Students of a class are made to stand in rows. If 4 students are extra in each row, then there would be 2 rows less. If four students are less in each row, then there would be 4 more rows. What is the number of students in the class?

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information given.

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

### Step 2: Set Up the First Equation
According to the problem, if 4 students are added to each row, there would be 2 rows less. This can be expressed as:
\[ (x + 4)(y - 2) = xy \]
Expanding this equation gives:
\[ xy - 2x + 4y - 8 = xy \]
Subtract \( xy \) from both sides:
\[ -2x + 4y - 8 = 0 \]
Rearranging gives us:
\[ 4y - 2x = 8 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
Next, if 4 students are removed from each row, there would be 4 more rows. This can be expressed as:
\[ (x - 4)(y + 4) = xy \]
Expanding this equation gives:
\[ xy + 4x - 4y - 16 = xy \]
Subtract \( xy \) from both sides:
\[ 4x - 4y - 16 = 0 \]
Rearranging gives us:
\[ 4x - 4y = 16 \quad \text{(Equation 2)} \]

### Step 4: Simplify Equation 2
We can simplify Equation 2 by dividing everything by 4:
\[ x - y = 4 \quad \text{(Equation 2 simplified)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( 4y - 2x = 8 \)
2. \( x - y = 4 \)

From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = y + 4 \]

### Step 6: Substitute into Equation 1
Substituting \( x = y + 4 \) into Equation 1:
\[ 4y - 2(y + 4) = 8 \]
Expanding this gives:
\[ 4y - 2y - 8 = 8 \]
Combining like terms:
\[ 2y - 8 = 8 \]
Adding 8 to both sides:
\[ 2y = 16 \]
Dividing by 2:
\[ y = 8 \]

### Step 7: Find \( x \)
Now substituting \( y = 8 \) back into the equation \( x = y + 4 \):
\[ x = 8 + 4 = 12 \]

### Step 8: Calculate Total Number of Students
The total number of students in the class is given by:
\[ \text{Total Students} = x \times y = 12 \times 8 = 96 \]

### Final Answer
The total number of students in the class is **96**.

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