In a barrack of soldiers there was stock of food for 190 days for 4000 soldiers. After 30 days 800 soldiers left the barrack. For how many days shall the left over food last for the remaining soldiers?

### Concept & Work Equivalence The core principle in provision problems is that the total amount of food (measured in "man-days") remains constant. We can equate the remaining provisions for the original men to the provisions for the new number of men. $$ M_1 \times D_1 = M_2 \times D_2 $$ ### Step-by-Step Solution * **Initial State:** Rations are sufficient for 4000 soldiers for 190 days. * **After 30 days:** The remaining food is sufficient for the original 4000 soldiers for the remaining $190 - 30 = 160$ days. * **New Group Size:** 800 soldiers leave, making the new group size $4000 - 800 = 3200$ soldiers. * **Calculation:** Let the remaining food last for $D$ days for these 3200 soldiers. * Equating the remaining man-days: $4000 \times 160 = 3200 \times D$ * $D = \frac{4000 \times 160}{3200}$ * $D = \frac{4000}{20} = 200$ days. ### Exam Strategy & Shortcut Always base the equation on the *remaining* days and the *current* number of men. Set up $M_{old} \times D_{remaining} = M_{new} \times D_{new}$. This single line $4000 \times 160 = 3200 \times x$ solves the problem in seconds. ### Common Pitfall A frequent error is calculating the total food initially and subtracting, which introduces large numbers and potential calculation errors. Equating the remainders is much cleaner. ### Final Answer Therefore, the correct answer is **200 days**.