Solving the Fort Food Provision Problem

This question is a classic example of problems involving proportional reasoning, specifically inverse proportion. When the number of people decreases, the same amount of food will last for a longer time, assuming the consumption rate per person remains constant.

Let's break down the problem step-by-step:

Understanding the Initial Provision

• Initially, there are 120 men in the fort.
• The food provision is for 200 days for these 120 men.
• The total amount of food can be thought of in terms of "man-days".
• Total food provision = \( \text{Number of men} \times \text{Number of days} \)
• Total food provision = \( 120 \times 200 \) man-days
• Total food provision = \( 24000 \) man-days

Food Consumed in the First Few Days

• After 5 days, the situation changes.
• For the first 5 days, the food was consumed by the initial 120 men.
• Food consumed in 5 days = \( \text{Number of men} \times \text{Number of days} \)
• Food consumed in 5 days = \( 120 \times 5 \) man-days
• Food consumed in 5 days = \( 600 \) man-days

Calculating Remaining Food

• The remaining food is the total provision minus the food consumed.
• Remaining food = Total food provision - Food consumed in 5 days
• Remaining food = \( 24000 - 600 \) man-days
• Remaining food = \( 23400 \) man-days

Calculating Remaining Men

• After 5 days, 30 men moved to another place.
• Remaining men = Initial men - Men who moved
• Remaining men = \( 120 - 30 \)
• Remaining men = \( 90 \) men

Finding How Long the Remaining Food Will Last

• Now, the remaining 90 men will consume the 23400 man-days of food.
• Let the remaining food last for \( D \) days for the 90 men.
• Remaining food = \( \text{Remaining men} \times \text{Number of remaining days} \)
• \( 23400 \text{ man-days} = 90 \text{ men} \times D \text{ days} \)
• To find \( D \), we rearrange the equation:
• \( D = \frac{\text{Remaining food (man-days)}}{\text{Remaining men}} \)
• \( D = \frac{23400}{90} \)

Calculation

Let's perform the division:

\( D = \frac{2340\cancel{0}}{9\cancel{0}} \)

\( D = \frac{2340}{9} \)

We can perform long division or simple division:

\( 2340 \div 9 \)

\( 9 \times 2 = 18 \)

\( 23 - 18 = 5 \)

Bring down 4, making it 54.

\( 9 \times 6 = 54 \)

\( 54 - 54 = 0 \)

Bring down 0, making it 0.

\( 9 \times 0 = 0 \)

\( 0 - 0 = 0 \)

So, \( \frac{2340}{9} = 260 \).

\( D = 260 \) days.

The remaining food will last for 260 days for the remaining 90 men.

Verification (Alternative Method)

Alternatively, consider the food remaining after 5 days. It was food for 120 men for \( 200 - 5 = 195 \) days.

Amount of food = \( 120 \times 195 \) man-days.

Now, this same amount of food needs to last for 90 men for \( D \) days.

\( 90 \times D = 120 \times 195 \)

\( D = \frac{120 \times 195}{90} \)

\( D = \frac{12\cancel{0} \times 195}{9\cancel{0}} \)

\( D = \frac{12 \times 195}{9} \)

Divide both 12 and 9 by their greatest common divisor, 3:

\( D = \frac{\cancel{12}^{4} \times 195}{\cancel{9}^{3}} \)

\( D = \frac{4 \times 195}{3} \)

Divide 195 by 3:

\( 195 \div 3 = 65 \)

\( D = 4 \times 65 \)

\( D = 260 \)

Both methods confirm that the remaining food will last for 260 days.

Summary of Steps

1. Calculate the total initial food provision in man-days.
2. Calculate the food consumed in the first 5 days in man-days.
3. Subtract the consumed food from the total to find the remaining food in man-days.
4. Calculate the number of men remaining after some moved away.
5. Divide the remaining food (in man-days) by the number of remaining men to find the number of days the food will last.

Fort Food Provision Problem Solution

Revision Table: Food Provision Calculations

Understanding how the number of people affects the duration of provisions is key to solving these problems.

Additional Information: Inverse Proportion Problems

Problems involving food provisions, work completed, or tasks finished by a group of people often involve inverse proportion. If the amount of work or food is fixed, increasing the number of people decreases the time it takes, and decreasing the number of people increases the time it takes.

• Key Idea: Total "units" (like man-days or work units) remain constant or change predictably.
• Example: If 10 men can finish a job in 5 days, the total work is \( 10 \times 5 = 50 \) man-days. If 20 men work on the same job, they will take \( 50 / 20 = 2.5 \) days.
• In this fort provision problem, the "total food" is like the total "work". When the number of consumers (men) changes, the duration the food lasts changes inversely.
• These problems can be solved by calculating the total 'consumption units' (like man-days) and then redistributing them based on the new number of consumers.

Understanding the concept of 'man-days' is crucial for solving such problems efficiently.