A car was brought in for servicing when its odometer read 2500 km. During the servicing, it was found that the odometer had a fault because of which it always skipped the digits 4 and 6. How much distance had the car actually travelled?

To solve this problem, we need to understand how the faulty odometer works and how much distance the car has actually traveled when the odometer shows 2500 km.

### Understanding the Faulty Odometer
The odometer skips the digits 4 and 6. This means that the odometer counts in a non-standard way:
- 0, 1, 2, 3, 5, 7, 8, 9

This sequence shows that there are 8 valid digits (0, 1, 2, 3, 5, 7, 8, 9) instead of the usual 10.

### Converting Odometer Reading to Actual Distance

Given that the faulty odometer skips the digits 4 and 6, the distance shown on the odometer doesn't correspond to the actual distance traveled. The actual distance is a number in base 10, but the odometer reading is essentially in a modified base-8 system.

Let's convert the odometer reading of 2500 (in this modified base-8 system) to its equivalent in base-10.

#### Step-by-Step Conversion

1. **Expand the number based on its place value in base-8**:
   \[
   2500_{\text{modified base-8}} = 2 \times 8^3 + 5 \times 8^2 + 0 \times 8^1 + 0 \times 8^0
   \]

2. **Calculate the value**:
   \[
   2 \times 512 + 5 \times 64 + 0 \times 8 + 0 \times 1 = 1024 + 320 + 0 + 0 = 1344
   \]

Thus, the actual distance traveled by the car is **1344 kilometers**.

Would you like more details or have any questions about the process?

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Here are 8 related questions to further explore this topic:

1. What would the odometer reading be if the actual distance traveled was 1500 km?
2. How would the result change if the odometer skipped different digits, like 2 and 7?
3. How do you convert from base-10 to this modified base-8 system?
4. How can you determine the fault in an odometer if you only have some odometer readings?
5. If the odometer skipped all odd digits, how would it affect the distance calculation?
6. What are the implications of using a faulty odometer in terms of legal regulations?
7. How would you check the accuracy of an odometer using GPS data?
8. How can you mathematically model other types of odometer faults?

**Tip**: Understanding different number systems can help in solving many problems in computing and electronics, where non-standard counting systems are often used.

Understand how to calculate the actual distance traveled by a car using a faulty odometer that skips digits 4 and 6. Learn the step-by-step process of converting from a modified base-8 system to base-10. Suitable for Grades 7-9.

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