The value of `(0.bar(63) + 0.bar(37))` is

Options

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

Answer

To solve the problem of finding the value of \(0.\overline{63} + 0.\overline{37}\), we will convert each repeating decimal into a fraction and then add them together. Here are the steps:

### Step 1: Convert \(0.\overline{63}\) to a fraction
Let \(x = 0.\overline{63}\).

To eliminate the repeating decimal, we can multiply \(x\) by 100 (since there are two digits in the repeating part):
\[ 100x = 63.\overline{63} \]

Now, we can subtract the original equation from this new equation:
\[ 100x - x = 63.\overline{63} - 0.\overline{63} \]
\[ 99x = 63 \]
\[ x = \frac{63}{99} \]

### Step 2: Simplify \(\frac{63}{99}\)
To simplify \(\frac{63}{99}\), we find the greatest common divisor (GCD) of 63 and 99, which is 9:
\[ \frac{63 \div 9}{99 \div 9} = \frac{7}{11} \]

So, \(0.\overline{63} = \frac{7}{11}\).

### Step 3: Convert \(0.\overline{37}\) to a fraction
Let \(y = 0.\overline{37}\).

Similarly, we multiply \(y\) by 100:
\[ 100y = 37.\overline{37} \]

Subtracting the original equation:
\[ 100y - y = 37.\overline{37} - 0.\overline{37} \]
\[ 99y = 37 \]
\[ y = \frac{37}{99} \]

### Step 4: Add the two fractions
Now we have:
\[ 0.\overline{63} + 0.\overline{37} = \frac{7}{11} + \frac{37}{99} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 11 and 99 is 99. We convert \(\frac{7}{11}\) to have a denominator of 99:
\[ \frac{7}{11} = \frac{7 \times 9}{11 \times 9} = \frac{63}{99} \]

Now we can add:
\[ \frac{63}{99} + \frac{37}{99} = \frac{63 + 37}{99} = \frac{100}{99} \]

### Final Result
Thus, the value of \(0.\overline{63} + 0.\overline{37} = \frac{100}{99}\).