If 1.525252 ……is converted to a fraction then what is the sum of its numerator and denominator?

Answer

To convert the repeating decimal \(1.525252...\) into a fraction and find the sum of its numerator and denominator, we can follow these steps:

### Step 1: Define the repeating decimal
Let \(x = 1.525252...\)

### Step 2: Eliminate the decimal part
To eliminate the repeating decimal, we can multiply \(x\) by 100 (since the repeating part has 2 digits):
\[ 100x = 152.525252... \]

### Step 3: Set up the equations
Now we have two equations:
1. \(x = 1.525252...\) (Equation 1)
2. \(100x = 152.525252...\) (Equation 2)

### Step 4: Subtract the first equation from the second
Subtract Equation 1 from Equation 2:
\[ 100x - x = 152.525252... - 1.525252... \]
This simplifies to:
\[ 99x = 151 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 99 to solve for \(x\):
\[ x = \frac{151}{99} \]

### Step 6: Identify the numerator and denominator
Here, the numerator is 151 and the denominator is 99.

### Step 7: Find the sum of the numerator and denominator
Now, we need to find the sum of the numerator and denominator:
\[ 151 + 99 = 250 \]

### Final Answer
The sum of the numerator and denominator is \(250\).

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