The problem is to solve the equation $F \times \frac{3}{5} = \frac{4}{10} \times [50 - \frac{4}{5}F]$ for $F$.

1. Problem Description

The problem is to solve the equation $F \times \frac{3}{5} = \frac{4}{10} \times [50 - \frac{4}{5}F]$ for $F$.

2. Solution Steps

First, we rewrite the equation:

$\frac{3}{5}F = \frac{4}{10} (50 - \frac{4}{5}F)$.

Simplify the equation:

$\frac{3}{5}F = \frac{2}{5} (50 - \frac{4}{5}F)$.

Multiply both sides by 5 to eliminate the fractions:

$3F = 2(50 - \frac{4}{5}F)$.

Distribute the 2 on the right side:

$3F = 100 - \frac{8}{5}F$.

Multiply both sides by 5 to eliminate the fraction:

$15F = 500 - 8F$.

Add $8F$ to both sides:

$15F + 8F = 500$.

Combine like terms:

$23F = 500$.

Divide both sides by 23:

$F = \frac{500}{23}$.

3. Final Answer

$F = \frac{500}{23}$