The average salary of all the workers in a workshop is ₹ 8000. The average salary of 7 technicians is ₹ 12000 and the average salary of the rest is ₹ 6000. The total number of workers in the workshop is

### Concept & Strategy This problem is a classic application of the **Weighted Average** or **Rule of Alligation**. It involves mixing two distinct groups (technicians and the rest) to form a combined whole (all workers). ### Step-by-Step Solution * **Given Data:** Overall average salary = ₹ $8000$ Average salary of technicians (Group 1) = ₹ $12000$ Number of technicians = $7$ Average salary of the rest (Group 2) = ₹ $6000$ * **Algebraic Approach:** Let the total number of workers be $x$. The total salary of all workers = $8000x$. The total salary of the 7 technicians = $7 \times 12000 = 84000$. The number of remaining workers = $x - 7$. The total salary of the remaining workers = $6000 \times (x - 7)$. Equating the total salaries: $$8000x = 84000 + 6000(x - 7)$$ $$8000x = 84000 + 6000x - 42000$$ $$8000x - 6000x = 42000$$ $$2000x = 42000$$ $$x = \frac{42000}{2000} = 21$$ ### Exam Strategy & Shortcut The fastest way to solve this is using the **Rule of Alligation**. Set up the averages of the two groups and the mean in the middle: Technicians (₹ $12000$) ----------- Rest (₹ $6000$) \ / Mean (₹ $8000$) / \ $(8000 - 6000)$ ----------- $(12000 - 8000)$ Ratio $\rightarrow 2000 : 4000 = 1 : 2$ This means the ratio of Technicians to Rest is $1 : 2$. If $1$ part represents $7$ technicians, then the total parts ($1 + 2 = 3$ parts) represent the total workers. Total workers = $3 \times 7 = 21$. ### Common Pitfall A common mistake is finding the number of "rest" workers ($14$) and forgetting to add back the $7$ technicians to find the *total* number of workers as the question asks. Always double-check what the final variable represents! ### Final Answer **Therefore, the correct answer is 21.**