Value of `0.bar(76) + 0. bar (34)` is

Options

A. `(110)/(99)`
B. `(100)/(99)`
C. `(90)/(99)`
D. `(42)/(99)`

Answer

To solve the problem of finding the value of \( 0.\overline{76} + 0.\overline{34} \), we can follow these steps:

### Step 1: Define the repeating decimals
Let:
- \( x = 0.\overline{76} \)
- \( y = 0.\overline{34} \)

### Step 2: Convert \( x \) to a fraction
To convert \( x = 0.\overline{76} \) into a fraction:
1. Multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 76.\overline{76} \]
2. Now, subtract \( x \) from \( 100x \):
\[ 100x - x = 76.\overline{76} - 0.\overline{76} \]
This simplifies to:
\[ 99x = 76 \]
3. Solve for \( x \):
\[ x = \frac{76}{99} \]

### Step 3: Convert \( y \) to a fraction
Now, convert \( y = 0.\overline{34} \) into a fraction:
1. Multiply \( y \) by 100:
\[ 100y = 34.\overline{34} \]
2. Subtract \( y \) from \( 100y \):
\[ 100y - y = 34.\overline{34} - 0.\overline{34} \]
This simplifies to:
\[ 99y = 34 \]
3. Solve for \( y \):
\[ y = \frac{34}{99} \]

### Step 4: Add the two fractions
Now we add \( x \) and \( y \):
\[ x + y = \frac{76}{99} + \frac{34}{99} \]
Since both fractions have the same denominator, we can combine them:
\[ x + y = \frac{76 + 34}{99} = \frac{110}{99} \]

### Step 5: Simplify the fraction
To simplify \( \frac{110}{99} \):
1. Find the greatest common divisor (GCD) of 110 and 99, which is 11.
2. Divide both the numerator and the denominator by 11:
\[ \frac{110 \div 11}{99 \div 11} = \frac{10}{9} \]

### Final Answer
Thus, the value of \( 0.\overline{76} + 0.\overline{34} \) is:
\[ \frac{10}{9} \]