time at home**: 1 p.m.
6. **Cycling speed**: 10 km/h.
7. **Walking speed**: 1 km/h.

### Steps to find the distance:

1. **Time spent after resting**: 
   From 7 a.m. to 1 p.m., the total time is 6 hours.
   Out of this, 35 minutes were spent resting, which is equivalent to \( \frac{35}{60} = \frac{7}{12} \) hours.
   Thus, the time spent walking and cycling combined is:
   \[
   6 - \frac{7}{12} = \frac{65}{12} \, \text{hours}.
   \]

2. **Walking time**: 
   Let the distance traveled on the bicycle be \( x \) km.
   Since the person walks back at 1 km/h and covered the same distance \( x \), the time taken to walk back is:
   \[
   \text{Walking time} = \frac{x}{1} = x \, \text{hours}.
   \]

3. **Cycling time**:
   The remaining time was spent cycling before the bicycle broke down. Since the cycling speed is 10 km/h, the time spent cycling is:
   \[
   \text{Cycling time} = \frac{x}{10} \, \text{hours}.
   \]

4. **Total time spent cycling and walking**:
   The total time for both cycling and walking (excluding rest time) is \( \frac{65}{12} \) hours. Therefore, we can set up the equation:
   \[
   \frac{x}{10} + x = \frac{65}{12}.
   \]

5. **Solve for \( x \)**:
   Multiply through by 60 to eliminate the denominators:
   \[
   6x + 60x = 65,
   \]
   \[
   66x = 65,
   \]
   \[
   x = \frac{65}{66} \, \text{km}.
   \]

Thus, the distance traveled by bicycle is approximately \( 0.9848 \) km.

However, this result doesn't seem to match the scale of the multiple-choice options provided. I will double-check the steps.The distance traveled on the bicycle is approximately **4.92 km**.

This matches one of the options provided in the question, so the correct answer is:

**4.92 km**.

Would you like further details or clarifications?

Here are 5 related questions for you:
1. How would the result change if the cycling speed were different?
2. How much time was spent walking compared to cycling?
3. What would happen to the total distance if the rest time increased?
4. How do you calculate distance using speed and time in other situations?
5. Could this problem be solved using a different method?

**Tip**: Always check your units when solving problems involving time and speed to avoid calculation mistakes.

I started on my bicycle at 7 a.m. to reach a certain place. After going a certain distance, my 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 km/h and my walking speed is 1 km/h, then on my bicycle I covered a distance of:

Solve a time-speed-distance problem where a cyclist covers a certain distance before their bicycle breaks down and they return on foot. This detailed step-by-step solution explains the calculations needed to determine the distance traveled before the breakdown.

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