In a garrison, there was some pilferage of ration everyday. If the pilferage was 12 kg a day, the ration would have lasted for 75 days. If the pilferage was restricted to 8 kg per day, the ration would have lasted for 90 days. For how many days would the ration have lasted, if there was no pilferage?

### Concept & Strategy The core concept is that the **Total Quantity of Ration** is fixed. The total ration consumed is the product of the number of days and the total daily usage (which is the garrison's legitimate daily consumption plus the daily pilferage). We can set up an equivalence equation between the two scenarios to find the garrison's true daily consumption. $$\text{Total Ration} = \text{Days} \times (\text{Daily Consumption} + \text{Pilferage})$$ ### Step-by-Step Solution * **Given:** * Scenario 1: Days = $75$, Pilferage = $12$ kg/day * Scenario 2: Days = $90$, Pilferage = $8$ kg/day * Let the garrison's legitimate daily consumption be $c$ kg/day. * **Calculation:** * Equate the total ration in both scenarios: * $75(c + 12) = 90(c + 8)$ * Simplify the equation by dividing both sides by $15$: * $5(c + 12) = 6(c + 8)$ * $5c + 60 = 6c + 48$ * $6c - 5c = 60 - 48 \Rightarrow c = 12$ kg/day. * Now, find the Total Ration using either scenario: * Total Ration = $75 \times (12 + 12) = 75 \times 24 = 1800$ kg. * Finally, calculate the days it would last with zero pilferage (where daily usage is just $c = 12$ kg/day): * Days without pilferage = $\frac{\text{Total Ration}}{\text{Daily Consumption}}$ * Days = $\frac{1800}{12} = 150$ days. ### Exam Strategy & Shortcut Use ratios to solve this faster without algebra. The inverse ratio of the Days ($75:90 = 5:6$) equals the direct ratio of the Daily Usage. Therefore, $(\text{Usage 1}) / (\text{Usage 2}) = 6 / 5$. We know Usage 1 is $4$ kg more than Usage 2 ($12$ kg pilferage vs $8$ kg). So, $6x - 5x = 4 \Rightarrow x = 4$. Usage 1 = $6 \times 4 = 24$ kg. Usage 2 = $5 \times 4 = 20$ kg. The legitimate consumption is Usage 1 minus pilferage: $24 - 12 = 12$ kg/day. Total ration = $24 \times 75 = 1800$. Days = $1800 / 12 = 150$. ### Common Pitfall A common error is confusing the *pilferage* with the *total consumption*. Students might mistakenly write $75 \times 12 = 900$ as the total ration, completely ignoring the garrison's own daily food requirements. The total daily depletion is the sum of both. ### Final Answer **Therefore, the correct answer is 150.**