If 45 students can consume a stock of food in 2 months, find for how many days the same stock of food will last for 27 students.

Answer

To solve the problem step by step, we will use the concept of inverse proportionality between the number of students and the time taken to consume the food stock.

### Step 1: Understand the relationship
We know that if more students are consuming the food, it will last for a shorter time. Therefore, the number of students (N) and the time (T) are inversely proportional. This means:
\[ N_1 \times T_1 = N_2 \times T_2 \]
where \( N_1 \) and \( T_1 \) are the number of students and time for the first scenario, and \( N_2 \) and \( T_2 \) are for the second scenario.

### Step 2: Set up the known values
From the problem, we know:
- \( N_1 = 45 \) students
- \( T_1 = 2 \) months (which is equal to 60 days since 1 month is approximately 30 days)
- \( N_2 = 27 \) students
- \( T_2 = ? \) (this is what we need to find)

### Step 3: Write the equation
Using the relationship from Step 1, we can write:
\[ N_1 \times T_1 = N_2 \times T_2 \]
Substituting the known values:
\[ 45 \times 60 = 27 \times T_2 \]

### Step 4: Calculate \( T_2 \)
Now, we can solve for \( T_2 \):
\[ 2700 = 27 \times T_2 \]
To find \( T_2 \), divide both sides by 27:
\[ T_2 = \frac{2700}{27} \]

### Step 5: Simplify the calculation
Calculating \( T_2 \):
\[ T_2 = 100 \text{ days} \]

### Conclusion
The same stock of food will last for 27 students for **100 days**.

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