The problem asks us to evaluate the expression $1\frac{1}{5} \times (-\frac{2}{3}) + 1\frac{1}{5}$.

1. Problem Description

The problem asks us to evaluate the expression $1\frac{1}{5} \times (-\frac{2}{3}) + 1\frac{1}{5}$.

2. Solution Steps

First, we convert the mixed numbers to improper fractions.

$1\frac{1}{5} = \frac{1 \times 5 + 1}{5} = \frac{6}{5}$

Now, we rewrite the expression with improper fractions:

$\frac{6}{5} \times (-\frac{2}{3}) + \frac{6}{5}$

Next, we perform the multiplication:

$\frac{6}{5} \times (-\frac{2}{3}) = -\frac{6 \times 2}{5 \times 3} = -\frac{12}{15}$

We can simplify the fraction $-\frac{12}{15}$ by dividing both the numerator and the denominator by 3:

$-\frac{12}{15} = -\frac{12 \div 3}{15 \div 3} = -\frac{4}{5}$

Now we have:

$-\frac{4}{5} + \frac{6}{5}$

Adding the two fractions:

$-\frac{4}{5} + \frac{6}{5} = \frac{-4 + 6}{5} = \frac{2}{5}$

3. Final Answer

$\frac{2}{5}$