The value of x in the following equation is :
`0.overset(*)3+0.overset(*)6+0.overset(*)7+0.overset(*)=x`

Options

A. `5.3`
B. `2(3)/(10)`
C. `2(2)/(3)`
D. `2.overset(*). overset(*)5`

Answer

To solve the equation \( 0.\overline{3} + 0.\overline{6} + 0.\overline{7} + 0.\overline{0} = x \), we will first convert the repeating decimals into fractions.

### Step 1: Convert \( 0.\overline{3} \) to a fraction
Let \( y = 0.\overline{3} \).
Then, \( 10y = 3.\overline{3} \).
Subtracting the first equation from the second gives:
\[ 10y - y = 3.\overline{3} - 0.\overline{3} \]
\[ 9y = 3 \implies y = \frac{3}{9} = \frac{1}{3} \]
So, \( 0.\overline{3} = \frac{1}{3} \).

### Step 2: Convert \( 0.\overline{6} \) to a fraction
Let \( z = 0.\overline{6} \).
Then, \( 10z = 6.\overline{6} \).
Subtracting gives:
\[ 10z - z = 6.\overline{6} - 0.\overline{6} \]
\[ 9z = 6 \implies z = \frac{6}{9} = \frac{2}{3} \]
So, \( 0.\overline{6} = \frac{2}{3} \).

### Step 3: Convert \( 0.\overline{7} \) to a fraction
Let \( w = 0.\overline{7} \).
Then, \( 10w = 7.\overline{7} \).
Subtracting gives:
\[ 10w - w = 7.\overline{7} - 0.\overline{7} \]
\[ 9w = 7 \implies w = \frac{7}{9} \]
So, \( 0.\overline{7} = \frac{7}{9} \).

### Step 4: Convert \( 0.\overline{0} \) to a fraction
Since \( 0.\overline{0} = 0 \), we can represent it as \( 0 = 0 \).

### Step 5: Add the fractions
Now we have:
\[ x = 0.\overline{3} + 0.\overline{6} + 0.\overline{7} + 0.\overline{0} = \frac{1}{3} + \frac{2}{3} + \frac{7}{9} + 0 \]
First, we add \( \frac{1}{3} + \frac{2}{3} \):
\[ \frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1 \]
Now we add \( 1 + \frac{7}{9} \):
To add these, we convert \( 1 \) into a fraction with a denominator of \( 9 \):
\[ 1 = \frac{9}{9} \]
So,
\[ x = \frac{9}{9} + \frac{7}{9} = \frac{16}{9} \]

### Step 6: Convert \( \frac{16}{9} \) to a decimal
To convert \( \frac{16}{9} \) to a decimal, we divide \( 16 \) by \( 9 \):
\[ 16 \div 9 = 1.777\ldots = 1.\overline{7} \]
Thus, \( x = \frac{16}{9} \) or approximately \( 1.777\ldots \).

### Final Answer
The value of \( x \) is \( \frac{16}{9} \) or approximately \( 1.777\ldots \).