What is the value of 0.007 `+ 0.00bar( 7 ) + 17. bar( 83) + 310.020bar(2)` ?

Options

A. 327.86638
B. 328.644
C. 327.86683
D. `327.866bar(8)`

Answer

To solve the question \( 0.007 + 0.00\overline{7} + 17.\overline{83} + 310.0\overline{2} \), we will convert each term into a fraction or a decimal and then sum them up.

### Step 1: Convert \( 0.007 \) to a decimal
This term is already in decimal form:
\[ 0.007 \]

### Step 2: Convert \( 0.00\overline{7} \) to a fraction
Let \( x = 0.00\overline{7} \). Then,
\[ 1000x = 7.\overline{7} \]
Subtracting the original equation from this gives:
\[ 1000x - x = 7.\overline{7} - 0.00\overline{7} \implies 999x = 7 \implies x = \frac{7}{999} \]
This fraction can be simplified:
\[ x = \frac{7}{999} = \frac{1}{143} \quad (\text{since } 7 \text{ and } 999 \text{ have a GCD of } 7) \]

### Step 3: Convert \( 17.\overline{83} \) to a fraction
Let \( y = 17.\overline{83} \). Then,
\[ 100y = 1783.\overline{83} \]
Subtracting the original equation gives:
\[ 100y - y = 1783.\overline{83} - 17.\overline{83} \implies 99y = 1766 \implies y = \frac{1766}{99} \]
This fraction can be simplified:
\[ y = 17 + \frac{83}{99} = 17 \frac{83}{99} \]

### Step 4: Convert \( 310.0\overline{2} \) to a fraction
Let \( z = 310.0\overline{2} \). Then,
\[ 1000z = 3102.\overline{2} \]
Subtracting the original equation gives:
\[ 1000z - z = 3102.\overline{2} - 310.0\overline{2} \implies 999z = 2792 \implies z = \frac{2792}{999} \]
This fraction can be simplified:
\[ z = 310 + \frac{2}{900} \quad (\text{since } 2792 \text{ and } 999 \text{ have a GCD of } 1) \]

### Step 5: Sum all the values
Now we will sum all the values:
\[ 0.007 + \frac{1}{143} + 17 \frac{83}{99} + 310 + \frac{2}{900} \]

Converting \( 0.007 \) to a fraction:
\[ 0.007 = \frac{7}{1000} \]

Now, we can find a common denominator for all fractions. The least common multiple of \( 1000, 143, 99, \) and \( 900 \) is \( 9000 \).

Converting each fraction to have a denominator of \( 9000 \):
1. \( 0.007 = \frac{7 \times 9}{1000 \times 9} = \frac{63}{9000} \)
2. \( \frac{1}{143} = \frac{1 \times 63}{143 \times 63} = \frac{63}{9000} \)
3. \( 17 \frac{83}{99} = \frac{1766}{99} = \frac{1766 \times 90}{99 \times 90} = \frac{158940}{9000} \)
4. \( 310 + \frac{2}{900} = \frac{310 \times 10}{9000} + \frac{20}{9000} = \frac{3100 + 20}{9000} = \frac{3120}{9000} \)

Now we can sum them:
\[ \frac{63 + 63 + 158940 + 3120}{9000} = \frac{162186}{9000} \]

### Final Answer
The value of \( 0.007 + 0.00\overline{7} + 17.\overline{83} + 310.0\overline{2} \) is:
\[ \frac{162186}{9000} \approx 18.021 \]