Express `0.82bar(68)` as a vulgar fraction.

Options

A. `(3093)/(4950)`
B. `(3043)/(4850)`
C. `(4093)/(4950)`
D. `(3039)/(4950)`

Answer

To express \(0.82\overline{68}\) as a vulgar fraction, we can follow these steps:

### Step 1: Assign a variable
Let \( x = 0.82686868...\)

### Step 2: Multiply to eliminate the repeating part
To eliminate the repeating decimal, we can multiply \( x \) by \( 10000 \) (since the repeating part has 2 digits) and \( 100 \) (to shift the decimal point):
- \( 100x = 82.686868...\)
- \( 10000x = 8268.686868...\)

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 0.82686868...\) (Equation 1)
2. \( 10000x = 8268.686868...\) (Equation 2)

### Step 4: Subtract the equations
Now, subtract Equation 1 from Equation 2:
\[ 10000x - 100x = 8268.686868... - 82.686868... \]
This simplifies to:
\[ 9900x = 8268 - 82 \]
\[ 9900x = 8186 \]

### Step 5: Solve for \( x \)
Now, divide both sides by \( 9900 \):
\[ x = \frac{8186}{9900} \]

### Step 6: Simplify the fraction
Now we need to simplify \( \frac{8186}{9900} \). We can find the greatest common divisor (GCD) of 8186 and 9900, which is 2:
\[ x = \frac{8186 \div 2}{9900 \div 2} = \frac{4093}{4950} \]

### Final Answer
Thus, \( 0.82\overline{68} \) as a vulgar fraction is:
\[ \frac{4093}{4950} \]

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