Solve. \[ \begin{array}{l}\frac{4 x}{5}-x=\frac{x}{15}-\frac{16}{3} \\ \text { Select the correct choice below and, if necessary, fill in the answer box to complete your choice. } \\ \text { B. The solution set is }\{x \mid x \text { is a real number }\} \text {. } \\ \text { C. The solution set is } \varnothing \text {. }\end{array} . \begin{array}{l}\text { A. (Type an integer or a simplified fraction.) }\end{array} \text {. } \]

To solve the equation \[ \frac{4 x}{5} - x = \frac{x}{15} - \frac{16}{3}, \] we start by simplifying both sides. First, rewrite \(x\) as \(\frac{5x}{5}\): \[ \frac{4x}{5} - \frac{5x}{5} = \frac{x}{15} - \frac{16}{3}. \] Now we can combine the left side: \[ \frac{4x - 5x}{5} = \frac{-x}{5}. \] Next, to solve for \(x\), let's combine like terms on the right side. We need a common denominator to combine \(\frac{x}{15}\) and \(-\frac{16}{3}\). The least common multiple of 15 and 3 is 15, so we write \(-\frac{16}{3}\) as \(-\frac{16 \cdot 5}{3 \cdot 5} = -\frac{80}{15}\): \[ \frac{x}{15} - \frac{80}{15} = \frac{x - 80}{15}. \] Now we rewrite our equation as: \[ \frac{-x}{5} = \frac{x - 80}{15}. \] Cross-multiplying gives us: \[ -15x = 5(x - 80). \] Expanding the right side: \[ -15x = 5x - 400. \] Now, combine like terms by adding \(15x\) to both sides: \[ 0 = 20x - 400. \] Next, add \(400\) to both sides: \[ 400 = 20x. \] Finally, divide both sides by \(20\): \[ x = 20. \] Now let's check our original equation with \(x = 20\): Left Side: \[ \frac{4(20)}{5} - 20 = \frac{80}{5} - 20 = 16 - 20 = -4. \] Right Side: \[ \frac{20}{15} - \frac{16}{3} = \frac{4}{3} - \frac{16}{3} = -\frac{12}{3} = -4. \] Since both sides equal, the solution is confirmed. The answer is: \[ \text{A. } 20. \] **Thus, the correct choice is: A. (Type an integer or a simplified fraction.) 20.**