The sum of `0.bar(6)` and `0.bar(7)` is

Options

A. `1.bar(3)`
B. `1.3`
C. `1.bar(4)`
D. an irrational number

Answer

To find the sum of \(0.\overline{6}\) and \(0.\overline{7}\), we can follow these steps:

### Step 1: Define the repeating decimals
Let:
- \(x = 0.\overline{6}\)
- \(y = 0.\overline{7}\)

### Step 2: Convert \(x\) into a fraction
To convert \(x = 0.\overline{6}\) into a fraction:
1. Multiply \(x\) by 10:
\[ 10x = 6.\overline{6} \]
2. Subtract the original \(x\) from this equation:
\[ 10x - x = 6.\overline{6} - 0.\overline{6} \]
This simplifies to:
\[ 9x = 6 \]
3. Solve for \(x\):
\[ x = \frac{6}{9} = \frac{2}{3} \]

### Step 3: Convert \(y\) into a fraction
Now, convert \(y = 0.\overline{7}\) into a fraction:
1. Multiply \(y\) by 10:
\[ 10y = 7.\overline{7} \]
2. Subtract the original \(y\) from this equation:
\[ 10y - y = 7.\overline{7} - 0.\overline{7} \]
This simplifies to:
\[ 9y = 7 \]
3. Solve for \(y\):
\[ y = \frac{7}{9} \]

### Step 4: Add the two fractions
Now, we need to find the sum \(x + y\):
\[ x + y = \frac{2}{3} + \frac{7}{9} \]

To add these fractions, we need a common denominator. The least common multiple of 3 and 9 is 9. So we convert \(\frac{2}{3}\) to a fraction with a denominator of 9:
\[ \frac{2}{3} = \frac{2 \times 3}{3 \times 3} = \frac{6}{9} \]

Now we can add:
\[ x + y = \frac{6}{9} + \frac{7}{9} = \frac{6 + 7}{9} = \frac{13}{9} \]

### Step 5: Convert the result back to a decimal
To express \(\frac{13}{9}\) as a decimal:
1. Divide 13 by 9:
\[ 13 \div 9 = 1.4444\ldots \]
This is \(1.\overline{4}\).

### Final Answer
Thus, the sum of \(0.\overline{6}\) and \(0.\overline{7}\) is:
\[ \boxed{1.\overline{4}} \]