Evaluate :
`2.bar(7)+1.bar(3)`

Explanation

To evaluate the expression \( 2.\overline{7} + 1.\overline{3} \), we will follow these steps:

### Step 1: Define \( 2.\overline{7} \) as a variable
Let \( x = 2.\overline{7} \). This means:
\[ x = 2.77777\ldots \]

### Step 2: Multiply \( x \) by 10
To eliminate the repeating decimal, multiply both sides by 10:
\[ 10x = 27.77777\ldots \]

### Step 3: Set up the subtraction
Now, subtract the original equation from this new equation:
\[ 10x - x = 27.77777\ldots - 2.77777\ldots \]
This simplifies to:
\[ 9x = 25 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 9:
\[ x = \frac{25}{9} \]
Thus, \( 2.\overline{7} = \frac{25}{9} \).

### Step 5: Define \( 1.\overline{3} \) as another variable
Let \( y = 1.\overline{3} \). This means:
\[ y = 1.33333\ldots \]

### Step 6: Multiply \( y \) by 10
Again, multiply both sides by 10:
\[ 10y = 13.33333\ldots \]

### Step 7: Set up the subtraction
Now, subtract the original equation from this new equation:
\[ 10y - y = 13.33333\ldots - 1.33333\ldots \]
This simplifies to:
\[ 9y = 12 \]

### Step 8: Solve for \( y \)
Now, divide both sides by 9:
\[ y = \frac{12}{9} = \frac{4}{3} \]
Thus, \( 1.\overline{3} = \frac{4}{3} \).

### Step 9: Add the two results
Now we can add the two fractions:
\[ 2.\overline{7} + 1.\overline{3} = \frac{25}{9} + \frac{4}{3} \]

### Step 10: Find a common denominator
The common denominator for 9 and 3 is 9. Rewrite \( \frac{4}{3} \) with a denominator of 9:
\[ \frac{4}{3} = \frac{4 \times 3}{3 \times 3} = \frac{12}{9} \]

### Step 11: Add the fractions
Now, add the two fractions:
\[ \frac{25}{9} + \frac{12}{9} = \frac{25 + 12}{9} = \frac{37}{9} \]

### Final Answer
Thus, the value of \( 2.\overline{7} + 1.\overline{3} \) is:
\[ \frac{37}{9} \]

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