`8.overset(*)(31)+0.overset(*)(6)+0.00overset(*)(2)` is equal to :

Options

A. `8.overset(*)9 overset(*)1 overset(*)2`
B. `8.9 overset(*)1 overset(*)2`
C. `8.9 overset(*)7 overset(*)9`
D. `8.9 overset(*)7 overset(*)9`

Answer

To solve the expression \( 8.\overline{31} + 0.\overline{6} + 0.00\overline{2} \), we will first convert each repeating decimal into a fraction.

### Step 1: Convert \( 8.\overline{31} \) to a fraction
Let \( x = 8.\overline{31} \).

1. Multiply both sides by 100 (since the repeating part has 2 digits):
\[ 100x = 831.\overline{31} \]
2. Now, subtract the original \( x \) from this equation:
\[ 100x - x = 831.\overline{31} - 8.\overline{31} \]
\[ 99x = 823 \]
3. Solve for \( x \):
\[ x = \frac{823}{99} \]

### Step 2: Convert \( 0.\overline{6} \) to a fraction
Let \( y = 0.\overline{6} \).

1. Multiply both sides by 10:
\[ 10y = 6.\overline{6} \]
2. Subtract the original \( y \):
\[ 10y - y = 6.\overline{6} - 0.\overline{6} \]
\[ 9y = 6 \]
3. Solve for \( y \):
\[ y = \frac{6}{9} = \frac{2}{3} \]

### Step 3: Convert \( 0.00\overline{2} \) to a fraction
Let \( z = 0.00\overline{2} \).

1. Multiply both sides by 1000:
\[ 1000z = 2.\overline{2} \]
2. Multiply both sides by 10:
\[ 10z = 0.02\overline{2} \]
3. Subtract the second equation from the first:
\[ 1000z - 10z = 2.\overline{2} - 0.02\overline{2} \]
\[ 990z = 2.2 \]
4. Solve for \( z \):
\[ z = \frac{2.2}{990} = \frac{22}{9900} = \frac{1}{450} \]

### Step 4: Combine all fractions
Now we need to add \( \frac{823}{99} + \frac{2}{3} + \frac{1}{450} \).

1. Find a common denominator. The least common multiple of \( 99, 3, \) and \( 450 \) is \( 9900 \).
2. Convert each fraction:
- For \( \frac{823}{99} \):
\[ \frac{823 \times 100}{99 \times 100} = \frac{82300}{9900} \]
- For \( \frac{2}{3} \):
\[ \frac{2 \times 3300}{3 \times 3300} = \frac{6600}{9900} \]
- For \( \frac{1}{450} \):
\[ \frac{1 \times 22}{450 \times 22} = \frac{22}{9900} \]

3. Now add them together:
\[ \frac{82300 + 6600 + 22}{9900} = \frac{88922}{9900} \]

### Step 5: Simplify the result
Now we can simplify \( \frac{88922}{9900} \) if possible.

1. Check for common factors:
- The GCD of \( 88922 \) and \( 9900 \) is \( 2 \).
- Divide both numerator and denominator by \( 2 \):
\[ \frac{88922 \div 2}{9900 \div 2} = \frac{44461}{4950} \]

### Final Answer
The final result is:
\[ \frac{44461}{4950} \]