Express ` 53overset(-)( 629)` as a fractions in the form ` (x) / ( y) ` where x, y ,` in 1 and y ne 0`

Answer

To express \( 53.\overline{629} \) as a fraction in the form \( \frac{x}{y} \), we will follow these steps:

### Step 1: Let \( x = 53.\overline{629} \)
We start by defining the repeating decimal as a variable:
\[ x = 53.629629629\ldots \]

### Step 2: Multiply both sides by 1000
Since the repeating part "629" has 3 digits, we will multiply both sides by \( 1000 \) to shift the decimal point three places to the right:
\[ 1000x = 53629.629629629\ldots \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 53.629629629\ldots \) (Equation 1)
2. \( 1000x = 53629.629629629\ldots \) (Equation 2)

### Step 4: Subtract Equation 1 from Equation 2
We will subtract Equation 1 from Equation 2 to eliminate the repeating part:
\[ 1000x - x = 53629.629629629\ldots - 53.629629629\ldots \]
This simplifies to:
\[ 999x = 53629 - 53 \]

### Step 5: Calculate the right-hand side
Now we calculate \( 53629 - 53 \):
\[ 53629 - 53 = 53576 \]
So we have:
\[ 999x = 53576 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by \( 999 \):
\[ x = \frac{53576}{999} \]

### Final Result
Thus, the expression \( 53.\overline{629} \) as a fraction is:
\[ \frac{53576}{999} \]