The simplification of `(0.bar(63) + 0.bar(37) + 0.bar(80))` yields the result

Options

A. `1.bar(80)`
B. `1.bar(81)`
C. `1.bar(79)`
D. `1.80`

Answer

To simplify the expression \(0.\overline{63} + 0.\overline{37} + 0.\overline{80}\), we can follow these steps:

### Step 1: Convert each repeating decimal to a fraction
The repeating decimal \(0.\overline{63}\) can be expressed as a fraction.

Let \(x = 0.\overline{63}\). Then,
\[ 100x = 63.\overline{63} \]
Subtracting the first equation from the second:
\[ 100x - x = 63.\overline{63} - 0.\overline{63} \]
\[ 99x = 63 \]
\[ x = \frac{63}{99} \]

### Step 2: Repeat for the other decimals
Now, we convert \(0.\overline{37}\) and \(0.\overline{80}\) similarly.

For \(0.\overline{37}\):
Let \(y = 0.\overline{37}\). Then,
\[ 100y = 37.\overline{37} \]
Subtracting gives:
\[ 99y = 37 \implies y = \frac{37}{99} \]

For \(0.\overline{80}\):
Let \(z = 0.\overline{80}\). Then,
\[ 100z = 80.\overline{80} \]
Subtracting gives:
\[ 99z = 80 \implies z = \frac{80}{99} \]

### Step 3: Add the fractions
Now we can add these fractions together:
\[ 0.\overline{63} + 0.\overline{37} + 0.\overline{80} = \frac{63}{99} + \frac{37}{99} + \frac{80}{99} \]
Since they have a common denominator, we can combine the numerators:
\[ = \frac{63 + 37 + 80}{99} = \frac{180}{99} \]

### Step 4: Simplify the fraction
Now we simplify \(\frac{180}{99}\):
\[ \frac{180 \div 9}{99 \div 9} = \frac{20}{11} \]

### Step 5: Convert to a mixed number
To express \(\frac{20}{11}\) as a decimal:
\[ 20 \div 11 = 1 \text{ remainder } 9 \]
So, \(\frac{20}{11} = 1 + \frac{9}{11}\).

### Step 6: Convert the remainder to a repeating decimal
Now, we convert \(\frac{9}{11}\) to a decimal:
\[ \frac{9}{11} = 0.\overline{81} \]
Thus,
\[ \frac{20}{11} = 1 + 0.\overline{81} = 1.\overline{81} \]

### Final Result
The simplification of \(0.\overline{63} + 0.\overline{37} + 0.\overline{80}\) yields:
\[ \boxed{1.\overline{81}} \]