The value of `(0.bar63)+0.bar37)` is

Options

A. 1
B. `(100)/(99)`
C. `(99)/(100)`
D. `(100)/(33)`

Answer

To find the value of \( (0.\overline{63} + 0.\overline{37}) \), we can follow these steps:

### Step 1: Convert the repeating decimals to fractions
First, we need to convert the repeating decimals \( 0.\overline{63} \) and \( 0.\overline{37} \) into fractions.

**For \( 0.\overline{63} \):**
Let \( x = 0.\overline{63} \).
Then, multiplying both sides by 100 (to shift the decimal point two places to the right):
\[ 100x = 63.\overline{63} \]
Now, subtract the original equation from this new equation:
\[ 100x - x = 63.\overline{63} - 0.\overline{63} \]
This simplifies to:
\[ 99x = 63 \]
Thus, we find:
\[ x = \frac{63}{99} \]
Now, simplify \( \frac{63}{99} \):
\[ x = \frac{7}{11} \]

**For \( 0.\overline{37} \):**
Let \( y = 0.\overline{37} \).
Similarly, multiplying both sides by 100:
\[ 100y = 37.\overline{37} \]
Subtracting the original equation gives:
\[ 100y - y = 37.\overline{37} - 0.\overline{37} \]
This simplifies to:
\[ 99y = 37 \]
Thus, we find:
\[ y = \frac{37}{99} \]

### Step 2: Add the fractions
Now we can add the two fractions:
\[ 0.\overline{63} + 0.\overline{37} = \frac{63}{99} + \frac{37}{99} \]
Since they have a common denominator, we can add the numerators:
\[ \frac{63 + 37}{99} = \frac{100}{99} \]

### Step 3: Simplify the result
The fraction \( \frac{100}{99} \) is already in its simplest form. This can also be expressed as:
\[ 1 + \frac{1}{99} \]

### Final Answer
Thus, the value of \( (0.\overline{63} + 0.\overline{37}) \) is:
\[ \frac{100}{99} \]