Evalaute `0.bar14+ 0. bar19`

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. 2
D. 10

Answer

To evaluate \( 0.\overline{14} + 0.\overline{19} \), we will follow these steps:

### Step 1: Convert the repeating decimals to fractions
The repeating decimal \( 0.\overline{14} \) can be expressed as a fraction using the formula:
\[ 0.\overline{xy} = \frac{xy}{99} \]
where \( xy \) is the repeating part. Here, \( xy = 14 \).

So, we have:
\[ 0.\overline{14} = \frac{14}{99} \]

Similarly, for \( 0.\overline{19} \):
\[ 0.\overline{19} = \frac{19}{99} \]

### Step 2: Add the fractions
Now, we can add the two fractions:
\[ 0.\overline{14} + 0.\overline{19} = \frac{14}{99} + \frac{19}{99} \]

Since the denominators are the same, we can combine the numerators:
\[ = \frac{14 + 19}{99} = \frac{33}{99} \]

### Step 3: Simplify the fraction
Now, we simplify \( \frac{33}{99} \):
\[ \frac{33}{99} = \frac{33 \div 33}{99 \div 33} = \frac{1}{3} \]

### Final Answer
Thus, the value of \( 0.\overline{14} + 0.\overline{19} \) is:
\[ \frac{1}{3} \]

---