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

Explanation

To evaluate the expression \(1.\overline{45} + 0.\overline{3}\), we will follow these steps:

### Step 1: Convert \(1.\overline{45}\) into a fraction
Let \(x = 1.\overline{45}\).

To eliminate the repeating decimal, we multiply \(x\) by 100 (since there are 2 digits repeating):
\[ 100x = 145.\overline{45} \]

Now, we can set up the equation:
\[ 100x - x = 145.\overline{45} - 1.\overline{45} \]
This simplifies to:
\[ 99x = 144 \]
Now, divide both sides by 99:
\[ x = \frac{144}{99} \]

### Step 2: Simplify \(\frac{144}{99}\)
To simplify \(\frac{144}{99}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 9:
\[ x = \frac{144 \div 9}{99 \div 9} = \frac{16}{11} \]

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

To eliminate the repeating decimal, we multiply \(y\) by 10 (since there is 1 digit repeating):
\[ 10y = 3.\overline{3} \]

Now, we can set up the equation:
\[ 10y - y = 3.\overline{3} - 0.\overline{3} \]
This simplifies to:
\[ 9y = 3 \]
Now, divide both sides by 9:
\[ y = \frac{3}{9} = \frac{1}{3} \]

### Step 4: Add the two fractions
Now we have:
\[ 1.\overline{45} + 0.\overline{3} = \frac{16}{11} + \frac{1}{3} \]

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

Convert each fraction:
\[ \frac{16}{11} = \frac{16 \times 3}{11 \times 3} = \frac{48}{33} \]
\[ \frac{1}{3} = \frac{1 \times 11}{3 \times 11} = \frac{11}{33} \]

Now, add the fractions:
\[ \frac{48}{33} + \frac{11}{33} = \frac{48 + 11}{33} = \frac{59}{33} \]

### Final Answer
Thus, the value of \(1.\overline{45} + 0.\overline{3}\) is \(\frac{59}{33}\).

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