To solve for \( b \) in the equation
\[ \frac{4}{5}b + 2 = \frac{1}{2} \left( \frac{2}{5} - b \right), \]
let's first simplify the right side of the equation.
- Distribute \( \frac{1}{2} \) on the right side:
\[ \frac{1}{2} \left( \frac{2}{5} - b \right) = \frac{1}{2} \cdot \frac{2}{5} - \frac{1}{2}b = \frac{1}{5} - \frac{1}{2}b. \]
This gives us the equation:
\[ \frac{4}{5}b + 2 = \frac{1}{5} - \frac{1}{2}b. \]
- Next, let’s eliminate the fractions by multiplying the entire equation by 10, the least common multiple of 5 and 2:
\[ 10 \left( \frac{4}{5}b + 2 \right) = 10 \left( \frac{1}{5} - \frac{1}{2}b \right). \]
This simplifies to:
\[ 8b + 20 = 2 - 5b. \]
- Now, let's add \( 5b \) to both sides to combine the \( b \) terms:
\[ 8b + 5b + 20 = 2. \]
This simplifies to:
\[ 13b + 20 = 2. \]
- Next, subtract 20 from both sides:
\[ 13b = 2 - 20, \] \[ 13b = -18. \]
- Finally, divide both sides by 13:
\[ b = -\frac{18}{13}. \]
Thus, the solution is:
\[ \boxed{-\frac{18}{13}}. \]