`2.bar(6)-0.bar(82)` is equal to

Options

A. `(181)/(999)`
B. `(182)/(99)`
C. `(82)/(99)`
D. None of these

Answer

To solve the expression \( 2.\overline{6} - 0.\overline{82} \), we will first convert the repeating decimals into fractions.

### Step 1: Convert \( 2.\overline{6} \) to a fraction

Let \( x = 2.\overline{6} \).

To eliminate the repeating part, we can multiply by 10:
\[ 10x = 26.\overline{6} \]

Now, subtract the original equation from this new equation:
\[ 10x - x = 26.\overline{6} - 2.\overline{6} \]
\[ 9x = 24 \]
\[ x = \frac{24}{9} = \frac{8}{3} \]

### Step 2: Convert \( 0.\overline{82} \) to a fraction

Let \( y = 0.\overline{82} \).

To eliminate the repeating part, we can multiply by 100:
\[ 100y = 82.\overline{82} \]

Now, subtract the original equation from this new equation:
\[ 100y - y = 82.\overline{82} - 0.\overline{82} \]
\[ 99y = 82 \]
\[ y = \frac{82}{99} \]

### Step 3: Subtract the two fractions

Now we need to calculate \( 2.\overline{6} - 0.\overline{82} \):
\[ x - y = \frac{8}{3} - \frac{82}{99} \]

To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 3 and 99 is 99.

Convert \( \frac{8}{3} \) to have a denominator of 99:
\[ \frac{8}{3} = \frac{8 \times 33}{3 \times 33} = \frac{264}{99} \]

Now we can subtract:
\[ \frac{264}{99} - \frac{82}{99} = \frac{264 - 82}{99} = \frac{182}{99} \]

### Final Answer

Thus, the result of \( 2.\overline{6} - 0.\overline{82} \) is:
\[ \frac{182}{99} \]