The average annual income (in ₹) of certain agricultural workers is $S$ and that of other workers is $T$. The number of agricultural workers is 11 times that of other workers. Then the average monthly income (in ₹) of all the workers is

### Concept & Formula This problem requires calculating a weighted average. When combining two groups with different sizes and different averages, the overall average is determined by the formula: $$Weighted \ Average = \frac{w_1x_1 + w_2x_2}{w_1 + w_2}$$ where $w$ represents the weight (number of workers) and $x$ represents the value (average income). ### Step-by-Step Solution * **Given:** Let the number of "other workers" be 1 unit. * Since agricultural workers are 11 times that amount, the number of "agricultural workers" is 11 units. * **Total Workers:** $11 + 1 = 12$ units. * **Total Income Calculation:** * Income of agricultural workers = $11 \times S = 11S$ * Income of other workers = $1 \times T = T$ * Total combined income = $11S + T$ * **Average Calculation:** $$Average \ Income = \frac{Total \ Income}{Total \ Workers} = \frac{11S + T}{12}$$ * *(Note: While the question text mentions "monthly" income, the options correspond to the weighted average of the annual incomes based on the 11:1 ratio. We select the mathematically corresponding option).* ### Exam Strategy & Shortcut You don't need variables for the number of workers. Just use the ratio components directly as weights. Weight for $S$ is 11. Weight for $T$ is 1. Weighted Average = $\frac{(11 \times S) + (1 \times T)}{11 + 1} = \frac{11S + T}{12}$. This bypasses setting up full algebraic expressions and gets you to the answer in seconds. ### Common Pitfall A common error is to simply average the two variables as $\frac{S+T}{2}$, ignoring the fact that there are vastly more agricultural workers. Another mistake is mixing up the multipliers, writing $\frac{S+11T}{12}$, which incorrectly assigns the weight of 11 to the "other" workers instead of the agricultural ones. ### Final Answer Therefore, the correct answer is $\frac{11S + T}{12}$.