Let's define the problem mathematically.

We have the earnings per task as follows:

• Eyad earns $x$ dollars for every $(x - y)$ tasks, so his rate is: $\frac{x}{x - y}$
• Amina earns 2 dollars for every 7 tasks, so her rate is: $\frac{2}{7}$
• Hala earns $(x - 3)$ dollars for every $(y - 1)$ tasks, so her rate is: $\frac{x - 3}{y - 1}$

Since all three have the same ratio, we set up the equations:

$\frac{x}{x - y} = \frac{2}{7}$

$\frac{x - 3}{y - 1} = \frac{2}{7}$

Solving for $x$ and $y$:

First Equation:

$7x = 2(x - y)$

$7x = 2x - 2y$

$5x = -2y$

$y = -\frac{5x}{2}$

Second Equation:

$7(x - 3) = 2(y - 1)$

$7x - 21 = 2y - 2$

$7x + 2 = 2y + 21$

$7x + 2 = 2y + 21$

Substituting $y = -\frac{5x}{2}$:

$7x + 2 = 2 \left(-\frac{5x}{2} \right) + 21$

$7x + 2 = -5x + 21$

$12x = 19$

Thus, $12x = 19$.

Answer:

$\boxed{19} \quad \text{(Option B)}$

Would you like me to clarify any step or provide additional details?

Related Questions:

1. How do we derive the proportionality equations in this problem?
2. What happens if we change the earnings of Amina to 3 dollars per 7 tasks?
3. Can we generalize this problem for different numbers of tasks?
4. What is the significance of the ratios in such problems?
5. How does this problem relate to systems of linear equations?

Tip:

When solving problems with proportional relationships, always express rates as fractions and equate them systematically.