The problem is to simplify the expression $\frac{1}{3}a^2b - \frac{3}{5}ab^2 + \frac{3}{8}a^2b - \frac{5}{3}ab$.

1. Problem Description

The problem is to simplify the expression $\frac{1}{3}a^2b - \frac{3}{5}ab^2 + \frac{3}{8}a^2b - \frac{5}{3}ab$.

2. Solution Steps

First, we need to identify the like terms in the expression. The terms $\frac{1}{3}a^2b$ and $\frac{3}{8}a^2b$ are like terms because they have the same variables raised to the same powers ($a^2b$). Also note that the term $\frac{-5}{3}ab$ seems to have no other like terms, and likewise $\frac{-3}{5}ab^2$ appears to have no other like terms.

Now, we combine the like terms:

$\frac{1}{3}a^2b + \frac{3}{8}a^2b = (\frac{1}{3} + \frac{3}{8})a^2b$

To add the fractions $\frac{1}{3}$ and $\frac{3}{8}$, we need to find a common denominator, which is

2

4. $\frac{1}{3} = \frac{1 \cdot 8}{3 \cdot 8} = \frac{8}{24}$

$\frac{3}{8} = \frac{3 \cdot 3}{8 \cdot 3} = \frac{9}{24}$

So, $\frac{1}{3} + \frac{3}{8} = \frac{8}{24} + \frac{9}{24} = \frac{8+9}{24} = \frac{17}{24}$

Therefore, $\frac{1}{3}a^2b + \frac{3}{8}a^2b = \frac{17}{24}a^2b$.

Now, rewrite the entire expression with the simplified like terms:

$\frac{17}{24}a^2b - \frac{3}{5}ab^2 - \frac{5}{3}ab$

3. Final Answer

The simplified expression is $\frac{17}{24}a^2b - \frac{3}{5}ab^2 - \frac{5}{3}ab$.