What is the value of `2.bar(6)-1.bar(9)` ?

Options

A. `0.bar(6)`
B. `0.bar(9)`
C. `0.bar(7)`
D. `0.7`

Answer

To find the value of \(2.\overline{6} - 1.\overline{9}\), we will first convert the repeating decimals into fractions and then perform the subtraction.

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

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

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

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

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

### Step 2: Convert \(1.\overline{9}\) into a fraction.

Let \(y = 1.\overline{9}\).

Again, multiply both sides by 10:
\[ 10y = 19.\overline{9} \]

Subtract the original equation from this new equation:
\[ 10y - y = 19.\overline{9} - 1.\overline{9} \]
\[ 9y = 18 \]

Now, solve for \(y\):
\[ y = \frac{18}{9} = 2 \]

### Step 3: Perform the subtraction \(2.\overline{6} - 1.\overline{9}\).

Now we have:
\[ 2.\overline{6} - 1.\overline{9} = x - y = \frac{8}{3} - 2 \]

Convert \(2\) into a fraction with a common denominator:
\[ 2 = \frac{6}{3} \]

Now perform the subtraction:
\[ \frac{8}{3} - \frac{6}{3} = \frac{2}{3} \]

### Step 4: Convert \(\frac{2}{3}\) into decimal form.

To convert \(\frac{2}{3}\) into decimal:
\[ \frac{2}{3} = 0.6666...\text{ (which can be written as } 0.\overline{6}) \]

### Final Answer:
Thus, the value of \(2.\overline{6} - 1.\overline{9}\) is:
\[ \boxed{0.\overline{6}} \]

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