Evaluate `0.bar(68) +0.bar(73)`.

Options

A. `1.bar(31)`
B. `1.bar(42)`
C. `1.bar(21)`
D. `1.0bar1`

Answer

To evaluate the expression \(0.\overline{68} + 0.\overline{73}\), we will follow a systematic approach to convert each repeating decimal into a fraction and then add them together.

### Step 1: Convert \(0.\overline{68}\) into a fraction
Let \(x = 0.\overline{68}\).

Multiply both sides by 100 (since the repeating part has 2 digits):
\[ 100x = 68.\overline{68} \]

Now, we have:
1. \(x = 0.\overline{68}\) (Equation 1)
2. \(100x = 68.\overline{68}\) (Equation 2)

### Step 2: Subtract Equation 1 from Equation 2
Subtract Equation 1 from Equation 2:
\[ 100x - x = 68.\overline{68} - 0.\overline{68} \]
This simplifies to:
\[ 99x = 68 \]

### Step 3: Solve for \(x\)
Now, divide both sides by 99:
\[ x = \frac{68}{99} \]

Thus, \(0.\overline{68} = \frac{68}{99}\).

### Step 4: Convert \(0.\overline{73}\) into a fraction
Let \(y = 0.\overline{73}\).

Multiply both sides by 100:
\[ 100y = 73.\overline{73} \]

Now, we have:
1. \(y = 0.\overline{73}\) (Equation 3)
2. \(100y = 73.\overline{73}\) (Equation 4)

### Step 5: Subtract Equation 3 from Equation 4
Subtract Equation 3 from Equation 4:
\[ 100y - y = 73.\overline{73} - 0.\overline{73} \]
This simplifies to:
\[ 99y = 73 \]

### Step 6: Solve for \(y\)
Now, divide both sides by 99:
\[ y = \frac{73}{99} \]

Thus, \(0.\overline{73} = \frac{73}{99}\).

### Step 7: Add the two fractions
Now we can add the two fractions:
\[ 0.\overline{68} + 0.\overline{73} = \frac{68}{99} + \frac{73}{99} \]

Since they have the same denominator, we can combine them:
\[ = \frac{68 + 73}{99} = \frac{141}{99} \]

### Step 8: Simplify the fraction
To simplify \(\frac{141}{99}\), we can divide both the numerator and denominator by their greatest common divisor (GCD). The GCD of 141 and 99 is 3:
\[ \frac{141 \div 3}{99 \div 3} = \frac{47}{33} \]

### Step 9: Convert to decimal
To convert \(\frac{47}{33}\) to a decimal, we perform the division:
\[ 47 \div 33 \approx 1.424242\ldots \]
This can be expressed as:
\[ 1.\overline{42} \]

### Final Answer
Thus, the final result of \(0.\overline{68} + 0.\overline{73}\) is:
\[ \boxed{1.\overline{42}} \]