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The problem asks for the combined average age of two groups of students. Let's break it down and solve it using both a basic and a short method.

### Problem Statement:
- Group 1: 12 students with an average age of 35 years.
- Group 2: 27 students with an average age of 30 years.

We need to find the average age of all 39 students combined.

### **Basic Method:**
1. **Calculate Total Age for Each Group:**
   - Total age for Group 1 = \( 12 \times 35 = 420 \) years
   - Total age for Group 2 = \( 27 \times 30 = 810 \) years

2. **Add the Total Ages:**
   - Combined total age = \( 420 + 810 = 1230 \) years

3. **Calculate the Combined Average Age:**
   - Combined average age = \( \frac{1230}{39} = 31.54 \) years

   So, the combined average age of all 39 students is **31.54 years**.

### **Short Method:**
The short method uses the formula for the combined average of two groups:

\[
\text{Combined Average Age} = \frac{(n_1 \times \text{Avg}_1) + (n_2 \times \text{Avg}_2)}{n_1 + n_2}
\]

Where:
- \( n_1 = 12 \), \( \text{Avg}_1 = 35 \)
- \( n_2 = 27 \), \( \text{Avg}_2 = 30 \)

\[
\text{Combined Average Age} = \frac{(12 \times 35) + (27 \times 30)}{12 + 27} = \frac{420 + 810}{39} = \frac{1230}{39} = 31.54 \text{ years}
\]

Thus, the combined average age is **31.54 years**.

### Summary:
Both methods give the same result: the combined average age is 31.54 years.

Would you like more details, or do you have any questions?

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**Here are 8 follow-up questions related to this topic:**

1. How would the combined average change if Group 1 had 15 students?
2. What happens to the combined average if Group 2's average age was 28 years?
3. Can you derive a formula to find the combined average of three groups?
4. What if the first group had an average age of 40 instead of 35?
5. How does the number of students in each group affect the combined average?
6. Can you explain why the combined average might not be an integer?
7. What is the combined average if both groups had the same average age?
8. How would the combined average change if the groups had the same total number of students?

**Tip:** When solving average problems involving multiple groups, remember that the total contribution of each group is weighted by the number of members in that group.

Learn how to calculate the combined average age of two groups of students. Understand the basic and short methods to find the combined average age when given group sizes and average ages. Suitable for Grades 7-9.

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