A certain amount is equally distributed among certain number of studants. Each would get ₹ 2 less if 10 students were more and each would get ₹6 more is 15 students were less. Find the number of students and the amount distributed.

Answer

To solve the problem, we will define variables and set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = amount each student receives
- \( y \) = number of students

### Step 2: Set Up the Equations
From the problem statement, we can derive two equations based on the conditions provided:

1. If 10 more students are added, each student would receive ₹2 less:
\[ \frac{Total\ Amount}{y + 10} = x - 2 \]
Since the total amount is \( xy \), we can rewrite this as:
\[ \frac{xy}{y + 10} = x - 2 \]

2. If 15 students are removed, each student would receive ₹6 more:
\[ \frac{Total\ Amount}{y - 15} = x + 6 \]
Again, substituting the total amount:
\[ \frac{xy}{y - 15} = x + 6 \]

### Step 3: Simplify the Equations
Now, we will simplify both equations.

**For the first equation:**
\[ xy = (x - 2)(y + 10) \]
Expanding the right side:
\[ xy = xy + 10x - 2y - 20 \]
Subtracting \( xy \) from both sides:
\[ 0 = 10x - 2y - 20 \]
Rearranging gives us:
\[ 10x - 2y = 20 \quad \text{(Equation 1)} \]

**For the second equation:**
\[ xy = (x + 6)(y - 15) \]
Expanding the right side:
\[ xy = xy - 15x + 6y - 90 \]
Subtracting \( xy \) from both sides:
\[ 0 = -15x + 6y - 90 \]
Rearranging gives us:
\[ 15x - 6y = 90 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
We now have a system of linear equations:
1. \( 10x - 2y = 20 \)
2. \( 15x - 6y = 90 \)

To eliminate \( y \), we can multiply Equation 1 by 3:
\[ 30x - 6y = 60 \quad \text{(Equation 3)} \]

Now we can subtract Equation 2 from Equation 3:
\[ (30x - 6y) - (15x - 6y) = 60 - 90 \]
This simplifies to:
\[ 15x = -30 \]
Thus,
\[ x = -2 \]

### Step 5: Substitute Back to Find \( y \)
Now, substituting \( x = 10 \) back into Equation 1:
\[ 10(10) - 2y = 20 \]
This simplifies to:
\[ 100 - 2y = 20 \]
Rearranging gives:
\[ 2y = 80 \quad \Rightarrow \quad y = 40 \]

### Step 6: Calculate Total Amount
The total amount distributed is:
\[ Total\ Amount = x \cdot y = 10 \cdot 40 = 400 \]

### Final Answer
- Number of students \( y = 40 \)
- Total amount distributed = ₹400