Let's solve the problem step by step.

Given:

• Average age of 30 students initially = $a$ years.
• One student's age was considered 15 years more than the actual age.
• After correction, the average decreased by 0.5 years.

Approach:

1. Initial Total Age: $\text{Total initial age} = 30 \times a$

2. Correction:

   • The incorrect age was 15 years more than the actual age.
   • Let the actual age of that student be $x$ years.
   • Incorrect age = $x + 15$ years.
   • Therefore, the total age after correction is: $30a - (x + 15) + x = 30a - 15$

3. New Average:

   • The new average after correction is given to be $a - 0.5$ years.
   • So, $\frac{30a - 15}{30} = a - 0.5$

4. Simplifying: $30a - 15 = 30(a - 0.5)$ $30a - 15 = 30a - 15$

5. Conclusion:

   • The equation simplifies to $30a - 15 = 30a - 15$, which is an identity and is always true.
   • This means the average $a$ could be any value, and we cannot determine a specific value for $a$ based on the given information.

Answer:

The correct answer is "Cannot be determined."

Would you like more details or have any other questions?

Here are some related questions to explore:

1. What happens if the incorrect age is less than the actual age?
2. How do changes in the sample size affect the average?
3. What if the error in age was 10 years instead of 15? How would that affect the average?
4. How can you verify an average when the data set is large?
5. How does the median differ from the average in such scenarios?

Tip: Always double-check data entries when calculating averages to avoid skewed results.