The average monthly salary of all workers in a workshop is Rs. 12,000. The average monthly salary of 7 technicians is Rs. 15,000 and the average salary of all the remaining workers is Rs. 9,000. What is the total number of workers in the workshop?

Introduction / Context:
This problem involves combining averages from two subgroups of a workforce to recover the total number of workers. We know the overall average salary, the average salary of technicians, and the average salary of the remaining group. From this information we can determine how many workers there are in total.

Given Data / Assumptions:

• Overall average salary of all workers = Rs. 12,000 per month.
• Average salary of 7 technicians = Rs. 15,000 per month.
• Average salary of the remaining workers = Rs. 9,000 per month.
• Let total number of workers (including the 7 technicians) be n.

Concept / Approach:
Let us denote:

• Total salary of all workers = 12,000 * n.
• Total salary of 7 technicians = 15,000 * 7.
• Number of remaining workers = n - 7, with total salary 9,000 * (n - 7).

Since the total salary must be equal whether we compute it from the overall average or from the subgroup totals, we can form an equation and solve for n.

Step-by-Step Solution:
Step 1: Express total salary using the overall average. Total salary = 12,000 * n. Step 2: Express total salary as the sum of salaries of technicians and others. Total salary = 15,000 * 7 + 9,000 * (n - 7). Step 3: Form the equation. 12,000 * n = 15,000 * 7 + 9,000 * (n - 7). Step 4: Simplify the right hand side. 15,000 * 7 = 105,000. 9,000 * (n - 7) = 9,000 * n - 63,000. So 12,000 * n = 105,000 + 9,000 * n - 63,000 = 9,000 * n + 42,000. Step 5: Solve for n. 12,000 * n - 9,000 * n = 42,000. 3,000 * n = 42,000. n = 42,000 / 3,000 = 14.

Verification / Alternative check:
If there are 14 workers in total, then there are 7 technicians and 7 other workers. Total salary of technicians = 7 * 15,000 = 105,000. Total salary of other workers = 7 * 9,000 = 63,000. Combined total salary = 105,000 + 63,000 = 168,000. Average salary = 168,000 / 14 = 12,000, which matches the given overall average, confirming that 14 is correct.

Why Other Options Are Wrong:
For n = 12 or 13 or 15, when you compute the total salary using both methods, the averages do not match. For example, if n = 12, the total salary by subgroup calculation would be 15,000 * 7 + 9,000 * 5 = 105,000 + 45,000 = 150,000, giving an overall average of 150,000 / 12 = 12,500, not 12,000. Similar inconsistencies occur for n = 13 and n = 15. Only n = 14 is consistent with all the data.

Common Pitfalls:
Some students try to average 15,000 and 9,000 directly without accounting for different group sizes, which is incorrect. Others forget that the total salary is simply average times number of workers and struggle with unnecessary variables. The safest method is to write total salary in two ways, set them equal, and solve the resulting linear equation.

Final Answer:
The total number of workers in the workshop is 14.