Evaluate Fraction Arithmetic Expression

The problem asks for the value of the expression:

$ \frac{5}{3} + \left( \frac{1}{1 + \frac{4}{9}} \right) - \frac{2}{5} $

Step-by-Step Calculation

1. Simplify the term in parentheses:

   First, calculate the denominator inside the parentheses:

   $ 1 + \frac{4}{9} = \frac{9}{9} + \frac{4}{9} = \frac{13}{9} $

   Now, calculate the entire term in parentheses:

   $ \frac{1}{\frac{13}{9}} = \frac{9}{13} $
2. Substitute back into the original expression:

   The expression becomes:

   $ \frac{5}{3} + \frac{9}{13} - \frac{2}{5} $
3. Find the Common Denominator:

   The denominators are 3, 13, and 5. The least common multiple (LCM) of 3, 13, and 5 is:

   $ \text{LCM}(3, 13, 5) = 3 \times 13 \times 5 = 195 $
4. Convert Fractions to Common Denominator:
   • $ \frac{5}{3} = \frac{5 \times 65}{3 \times 65} = \frac{325}{195} $
   • $ \frac{9}{13} = \frac{9 \times 15}{13 \times 15} = \frac{135}{195} $
   • $ \frac{2}{5} = \frac{2 \times 39}{5 \times 39} = \frac{78}{195} $
5. Perform Addition and Subtraction:

   Combine the fractions using the common denominator:

   $ \frac{325}{195} + \frac{135}{195} - \frac{78}{195} = \frac{325 + 135 - 78}{195} $

   Calculate the numerator:

   $ \frac{460 - 78}{195} = \frac{382}{195} $

Final Value

The value of the given expression is $\frac{382}{195}$.