Prices register an increase of $10\%$ on foodgrains and $15\%$ on other items of expenditure. If the ratio of an employee's expenditure on foodgrains and other items be $2 : 5$, by how much should his salary be increased in order that he may maintain the same level of consumption as before, his present salary being ₹ $2590$?

### Concept & Strategy To find the required increase in salary to maintain the same consumption, we can use the concept of the weighted average. Instead of calculating the absolute values for each expenditure category, we can find the overall percentage increase required for the total salary. $$ \text{Weighted Average Increase} = \frac{(W_1 \times R_1) + (W_2 \times R_2)}{W_1 + W_2} $$ Where $W$ represents the ratio weight of the expenditure and $R$ represents the percentage increase. ### Step-by-Step Solution 1. **Given:** * Ratio of expenditure (Foodgrains : Other) = $2 : 5$ * Increase on foodgrains ($R_1$) = $10\%$ * Increase on other items ($R_2$) = $15\%$ * Total present salary = ₹ $2590$ 2. **Calculate the overall percentage increase required:** * $$ \text{Overall } \% \text{ Increase} = \frac{(2 \times 10) + (5 \times 15)}{2 + 5} $$ * $$ = \frac{20 + 75}{7} = \frac{95}{7}\% $$ 3. **Calculate the absolute increase amount:** * Required Increase = $\frac{95}{7}\%$ of Total Salary * $$ \text{Increase} = \left( \frac{95}{7 \times 100} \right) \times 2590 $$ * $$ = \left( \frac{95}{700} \right) \times 2590 $$ * $$ = \frac{95 \times 259}{70} $$ * $$ = \frac{95 \times 37}{10} $$ (Canceling out by $7$) * $$ = \frac{3515}{10} = 351.5 $$ 4. **Evaluate Options:** * The calculated increase is ₹ $351.50$. * Looking at the options: (a) $323.75$, (b) $350$, (c) $360.50$. None match exactly. ### Exam Strategy & Shortcut Instead of finding the overall percentage first, you can split the salary into actual amounts based on the $2:5$ ratio. Total parts = $7$. One part = $\frac{2590}{7} = 370$. Foodgrain expense = $2 \times 370 = 740$. Other expense = $5 \times 370 = 1850$. Increase = ($10\%$ of $740$) + ($15\%$ of $1850$) = $74 + 277.5 = 351.5$. This avoids fraction multiplication at the end and is highly intuitive. ### Common Pitfall A common mistake is taking a simple average of the percentages ($\frac{10 + 15}{2} = 12.5\%$) instead of the weighted average. This ignores the fact that a much larger portion of the budget ($5$ parts out of $7$) experiences the higher $15\%$ inflation rate. ### Final Answer **Therefore, the correct answer is None of these.**