The value of `0.bar(45)+ 0.bar(36)` is

Options

A. `(67)/(33)`
B. `(24)/(11)`
C. `(9)/(10)`
D. `(167)/(110)`

Answer

To solve the problem of finding the value of \(0.\overline{45} + 0.\overline{36}\), we will follow these steps:

### Step 1: Define the repeating decimals
Let:
- \( x = 0.\overline{45} \)
- \( y = 0.\overline{36} \)

### Step 2: Convert \( x \) to a fraction
To convert \( x \) into a fraction:
1. Multiply \( x \) by 100 (since the repeating part has 2 digits):
\[ 100x = 45.\overline{45} \]
2. Now, subtract \( x \) from \( 100x \):
\[ 100x - x = 45.\overline{45} - 0.\overline{45} \]
This simplifies to:
\[ 99x = 45 \]
3. Solving for \( x \):
\[ x = \frac{45}{99} \]
4. Simplifying \( \frac{45}{99} \):
\[ x = \frac{5}{11} \]

### Step 3: Convert \( y \) to a fraction
Now, let's convert \( y \) into a fraction:
1. Multiply \( y \) by 100:
\[ 100y = 36.\overline{36} \]
2. Subtract \( y \) from \( 100y \):
\[ 100y - y = 36.\overline{36} - 0.\overline{36} \]
This simplifies to:
\[ 99y = 36 \]
3. Solving for \( y \):
\[ y = \frac{36}{99} \]
4. Simplifying \( \frac{36}{99} \):
\[ y = \frac{4}{11} \]

### Step 4: Add \( x \) and \( y \)
Now we add \( x \) and \( y \):
\[ x + y = \frac{5}{11} + \frac{4}{11} = \frac{5 + 4}{11} = \frac{9}{11} \]

### Final Answer
The value of \( 0.\overline{45} + 0.\overline{36} \) is:
\[ \frac{9}{11} \]

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