If the repeating decimal `0.237bar(37)…` is written as a fraction in lowest terms, the sum of the numerator and denominator is

Options

A. 16
B. 47
C. 245
D. 334

Explanation

To convert the repeating decimal \(0.237373737...\) into a fraction and find the sum of the numerator and denominator, we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.237373737...\)

### Step 2: Isolate the repeating part
To eliminate the repeating part, we first multiply \( x \) by \( 1000 \) (since the repeating part has 2 digits):
\[ 1000x = 237.373737... \]

### Step 3: Multiply to isolate the repeating decimal
Next, we multiply \( x \) by \( 10 \) to shift the decimal point:
\[ 10x = 2.373737... \]

### Step 4: Set up the equation
Now we can set up the equation using the two expressions we have:
\[ 1000x - 10x = 237.373737... - 2.373737... \]
This simplifies to:
\[ 990x = 235 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{235}{990} \]

### Step 6: Simplify the fraction
Next, we simplify the fraction \( \frac{235}{990} \). We can find the greatest common divisor (GCD) of 235 and 990, which is 5:
\[ \frac{235 \div 5}{990 \div 5} = \frac{47}{198} \]

### Step 7: Find the sum of the numerator and denominator
Now, we find the sum of the numerator and denominator:
\[ 47 + 198 = 245 \]

### Final Answer
Thus, the sum of the numerator and denominator is \( \boxed{245} \).

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