`2.8bar(768)` expressed as a rational number is

Options

A. ` 2 878/999`
B. `2 9/10`
C. `2 292/333`
D. `2 (4394)/(4995)`

Answer

To express the number \( 2.8\overline{768} \) as a rational number, we can follow these steps:

### Step 1: Define the number
Let \( x = 2.876876876... \) (where \( 768 \) repeats indefinitely).

### Step 2: Eliminate the repeating decimal
To eliminate the repeating part, we can multiply \( x \) by \( 1000 \) (since \( 768 \) has three digits):
\[ 1000x = 2876.876876876... \]

### Step 3: Set up another equation
Next, we can multiply \( x \) by \( 10 \) to shift the decimal point:
\[ 10x = 28.768768768... \]

### Step 4: Subtract the two equations
Now we will subtract the second equation from the first:
\[ 1000x - 10x = 2876.876876876... - 28.768768768... \]
This simplifies to:
\[ 990x = 2848.108108108... \]

### Step 5: Solve for \( x \)
Now we can solve for \( x \):
\[ 990x = 2848.108108108... \]
To isolate \( x \), we divide both sides by \( 990 \):
\[ x = \frac{2848.108108108...}{990} \]

### Step 6: Convert the decimal to a fraction
To convert \( 2848.108108108... \) to a fraction, we can express it as:
\[ 2848 + \frac{108}{990} \]
Now, simplify \( \frac{108}{990} \):
\[ \frac{108 \div 9}{990 \div 9} = \frac{12}{110} = \frac{6}{55} \]
Thus, we have:
\[ x = 2848 + \frac{6}{55} \]

### Step 7: Combine into a single fraction
To express \( 2848 \) as a fraction:
\[ 2848 = \frac{2848 \times 55}{55} = \frac{156640}{55} \]
Now, combine:
\[ x = \frac{156640 + 6}{55} = \frac{156646}{55} \]

### Step 8: Final simplification
Now we can check if \( \frac{156646}{55} \) can be simplified further. Since \( 156646 \) and \( 55 \) have no common factors, this is the final rational number representation.

### Final Answer
Thus, \( 2.8\overline{768} \) expressed as a rational number is:
\[ \frac{156646}{55} \]

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