Express `3.overset(-)(2)overset(-)(5)overset(-)(3)` as a fractions in the form ` (x)/(y) ` where ` x, y in I and y ne 0 `

Answer

To express the repeating decimal \( 3.\overline{253} \) as a fraction in the form \( \frac{x}{y} \), where \( x \) and \( y \) are integers and \( y \neq 0 \), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 3.\overline{253} \).

### Step 2: Multiply by a power of 10
Since the repeating part "253" has 3 digits, multiply both sides of the equation by \( 1000 \) (which is \( 10^3 \)):
\[ 1000x = 3253.\overline{253} \]

### Step 3: Set up the equation
Now, we have two equations:
1. \( x = 3.\overline{253} \)
2. \( 1000x = 3253.\overline{253} \)

### Step 4: Subtract the first equation from the second
Subtract the first equation from the second:
\[ 1000x - x = 3253.\overline{253} - 3.\overline{253} \]
This simplifies to:
\[ 999x = 3253 - 3 \]

### Step 5: Simplify the right side
Calculate \( 3253 - 3 \):
\[ 999x = 3250 \]

### Step 6: Solve for \( x \)
Now, divide both sides by \( 999 \):
\[ x = \frac{3250}{999} \]

### Step 7: Final expression
Thus, the repeating decimal \( 3.\overline{253} \) can be expressed as the fraction:
\[ \frac{3250}{999} \]