In a besieged town provision problem, if a town has provisions for a certain number of men for a certain number of days, the problem uses inverse proportion: more men means fewer days, fewer men means more days. The key formula is: Men₁ × Days₁ = Men₂ × Days₂.
Provision problems use inverse proportion: more men → fewer days
Formula: M₁ × D₁ = M₂ × D₂
Total provisions = Men × Days (kept constant)
If days have already passed, subtract consumed provisions first
Remaining days = Remaining provisions ÷ New number of men
These problems appear in SSC, UPSC, bank exams, and school competitions
The concept also applies to work problems: more workers = less time
Always check units: provisions per man per day must be consistent
If provisions last for N men for D days, then the total food = N × D units.
When more men arrive or some leave, the total food remains the same but the duration changes.
Inverse Proportion Principle: More men → fewer days Fewer men → more days
Formula: M₁ × D₁ = M₂ × D₂ Where M = number of men, D = number of days
Problem: A besieged town has provisions for 1200 soldiers for 60 days. After 15 days, 300 more soldiers join. How many more days will the provisions last?
Step 1: Find provisions consumed in 15 days Consumed = 1200 × 15 = 18,000 soldier-days
Step 2: Find remaining provisions Total = 1200 × 60 = 72,000 soldier-days Remaining = 72,000 − 18,000 = 54,000 soldier-days
Step 3: New number of soldiers = 1200 + 300 = 1500
Step 4: Days remaining = 54,000 ÷ 1500 = 36 days
Answer: The provisions will last 36 more days.
Problem: A town has provisions for 800 men for 45 days. After 10 days, 200 men leave. How long will the remaining provisions last?
Step 1: Provisions used in 10 days = 800 × 10 = 8,000 Total = 800 × 45 = 36,000 Remaining = 36,000 − 8,000 = 28,000 units
Step 2: Remaining men = 800 − 200 = 600
Step 3: Days = 28,000 ÷ 600 = 46.67 days ≈ 46 days 16 hours
Answer: Provisions last approximately 46 more days.
For problems where no days have passed yet: M₁ × D₁ = M₂ × D₂
Example: Provisions for 500 men for 40 days. If 100 extra men arrive, find new duration. 500 × 40 = 600 × D₂ 20,000 = 600 × D₂ D₂ = 20,000 ÷ 600 = 33.33 days
Note: This only works if no days have already passed. Otherwise, use the step-by-step method above.
Inverse proportion is used. More men means fewer days, and fewer men means more days. The formula is M₁ × D₁ = M₂ × D₂.
Using M₁D₁ = M₂D₂: 600 × 30 = 400 × D₂. D₂ = 18,000 ÷ 400 = 45 days.
Step 1: Calculate provisions consumed in elapsed days (men × days elapsed). Step 2: Subtract from total provisions. Step 3: Divide remaining provisions by new number of men.
Because men and days are inversely related — if you double the men, the provisions last half as long. The product of men × days remains constant.
Remaining provisions: 1000 × (50−20) = 1000 × 30 = 30,000. New men = 1500. Days = 30,000 ÷ 1500 = 20 more days.
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