The value of `overline(23)` + 0.`overline(22)` is

Options

A. 0.45
B. `0.overline(45)`
C. `(45)/(99)`
D. both (b) and c

Answer

To solve the problem of finding the value of \( \overline{0.23} + \overline{0.22} \), we will follow these steps:

### Step 1: Convert \( \overline{0.23} \) to a Fraction
The repeating decimal \( 0.23\overline{23} \) can be expressed as a fraction.

Let \( x = 0.23\overline{23} \).

To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 23.23\overline{23} \]

Now, subtract the original \( x \) from this equation:
\[ 100x - x = 23.23\overline{23} - 0.23\overline{23} \]
\[ 99x = 23 \]
\[ x = \frac{23}{99} \]

### Step 2: Convert \( \overline{0.22} \) to a Fraction
Similarly, we convert \( 0.22\overline{22} \) to a fraction.

Let \( y = 0.22\overline{22} \).

Again, multiply \( y \) by 100:
\[ 100y = 22.22\overline{22} \]

Subtract the original \( y \):
\[ 100y - y = 22.22\overline{22} - 0.22\overline{22} \]
\[ 99y = 22 \]
\[ y = \frac{22}{99} \]

### Step 3: Add the Two Fractions
Now we add the two fractions we found:
\[ \overline{0.23} + \overline{0.22} = \frac{23}{99} + \frac{22}{99} \]
Since the denominators are the same, we can add the numerators:
\[ \frac{23 + 22}{99} = \frac{45}{99} \]

### Step 4: Simplify the Fraction
Next, we simplify \( \frac{45}{99} \):
Both 45 and 99 can be divided by 9:
\[ \frac{45 \div 9}{99 \div 9} = \frac{5}{11} \]

### Step 5: Convert to Decimal Form
To convert \( \frac{5}{11} \) back to decimal form, we can perform the division:
\[ 5 \div 11 = 0.454545\ldots = 0.\overline{45} \]

### Final Answer
Thus, the value of \( \overline{0.23} + \overline{0.22} \) is:
\[ \frac{5}{11} \text{ or } 0.\overline{45} \]