Some money is to be distributed equally among children of a locality. If there are 8 children less, every one will get 10 more and if there are 16 children more, every one will get 10 less. What is the total amount of money to be distributed ?

Answer

To solve the problem step by step, we will define variables and set up equations based on the conditions provided in the question.

### Step 1: Define Variables
Let:
- \( x \) = total number of children
- \( y \) = amount of money each child receives

### Step 2: Write the Total Money Equation
The total amount of money to be distributed can be expressed as:
\[ \text{Total Money} = x \times y \]

### Step 3: Set Up the First Condition
According to the first condition, if there are 8 children less, each child will receive 10 more. Therefore:
- Number of children = \( x - 8 \)
- Amount each child receives = \( y + 10 \)

The total money in this case can be expressed as:
\[ (x - 8)(y + 10) = xy \]

### Step 4: Expand the First Condition Equation
Expanding the equation:
\[ xy + 10x - 8y - 80 = xy \]
Subtract \( xy \) from both sides:
\[ 10x - 8y - 80 = 0 \]
Rearranging gives us:
\[ 10x - 8y = 80 \quad \text{(Equation 1)} \]

### Step 5: Set Up the Second Condition
According to the second condition, if there are 16 children more, each child will receive 10 less. Therefore:
- Number of children = \( x + 16 \)
- Amount each child receives = \( y - 10 \)

The total money in this case can be expressed as:
\[ (x + 16)(y - 10) = xy \]

### Step 6: Expand the Second Condition Equation
Expanding the equation:
\[ xy - 10x + 16y - 160 = xy \]
Subtract \( xy \) from both sides:
\[ -10x + 16y - 160 = 0 \]
Rearranging gives us:
\[ -10x + 16y = 160 \quad \text{(Equation 2)} \]

### Step 7: Solve the System of Equations
Now we have two equations:
1. \( 10x - 8y = 80 \)
2. \( -10x + 16y = 160 \)

Adding these two equations:
\[ (10x - 8y) + (-10x + 16y) = 80 + 160 \]
This simplifies to:
\[ 8y = 240 \]
Dividing both sides by 8:
\[ y = 30 \]

### Step 8: Substitute \( y \) Back to Find \( x \)
Now substitute \( y = 30 \) back into Equation 1:
\[ 10x - 8(30) = 80 \]
This simplifies to:
\[ 10x - 240 = 80 \]
Adding 240 to both sides:
\[ 10x = 320 \]
Dividing by 10:
\[ x = 32 \]

### Step 9: Calculate Total Amount of Money
Now we can find the total amount of money:
\[ \text{Total Money} = x \times y = 32 \times 30 = 960 \]

### Final Answer
The total amount of money to be distributed is **960 rupees**.

---