Evaluate :
`1.bar(45)+0.bar(3)`

Explanation

To evaluate the expression \( 1.\overline{45} + 0.\overline{3} \), we will convert each repeating decimal into a fraction and then add them together.

### Step-by-Step Solution:

1. **Convert \( 1.\overline{45} \) to a Fraction:**
- Let \( x = 1.\overline{45} \).
- This means \( x = 1.454545...\).
- Since the repeating part has 2 digits, multiply both sides by 100:
\[ 100x = 145.454545... \]
- Now, subtract the original equation from this new equation:
\[ 100x - x = 145.454545... - 1.454545... \]
\[ 99x = 144 \]
- Solving for \( x \):
\[ x = \frac{144}{99} \]

2. **Convert \( 0.\overline{3} \) to a Fraction:**
- Let \( y = 0.\overline{3} \).
- This means \( y = 0.3333...\).
- Since the repeating part has 1 digit, multiply both sides by 10:
\[ 10y = 3.3333... \]
- Now, subtract the original equation from this new equation:
\[ 10y - y = 3.3333... - 0.3333... \]
\[ 9y = 3 \]
- Solving for \( y \):
\[ y = \frac{3}{9} = \frac{1}{3} \]

3. **Add the Two Fractions:**
- Now we need to add \( x \) and \( y \):
\[ x + y = \frac{144}{99} + \frac{1}{3} \]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 99 and 3 is 99.
- Convert \( \frac{1}{3} \) to have a denominator of 99:
\[ \frac{1}{3} = \frac{1 \times 33}{3 \times 33} = \frac{33}{99} \]
- Now we can add:
\[ x + y = \frac{144}{99} + \frac{33}{99} = \frac{144 + 33}{99} = \frac{177}{99} \]

4. **Simplify the Result:**
- We can simplify \( \frac{177}{99} \) by finding the greatest common divisor (GCD) of 177 and 99. The GCD is 9.
- Dividing both the numerator and denominator by 9:
\[ \frac{177 \div 9}{99 \div 9} = \frac{19.6667}{11} = \frac{59}{33} \]

Thus, the final answer is:
\[ \frac{59}{33} \]