Express `0.bar(34) + 0.3bar(4)` as a single decimal.

Options

A. `0.67bar(88)`
B. `0.6bar(89)`
C. `0.68bar(78)`
D. `0.6bar(87)`

Explanation

To express \(0.\overline{34} + 0.3\overline{4}\) as a single decimal, we will follow these steps:

### Step 1: Convert \(0.\overline{34}\) to a fraction
Let \(x = 0.\overline{34}\).

To eliminate the repeating decimal, multiply both sides by 100 (since the repeating part has 2 digits):
\[ 100x = 34.\overline{34} \]
Now, subtract the original \(x\) from this equation:
\[ 100x - x = 34.\overline{34} - 0.\overline{34} \]
This simplifies to:
\[ 99x = 34 \]
So,
\[ x = \frac{34}{99} \]

### Step 2: Convert \(0.3\overline{4}\) to a fraction
Let \(y = 0.3\overline{4}\).

To eliminate the repeating decimal, multiply both sides by 10 (since the repeating part has 1 digit):
\[ 10y = 3.4\overline{4} \]
Now, subtract the original \(y\) from this equation:
\[ 10y - y = 3.4\overline{4} - 0.3\overline{4} \]
This simplifies to:
\[ 9y = 3.1 \]
So,
\[ y = \frac{31}{90} \]

### Step 3: Add the two fractions
Now we need to add \(x\) and \(y\):
\[ x + y = \frac{34}{99} + \frac{31}{90} \]

To add these fractions, we need a common denominator. The least common multiple of 99 and 90 is 990.

Convert each fraction:
\[ \frac{34}{99} = \frac{34 \times 10}{99 \times 10} = \frac{340}{990} \]
\[ \frac{31}{90} = \frac{31 \times 11}{90 \times 11} = \frac{341}{990} \]

Now add them:
\[ \frac{340}{990} + \frac{341}{990} = \frac{681}{990} \]

### Step 4: Simplify the fraction
Now we simplify \(\frac{681}{990}\). The greatest common divisor (GCD) of 681 and 990 is 9.
\[ \frac{681 \div 9}{990 \div 9} = \frac{75.6667}{110} = \frac{75.6667}{110} \]

### Step 5: Convert to decimal
Now, we convert \(\frac{681}{990}\) to a decimal:
\[ 681 \div 990 \approx 0.6878787878... \]
This can be expressed as:
\[ 0.687\overline{87} \]

### Final Answer
Thus, \(0.\overline{34} + 0.3\overline{4} = 0.687\overline{87}\).