A garrison of 900 men had provisions for 42 days. However, a reinforcenment of 500 men arrived. For how many days will the food last now?

Answer

To solve the problem step by step, we will use the concept of inverse proportion. Here’s how we can approach it:

### Step 1: Understand the Initial Situation
The initial garrison has 900 men and provisions that last for 42 days.

### Step 2: Calculate Total Man-Days of Provisions
First, we calculate the total amount of food available in terms of man-days. This is done by multiplying the number of men by the number of days the provisions last.

\[ \text{Total man-days} = \text{Number of men} \times \text{Number of days} = 900 \times 42 \]

### Step 3: Calculate Total Man-Days
Now, we compute the total man-days:

\[ 900 \times 42 = 37800 \text{ man-days} \]

### Step 4: Determine the New Number of Men
Next, we need to account for the reinforcements. The total number of men after the reinforcement is:

\[ \text{Total men} = 900 + 500 = 1400 \text{ men} \]

### Step 5: Calculate the New Duration of Provisions
Now, we need to find out how many days the provisions will last for 1400 men. We can use the total man-days calculated earlier and divide it by the new number of men:

\[ \text{Number of days} = \frac{\text{Total man-days}}{\text{Total men}} = \frac{37800}{1400} \]

### Step 6: Perform the Calculation
Now we calculate the number of days:

\[ \frac{37800}{1400} = 27 \text{ days} \]

### Final Answer
The food will last for **27 days** for the 1400 men.

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