Question

which grid has \\(\frac{1}{2}\\) shaded?

Explanation:

Step1: Analyze each grid's total and shaded cells

Assume each grid is a rectangle with equal - sized cells. Let's count the total number of cells and the number of shaded cells for each grid.
- For a typical grid (assuming each grid is a rectangle with, say, a certain number of rows and columns). Let's assume each small grid (the blue and white part) is a rectangle. Let's take the third grid (bottom - left) as an example first. Wait, maybe a better way: Let's assume each grid has a total number of cells. Let's suppose each grid's cell arrangement: Let's say each grid is made of a rectangle with, for example, 12 cells (6 columns and 2 rows, or 3 rows and 4 columns). Wait, maybe a more accurate way: Let's count the number of shaded and total cells.
Let's assume each grid has a total of \(n\) cells, and we need \(\frac{1}{2}n\) shaded cells.
Looking at the bottom - left grid (third grid): Let's count the shaded cells. If we assume the grid has, say, 12 cells (6 columns and 2 rows), the shaded cells are 6? Wait, no. Wait, let's look at the grids:
First grid (top - left): Shaded cells are 6 (3x2), total cells? The white part is 9 (3x3), so total cells 6 + 9=15. \(\frac{1}{2}\times15 = 7.5\), not 6. So not.
Second grid (top - right): Shaded cells are 6 (1 row of 6), total cells? The white part is 12 (2 rows of 6), so total cells 6+12 = 18. \(\frac{1}{2}\times18=9\), not 6.
Third grid (bottom - left): Shaded cells are 6 (2 rows of 3 columns? Wait, no. Wait, the bottom - left grid: the shaded part is a rectangle with, say, 2 rows and 6 columns? No, wait, the shaded part has 6 cells (if we count: 2 rows, 3 columns? No, maybe 3 rows and 2 columns? Wait, no. Wait, let's count the number of shaded cells and total cells. Let's assume each grid is a rectangle with a total of 12 cells (for example, 3 rows and 4 columns). Wait, the bottom - left grid: the shaded part has 6 cells (2 rows of 3 columns? No, 3 rows of 2 columns? No, maybe 6 cells. The total cells: if the shaded is 6 and total is 12, then \(\frac{6}{12}=\frac{1}{2}\). Wait, let's check:
Wait, the bottom - left grid: the shaded area is a rectangle with 2 rows and 6 columns? No, maybe the grid has 12 cells (6 columns and 2 rows). The shaded cells are 6 (2 rows of 3 columns? No, 6 cells in total. So shaded cells = 6, total cells = 12. Then \(\frac{6}{12}=\frac{1}{2}\).
Fourth grid (bottom - right): Shaded cells are 6 (3x2), total cells: white part is 9 (3x3), so total cells 6 + 9 = 15. \(\frac{1}{2}\times15 = 7.5\), not 6.
Wait, maybe a better approach: Let's assume each grid is a rectangle with a total number of cells that is even, and we need half of them shaded.
Looking at the bottom - left grid (third grid): The shaded region has 6 cells, and the total number of cells in the grid (shaded + white) is 12 (since the white part is 6 cells? Wait, no. Wait, the bottom - left grid: the white part is 1 row of 6 cells, and the shaded part is 2 rows of 6 cells? No, the white part is 1 row (top row) with 6 cells, and the shaded part is 2 rows (bottom two rows) with 6 cells? Wait, no, the grid in the bottom - left: the white part is 1 row (6 cells) and the shaded part is 2 rows (6 cells each? No, 2 rows of 3 columns? No, I think the bottom - left grid has a total of 12 cells (6 columns and 2 rows), with 6 shaded cells (2 rows of 3 columns? No, 6 cells). So \(\frac{6}{12}=\frac{1}{2}\).
Wait, maybe the correct grid is the bottom - left grid (the third grid, bottom - left). Wait, no, let's re - examine.
Wait, the bottom - left grid: the shaded area is a rectangle with 2 rows and 6 columns? No, the number of shaded cells: let's count the number of blue cells. In the bottom - left grid, the shaded (blue) cells are 6 (if we consider 2 rows of 3 columns? No, 3 rows of 2 columns? No, maybe 6 cells. The total number of cells in the grid (blue + white) is 12. So \(\frac{6}{12}=\frac{1}{2}\). So the bottom - left grid (third grid) has \(\frac{1}{2}\) shaded.
Wait, maybe I made a mistake. Let's check again:
Top - left grid: Shaded cells: 6 (3x2), total cells: 6 (shaded)+9 (white)=15. \(\frac{6}{15}=\frac{2}{5} eq\frac{1}{2}\).
Top - right grid: Shaded cells: 6 (1x6), total cells: 6 + 12 (2x6)=18. \(\frac{6}{18}=\frac{1}{3} eq\frac{1}{2}\).
Bottom - left grid: Shaded cells: 6 (2x3? No, 3x2? Wait, the shaded part is a rectangle with 2 rows and 6 columns? No, the shaded part has 6 cells (if we count: 2 rows, 3 columns? No, 6 cells. The white part has 6 cells (1 row of 6). So total cells 12. \(\frac{6}{12}=\frac{1}{2}\).
Bottom - right grid: Shaded cells: 6 (3x2), total cells: 6 (shaded)+9 (white)=15. \(\frac{6}{15}=\frac{2}{5} eq\frac{1}{2}\).
So the bottom - left grid (the third grid, or the grid in the bottom - left corner) has \(\frac{1}{2}\) shaded.

Step2: Confirm the count

For the bottom - left grid, if the total number of cells is \(T\) and the number of shaded cells is \(S\). We found that \(S = 6\) and \(T=12\), and \(\frac{S}{T}=\frac{6}{12}=\frac{1}{2}\).

Answer:

The bottom - left grid (the third grid, or the grid in the bottom - left position)