The decimal fraction `2.3bar(49)` is equal to

Options

A. `2326/999`
B. `2326/990`
C. `2347/999`
D. `2347/990`

Answer

To convert the decimal fraction \(2.3\overline{49}\) into a fraction, we can follow these steps:

### Step 1: Let \( x = 2.3\overline{49} \)
We start by letting \( x \) represent the decimal fraction we want to convert.

### Step 2: Multiply by 10 to shift the decimal point
Since there is one digit (3) before the repeating part (49), we multiply \( x \) by 10:
\[ 10x = 23.4\overline{49} \]

### Step 3: Multiply by 1000 to shift the repeating part
Next, we need to eliminate the repeating decimal. Since "49" has 2 digits, we multiply by 1000:
\[ 1000x = 2349.\overline{49} \]

### Step 4: Set up the equation
Now we have two equations:
1. \( 10x = 23.4\overline{49} \)
2. \( 1000x = 2349.\overline{49} \)

### Step 5: Subtract the first equation from the second
Subtract the first equation from the second to eliminate the repeating part:
\[ 1000x - 10x = 2349.\overline{49} - 23.4\overline{49} \]
This simplifies to:
\[ 990x = 2349 - 23.4 \]

### Step 6: Calculate the right side
Now, we calculate \( 2349 - 23.4 \):
\[ 2349 - 23.4 = 2325.6 \]
So we have:
\[ 990x = 2325.6 \]

### Step 7: Solve for \( x \)
Now, divide both sides by 990:
\[ x = \frac{2325.6}{990} \]

### Step 8: Convert the decimal to a fraction
To convert \( 2325.6 \) to a fraction, we can express it as:
\[ 2325.6 = \frac{23256}{10} \]
Thus:
\[ x = \frac{23256}{10 \times 990} = \frac{23256}{9900} \]

### Step 9: Simplify the fraction
Now we simplify \( \frac{23256}{9900} \). We can find the greatest common divisor (GCD) of 23256 and 9900, which is 18. Dividing both the numerator and denominator by 18 gives:
\[ x = \frac{1292}{550} \]

### Final Answer
Thus, the decimal fraction \( 2.3\overline{49} \) is equal to:
\[ \frac{1292}{550} \]