When 0.252525`"………."` is converted into a fraction, then find the result.

Options

A. `(25)/(99)`
B. `(25)/(90)`
C. `(25)/(999)`
D. `(25)/(9999)`

Answer

To convert the repeating decimal \(0.252525...\) into a fraction, we can follow these steps:

### Step 1: Define the repeating decimal as a variable
Let \( x = 0.252525...\)

### Step 2: Eliminate the repeating part
Since the repeating part "25" has two digits, we multiply both sides of the equation by \(100\) (which is \(10^2\)):
\[ 100x = 25.252525... \]

### Step 3: Set up the equation
Now we can express \(100x\) in terms of \(x\):
\[ 100x = 25 + 0.252525... \]
Notice that \(0.252525...\) is the same as \(x\):
\[ 100x = 25 + x \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate \(x\):
\[ 100x - x = 25 \]
\[ 99x = 25 \]

### Step 5: Solve for \(x\)
Now, divide both sides by \(99\):
\[ x = \frac{25}{99} \]

### Conclusion
Thus, the repeating decimal \(0.252525...\) can be expressed as the fraction:
\[ \frac{25}{99} \]

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