The problem asks to evaluate the expression $-\frac{2}{5} + 2\frac{3}{10} + 1\frac{1}{3}$.

1. Problem Description

The problem asks to evaluate the expression $-\frac{2}{5} + 2\frac{3}{10} + 1\frac{1}{3}$.

2. Solution Steps

First, convert the mixed numbers to improper fractions.

$2\frac{3}{10} = \frac{2 \times 10 + 3}{10} = \frac{20 + 3}{10} = \frac{23}{10}$

$1\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3}$

Now the expression becomes:

$-\frac{2}{5} + \frac{23}{10} + \frac{4}{3}$

Find a common denominator. The least common multiple of 5, 10, and 3 is

3

0. $-\frac{2}{5} = -\frac{2 \times 6}{5 \times 6} = -\frac{12}{30}$

$\frac{23}{10} = \frac{23 \times 3}{10 \times 3} = \frac{69}{30}$

$\frac{4}{3} = \frac{4 \times 10}{3 \times 10} = \frac{40}{30}$

Substitute the fractions with common denominators into the expression:

$-\frac{12}{30} + \frac{69}{30} + \frac{40}{30} = \frac{-12 + 69 + 40}{30} = \frac{57 + 40}{30} = \frac{97}{30}$

Convert the improper fraction $\frac{97}{30}$ to a mixed number.

$97 \div 30 = 3$ with a remainder of $7$. So, $\frac{97}{30} = 3\frac{7}{30}$.

3. Final Answer

$3\frac{7}{30}$