`0.bar3+0.bar(45)=`_____

Options

A. `0.bar(75)`
B. `0.bar(48)`
C. `0.bar(76)`
D. `0.bar(78)`

Answer

To solve the problem \(0.\overline{3} + 0.\overline{45}\), we will convert these repeating decimals into fractions and then add them together. Here’s the step-by-step solution:

### Step 1: Convert \(0.\overline{3}\) to a Fraction
Let \(a = 0.\overline{3}\).

To convert this repeating decimal to a fraction:
1. Multiply \(a\) by 10 to shift the decimal point:
\[ 10a = 3.\overline{3} \]
2. Now, subtract the original \(a\) from this equation:
\[ 10a - a = 3.\overline{3} - 0.\overline{3} \]
\[ 9a = 3 \]
3. Solve for \(a\):
\[ a = \frac{3}{9} = \frac{1}{3} \]

### Step 2: Convert \(0.\overline{45}\) to a Fraction
Let \(b = 0.\overline{45}\).

To convert this repeating decimal to a fraction:
1. Multiply \(b\) by 100 (since there are two digits in the repeating part):
\[ 100b = 45.\overline{45} \]
2. Now, subtract the original \(b\) from this equation:
\[ 100b - b = 45.\overline{45} - 0.\overline{45} \]
\[ 99b = 45 \]
3. Solve for \(b\):
\[ b = \frac{45}{99} \]
Simplifying this fraction:
\[ b = \frac{5}{11} \]

### Step 3: Add the Two Fractions
Now we need to add \(a\) and \(b\):
\[ 0.\overline{3} + 0.\overline{45} = \frac{1}{3} + \frac{5}{11} \]

To add these fractions, we need a common denominator. The least common multiple of 3 and 11 is 33.

1. Convert \(\frac{1}{3}\) to have a denominator of 33:
\[ \frac{1}{3} = \frac{11}{33} \]
2. Convert \(\frac{5}{11}\) to have a denominator of 33:
\[ \frac{5}{11} = \frac{15}{33} \]

Now we can add them:
\[ \frac{11}{33} + \frac{15}{33} = \frac{26}{33} \]

### Step 4: Convert the Result Back to Decimal
Now we convert \(\frac{26}{33}\) back to a decimal:
1. Divide 26 by 33:
\[ 26 \div 33 \approx 0.787878\ldots = 0.\overline{78} \]

### Final Answer
Thus, the final result is:
\[ 0.\overline{3} + 0.\overline{45} = 0.\overline{78} \]

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