Let us analyze the problem in detail:

Problem Setup:

The square represents a total area of 1 square unit. The dark gray portion's area must be determined as a fraction of the total square's area. To do this, we analyze:

1. The horizontal division: The square is divided into 5 equal columns, and 4 out of these 5 columns are shaded.
2. The vertical division: Each column is further divided into 2 rows, and only 1 row per column is shaded.

This means the shaded area is a fraction of the total area of the square, calculated as: $\text{Shaded Area} = \left(\frac{\text{shaded columns}}{\text{total columns}}\right) \times \left(\frac{\text{shaded rows}}{\text{total rows}}\right) = \frac{4}{5} \times \frac{1}{2}.$

Solution:

The equation representing the shaded area is: $\frac{4}{5} \times \frac{1}{2} = \frac{4 \times 1}{5 \times 2} = \frac{4}{10}.$

This matches the correct equation from the options: $\frac{4}{5} \times \frac{1}{2} = \frac{4}{10}$.

Would you like further details on solving this step-by-step? Here are some related questions to expand your understanding:

1. How do you calculate the area of overlapping fractions like this for other shapes (e.g., circles)?
2. How can we simplify $\frac{4}{10}$ further, and what does it represent?
3. What would happen if we changed the grid's rows or columns? How would the equation change?
4. How can you use multiplication of fractions to solve for portions in real-world scenarios?
5. What does "unit area" mean, and why is it useful in problems like this?

Tip:

When working with fraction-based area problems, always break the grid into its parts (rows and columns) to determine proportions for multiplication.