The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

Answer

To solve the problem, we need to set up equations based on the information given. Let's break it down step by step.

### Step 1: Define Variables
Let:
- \( x \) = number of students in each row
- \( y \) = number of rows

### Step 2: Formulate the First Equation
According to the problem, if 3 students are added to each row, there would be 1 row less. This can be expressed mathematically as:
\[ (x + 3)(y - 1) = xy \]
Expanding this gives:
\[ xy - x + 3y - 3 = xy \]
Cancelling \( xy \) from both sides, we get:
\[ -x + 3y - 3 = 0 \]
Rearranging this, we have:
\[ x - 3y + 3 = 0 \quad \text{(Equation 1)} \]

### Step 3: Formulate the Second Equation
Next, if 3 students are removed from each row, there would be 2 rows more. This can be expressed as:
\[ (x - 3)(y + 2) = xy \]
Expanding this gives:
\[ xy + 2x - 3y - 6 = xy \]
Cancelling \( xy \) from both sides, we get:
\[ 2x - 3y - 6 = 0 \]
Rearranging this, we have:
\[ 2x - 3y - 6 = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( x - 3y + 3 = 0 \)
2. \( 2x - 3y - 6 = 0 \)

We can solve these equations simultaneously. Let's solve Equation 1 for \( x \):
\[ x = 3y - 3 \]

Now, substitute this expression for \( x \) into Equation 2:
\[ 2(3y - 3) - 3y - 6 = 0 \]
Expanding this gives:
\[ 6y - 6 - 3y - 6 = 0 \]
Combining like terms results in:
\[ 3y - 12 = 0 \]
Solving for \( y \):
\[ 3y = 12 \quad \Rightarrow \quad y = 4 \]

### Step 5: Find \( x \)
Now substitute \( y = 4 \) back into the expression for \( x \):
\[ x = 3(4) - 3 = 12 - 3 = 9 \]

### Step 6: Calculate Total Number of Students
The total number of students in the class is given by:
\[ \text{Total Students} = x \times y = 9 \times 4 = 36 \]

### Final Answer
The number of students in the class is **36**.

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