Camp rationing with arrivals: Food was sufficient for 2000 people for 54 days. After 15 days, more people arrived and the food lasted only 20 more days. How many people arrived?
Correct Answer: 1900
Introduction / Context:Track stock in person-days, subtract the initial consumption, then divide the remainder by the new total headcount to match the shortened remaining duration. Solve for the unknown additional headcount algebraically and exactly.
Given Data / Assumptions:
- Initial stock: 2000 people for 54 days ⇒ 108000 person-days.
- First 15 days at 2000 people.
- After that, food lasted 20 more days with the enlarged population.
Concept / Approach:Remaining stock after 15 days = 108000 − (2000*15). Let new headcount be 2000 + x. Then (2000 + x)*20 must equal the remaining stock. Solve for x (arrivals).
Step-by-Step Solution:
Remaining stock = 108000 − 30000 = 78000 person-days (2000 + x) * 20 = 78000 ⇒ 2000 + x = 3900 ⇒ x = 1900Verification / Alternative check:New headcount = 3900; 3900 * 20 = 78000 matches remaining stock. Numbers balance exactly.
Why Other Options Are Wrong:2400, 1940, 2220, 2100 do not satisfy the exact person-day equality for 20-day residual life.
Common Pitfalls:Forgetting to subtract the first 15 days of consumption or mixing people added with total headcount after arrival.
Final Answer:1900