`2.6bar1+9.2bar4+10.6bar3=?`

Options

A. `22.bar(48)`
B. `21.bar(48)`
C. `22.4bar(8)`
D. None of these

Answer

To solve the problem \(2.6\overline{1} + 9.2\overline{4} + 10.6\overline{3}\), we need to convert each repeating decimal into a fraction or a non-repeating decimal.

### Step 1: Convert \(2.6\overline{1}\) to a fraction
Let \(x = 2.6\overline{1}\).

Multiply by 10 to shift the decimal:
\[ 10x = 26.1\overline{1} \]

Now, subtract the original \(x\) from this equation:
\[ 10x - x = 26.1\overline{1} - 2.6\overline{1} \]
\[ 9x = 23.5 \]
\[ x = \frac{23.5}{9} \]

To convert \(23.5\) to a fraction:
\[ 23.5 = \frac{235}{10} = \frac{47}{2} \]
So,
\[ x = \frac{47/2}{9} = \frac{47}{18} \]

### Step 2: Convert \(9.2\overline{4}\) to a fraction
Let \(y = 9.2\overline{4}\).

Multiply by 10:
\[ 10y = 92.4\overline{4} \]

Subtract the original \(y\):
\[ 10y - y = 92.4\overline{4} - 9.2\overline{4} \]
\[ 9y = 83.2 \]
\[ y = \frac{83.2}{9} \]

Convert \(83.2\) to a fraction:
\[ 83.2 = \frac{832}{10} = \frac{416}{5} \]
So,
\[ y = \frac{416/5}{9} = \frac{416}{45} \]

### Step 3: Convert \(10.6\overline{3}\) to a fraction
Let \(z = 10.6\overline{3}\).

Multiply by 10:
\[ 10z = 106.3\overline{3} \]

Subtract the original \(z\):
\[ 10z - z = 106.3\overline{3} - 10.6\overline{3} \]
\[ 9z = 95.7 \]
\[ z = \frac{95.7}{9} \]

Convert \(95.7\) to a fraction:
\[ 95.7 = \frac{957}{10} = \frac{9570}{100} \]
So,
\[ z = \frac{957/10}{9} = \frac{957}{90} \]

### Step 4: Add the fractions
Now we have:
\[ x = \frac{47}{18}, \quad y = \frac{416}{45}, \quad z = \frac{957}{90} \]

To add these fractions, we need a common denominator. The least common multiple of \(18\), \(45\), and \(90\) is \(90\).

Convert each fraction:
\[ x = \frac{47 \times 5}{90} = \frac{235}{90} \]
\[ y = \frac{416 \times 2}{90} = \frac{832}{90} \]
\[ z = \frac{957}{90} \]

Now, add them:
\[ \frac{235}{90} + \frac{832}{90} + \frac{957}{90} = \frac{235 + 832 + 957}{90} = \frac{2024}{90} \]

### Step 5: Simplify the fraction
Now simplify \(\frac{2024}{90}\):
\[ 2024 \div 2 = 1012, \quad 90 \div 2 = 45 \]
So,
\[ \frac{2024}{90} = \frac{1012}{45} \]

### Step 6: Convert to decimal
Now, convert \(\frac{1012}{45}\) to a decimal:
\[ 1012 \div 45 \approx 22.48 \]

Thus, the final answer is:
\[ \boxed{22.48\overline{8}} \]