A garrison of 2000 men has provision of ration for 66 days. At the end of a fortnight, reinforcement arrives and it is found that ration will last only for 20 days more. The strength of the reinforcement is

### Concept & Remaining Provisions The total quantity of remaining ration is constant regardless of how many people consume it. We equate the (original men $\times$ remaining scheduled days) to (new total men $\times$ actual days it lasted). $$ M_1 \times D_{remaining} = M_{total} \times D_{new} $$ ### Step-by-Step Solution * **Initial state:** 2000 men have rations for 66 days. * **Time passed:** A fortnight is 14 days. After 14 days, the remaining food would last the 2000 men for $66 - 14 = 52$ days. * **Remaining man-days:** $2000 \times 52$. * **New scenario:** Let the total number of men after reinforcement be $M_{total}$. The remaining food lasts for 20 days. * **Equation:** $2000 \times 52 = M_{total} \times 20$ * $M_{total} = \frac{2000 \times 52}{20} = 100 \times 52 = 5200$ men. * **Finding Reinforcement:** The strength of the reinforcement is the difference between the total men and the original men. * Reinforcement $= 5200 - 2000 = 3200$ men. ### Exam Strategy & Shortcut Instead of finding total men first, you can directly find the reinforcement ($R$). The food meant for 2000 men for $(52 - 20) = 32$ days is eaten by the reinforcement ($R$) in 20 days. $2000 \times 32 = R \times 20 \implies R = 3200$. ### Common Pitfall Forgetting what a "fortnight" means (14 days) or solving for the total number of men (5200) and incorrectly choosing an option, forgetting to subtract the initial 2000 men to find the reinforcement. ### Final Answer Therefore, the correct answer is **3200**.