`2.bar(43)+3.bar(62)+3.bar(18)=?`

Options

A. `8.bar(24)`
B. `9.bar(24)`
C. `8.bar(23)`
D. None of these

Answer

To solve the expression \(2.\overline{43} + 3.\overline{62} + 3.\overline{18}\), we will first convert each repeating decimal into a fraction.

### Step-by-Step Solution:

1. **Convert \(2.\overline{43}\) to a fraction:**
- Let \(x = 2.\overline{43}\).
- Multiply by 100 (since "43" has 2 digits): \(100x = 243.\overline{43}\).
- Subtract the original equation from this:
\[ 100x - x = 243.\overline{43} - 2.\overline{43} \]
\[ 99x = 241 \]
- Therefore, \(x = \frac{241}{99}\).
- So, \(2.\overline{43} = 2 + \frac{241}{99} = \frac{198 + 241}{99} = \frac{439}{99}\).

2. **Convert \(3.\overline{62}\) to a fraction:**
- Let \(y = 3.\overline{62}\).
- Multiply by 100: \(100y = 362.\overline{62}\).
- Subtract the original equation:
\[ 100y - y = 362.\overline{62} - 3.\overline{62} \]
\[ 99y = 359 \]
- Therefore, \(y = \frac{359}{99}\).
- So, \(3.\overline{62} = 3 + \frac{359}{99} = \frac{297 + 359}{99} = \frac{656}{99}\).

3. **Convert \(3.\overline{18}\) to a fraction:**
- Let \(z = 3.\overline{18}\).
- Multiply by 100: \(100z = 318.\overline{18}\).
- Subtract the original equation:
\[ 100z - z = 318.\overline{18} - 3.\overline{18} \]
\[ 99z = 315 \]
- Therefore, \(z = \frac{315}{99}\).
- So, \(3.\overline{18} = 3 + \frac{315}{99} = \frac{297 + 315}{99} = \frac{612}{99}\).

4. **Now, add all three fractions:**
\[ 2.\overline{43} + 3.\overline{62} + 3.\overline{18} = \frac{439}{99} + \frac{656}{99} + \frac{612}{99} \]
- Combine the numerators:
\[ = \frac{439 + 656 + 612}{99} = \frac{1707}{99} \]

5. **Simplify the fraction if possible:**
- \(1707\) and \(99\) have a common factor of \(9\):
\[ \frac{1707 \div 9}{99 \div 9} = \frac{189}{11} \]

6. **Convert back to a mixed number if needed:**
- \(189 \div 11 = 17\) remainder \(2\), so:
\[ \frac{189}{11} = 17 \frac{2}{11} \]

### Final Answer:
\[ 2.\overline{43} + 3.\overline{62} + 3.\overline{18} = \frac{189}{11} \text{ or } 17 \frac{2}{11} \]