To solve the expression \([1 \frac{4}{5} + ( \frac{1}{4} + \frac{1}{5}) ] - 2 \frac{2}{5}\), we will follow these steps:
- Convert mixed numbers to improper fractions.
- Simplify the expression inside the brackets.
- Subtract the final term.
Step 1: Convert mixed numbers to improper fractions
Convert \(1 \frac{4}{5}\) and \(2 \frac{2}{5}\):
\[ 1 \frac{4}{5} = \frac{5 \cdot 1 + 4}{5} = \frac{9}{5} \] \[ 2 \frac{2}{5} = \frac{5 \cdot 2 + 2}{5} = \frac{12}{5} \]
Step 2: Simplify the expression inside the brackets
Calculate \(\frac{1}{4} + \frac{1}{5}\):
To add \(\frac{1}{4}\) and \(\frac{1}{5}\), we need a common denominator, which is 20.
\[ \frac{1}{4} = \frac{5}{20} \] \[ \frac{1}{5} = \frac{4}{20} \] \[ \frac{1}{4} + \frac{1}{5} = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} \]
Now plug this back into the expression:
\[ 1 \frac{4}{5} + \left( \frac{1}{4} + \frac{1}{5} \right) = \frac{9}{5} + \frac{9}{20} \]
To add \(\frac{9}{5}\) and \(\frac{9}{20}\), we again need a common denominator. The least common denominator of 5 and 20 is 20.
Convert \(\frac{9}{5}\) to a denominator of 20:
\[ \frac{9}{5} = \frac{36}{20} \]
Now add:
\[ \frac{36}{20} + \frac{9}{20} = \frac{45}{20} \]
Step 3: Subtract \(2 \frac{2}{5}\)
Now substitute back into the expression:
\[ [1 \frac{4}{5} + ( \frac{1}{4} + \frac{1}{5}) ] - 2 \frac{2}{5} = \frac{45}{20} - \frac{12}{5} \]
Convert \(\frac{12}{5}\) to a denominator of 20:
\[ \frac{12}{5} = \frac{48}{20} \]
Now subtract:
\[ \frac{45}{20} - \frac{48}{20} = \frac{-3}{20} \]
Final Result
The final result of the expression \([1 \frac{4}{5} + ( \frac{1}{4} + \frac{1}{5}) ] - 2 \frac{2}{5}\) is:
\[ \boxed{-\frac{3}{20}} \]