Evaluate `0.bar(65) + 2.bar(61)`.

Options

A. `1.bar(31)`
B. `3.bar(27)`
C. `1.bar(21)`
D. `1.0bar1`

Answer

To evaluate the expression \(0.\overline{65} + 2.\overline{61}\), we will convert the repeating decimals into fractions and then perform the addition.

### Step 1: Convert \(0.\overline{65}\) to a fraction
Let \(x = 0.\overline{65}\).

To eliminate the repeating part, multiply both sides by 100 (since the repeat length is 2):
\[ 100x = 65.\overline{65} \]

Now, subtract the original \(x\) from this equation:
\[ 100x - x = 65.\overline{65} - 0.\overline{65} \]
\[ 99x = 65 \]

Now, solve for \(x\):
\[ x = \frac{65}{99} \]

### Step 2: Convert \(2.\overline{61}\) to a fraction
Let \(y = 2.\overline{61}\).

Again, multiply both sides by 100:
\[ 100y = 261.\overline{61} \]

Subtract the original \(y\):
\[ 100y - y = 261.\overline{61} - 2.\overline{61} \]
\[ 99y = 259 \]

Now, solve for \(y\):
\[ y = \frac{259}{99} \]

### Step 3: Add the two fractions
Now we can add \(0.\overline{65}\) and \(2.\overline{61}\):
\[ 0.\overline{65} + 2.\overline{61} = \frac{65}{99} + \frac{259}{99} \]

Since the denominators are the same, we can add the numerators:
\[ = \frac{65 + 259}{99} = \frac{324}{99} \]

### Step 4: Simplify the fraction
Now, simplify \(\frac{324}{99}\):
\[ \frac{324 \div 9}{99 \div 9} = \frac{36}{11} \]

### Final Answer
Thus, the value of \(0.\overline{65} + 2.\overline{61}\) is \(\frac{36}{11}\).

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