The value of `(0.44bar(67) +0.14 bar(44))` is :

Options

A. `0.59`
B. `0.59bar(12)`
C. `0.bar(59)`
D. `0.5bar(9)`

Answer

To solve the expression \( (0.44\overline{67} + 0.14\overline{44}) \), we need to convert each repeating decimal into a fraction and then add them together. Let's go through the steps:

### Step 1: Convert \( 0.44\overline{67} \) to a fraction

Let \( y = 0.44\overline{67} \).

1. **Multiply by 100** to shift the decimal point two places to the right:
\[ 100y = 44.67676767\ldots \]

2. **Multiply by 10,000** to shift the decimal point four places to the right:
\[ 10,000y = 4467.67676767\ldots \]

3. **Subtract the first equation from the second**:
\[ 10,000y - 100y = 4467.67676767\ldots - 44.67676767\ldots \]
This simplifies to:
\[ 9900y = 4467 - 44 \]
\[ 9900y = 4423 \]
\[ y = \frac{4423}{9900} \]

### Step 2: Convert \( 0.14\overline{44} \) to a fraction

Let \( x = 0.14\overline{44} \).

1. **Multiply by 100**:
\[ 100x = 14.444444\ldots \]

2. **Multiply by 10,000**:
\[ 10,000x = 1444.444444\ldots \]

3. **Subtract the first equation from the second**:
\[ 10,000x - 100x = 1444.444444\ldots - 14.444444\ldots \]
This simplifies to:
\[ 9900x = 1444 - 14 \]
\[ 9900x = 1430 \]
\[ x = \frac{1430}{9900} \]

### Step 3: Add the two fractions

Now we need to add \( y \) and \( x \):
\[ y + x = \frac{4423}{9900} + \frac{1430}{9900} = \frac{4423 + 1430}{9900} = \frac{5853}{9900} \]

### Step 4: Simplify the fraction

To simplify \( \frac{5853}{9900} \), we can find the greatest common divisor (GCD) of 5853 and 9900.

After finding the GCD, we can simplify:
\[ \frac{5853 \div 3}{9900 \div 3} = \frac{1951}{3300} \]

### Step 5: Convert to decimal

Now we can convert \( \frac{1951}{3300} \) to a decimal:
\[ 1951 \div 3300 = 0.591212121\ldots \]

Since \( 12 \) is repeating, we can express this as:
\[ 0.5912\overline{12} \]

### Final Answer

The value of \( (0.44\overline{67} + 0.14\overline{44}) \) is:
\[ \boxed{0.5912\overline{12}} \]