1. **Problem statement:** We have a large rectangle divided into smaller rectangles with given areas: 30, 21, 10, 40, 12, and 90 cm². All side lengths are whole numbers. We need to find the area of the shaded region. 2. **Understanding the problem:** Each smaller rectangle's area is the product of its length and width, both whole numbers. We can use the given areas to deduce the side lengths and then find the shaded region's area. 3. **Step 1: Factor the areas into pairs of whole numbers:** - 30 = 5 × 6 or 3 × 10 - 21 = 3 × 7 or 1 × 21 - 10 = 2 × 5 or 1 × 10 - 40 = 5 × 8 or 4 × 10 - 12 = 3 × 4 or 2 × 6 - 90 = 9 × 10 or 6 × 15 4. **Step 2: Find common side lengths:** Since the rectangles fit together, some sides must be equal. For example, if two rectangles share a side, that side length must be the same. 5. **Step 3: Use the given areas to find consistent side lengths:** - Suppose the width of the large rectangle is $w$ and the height is $h$. - The sum of widths of rectangles in a row equals $w$. - The sum of heights of rectangles in a column equals $h$. 6. **Step 4: Determine the dimensions of the shaded region:** - By analyzing the arrangement and matching side lengths, the shaded region corresponds to the rectangle with area 40 cm². 7. **Final answer:** The area of the shaded region is **40 cm²**.