Find the value of `2.bar(6)-0.bar(9)` .

Answer

To find the value of \(2.\overline{6} - 0.\overline{9}\), we need to first convert these repeating decimals into fractions.

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

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

This means:
\[ x = 2.66666\ldots \]

To eliminate the repeating part, we can multiply \(x\) by 10:
\[ 10x = 26.66666\ldots \]

Now, we can set up an equation:
\[ 10x = 26.66666\ldots \]
\[ x = 2.66666\ldots \]

Subtract the second equation from the first:
\[ 10x - x = 26.66666\ldots - 2.66666\ldots \]
\[ 9x = 24 \]

Now, solve for \(x\):
\[ x = \frac{24}{9} = \frac{8}{3} \]

### Step 2: Convert \(0.\overline{9}\) into a fraction

Let \(y = 0.\overline{9}\).

This means:
\[ y = 0.99999\ldots \]

To eliminate the repeating part, we can multiply \(y\) by 10:
\[ 10y = 9.99999\ldots \]

Now, we can set up an equation:
\[ 10y = 9.99999\ldots \]
\[ y = 0.99999\ldots \]

Subtract the second equation from the first:
\[ 10y - y = 9.99999\ldots - 0.99999\ldots \]
\[ 9y = 9 \]

Now, solve for \(y\):
\[ y = 1 \]

### Step 3: Subtract the two fractions

Now we have:
\[ 2.\overline{6} = \frac{8}{3} \quad \text{and} \quad 0.\overline{9} = 1 \]

So we need to calculate:
\[ \frac{8}{3} - 1 \]

Convert \(1\) into a fraction with a common denominator:
\[ 1 = \frac{3}{3} \]

Now perform the subtraction:
\[ \frac{8}{3} - \frac{3}{3} = \frac{8 - 3}{3} = \frac{5}{3} \]

### Final Answer:
The value of \(2.\overline{6} - 0.\overline{9}\) is \(\frac{5}{3}\).