The problem given is related to the concept of alligation, often used in mixtures and averages. Let's break it down and solve it step by step.

Given Information:

1. The average age of the students is $6$ years.
2. The average age of the 12 teachers is $40$ years.
3. The average age of the entire group (students + teachers) is $7$ years.
4. We need to find the total number of students.

Let:

• The number of students be $x$.
• The total number of people (students + teachers) = $x + 12$.

The weighted average formula can be used here:

$\text{Average Age of Group} = \frac{\text{Sum of ages of students} + \text{Sum of ages of teachers}}{\text{Total number of people}}$

From the problem:

• Sum of ages of students = $6x$ (since each student’s age is 6 and there are $x$ students).
• Sum of ages of teachers = $40 \times 12 = 480$.

Now, we use the weighted average:

$7 = \frac{6x + 480}{x + 12}$

Solve the equation:

Multiplying both sides by $x + 12$ to eliminate the denominator:

$7(x + 12) = 6x + 480$

Simplifying:

$7x + 84 = 6x + 480$

Now, subtract $6x$ from both sides:

$x + 84 = 480$

Finally, subtract 84 from both sides:

$x = 396$

Conclusion:

The total number of students is 396.

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

Related Questions:

1. How can you use alligation for problems involving more than two groups?
2. What happens if the average age of the teachers is different, say 45 years instead of 40 years?
3. How would the solution change if the total number of teachers was unknown, but the total number of people was provided?
4. How can the alligation method be applied to mixing solutions of different concentrations?
5. What other mathematical topics frequently involve averages and mixtures?

Tip:

In alligation, balancing the differences between averages and the quantities is key to simplifying and solving the problem effectively.