Which comparison about the shapes is true?

Question

6 triangles and 2 rectangles.
For every 2 rectangles there are 8 triangles.
For every 6 triangles there are 8 rectangles.
For every 2 rectangles there are 4 triangles.
For every 6 triangles there are 2 rectangles.

Answer 1

To analyze the relationships between the numbers of triangles and rectangles, we can convert the statements into mathematical proportions.

1. Original Count: We have 6 triangles and 2 rectangles.
2. Ratios:
   • The ratio of triangles to rectangles in the original count is 6 triangles : 2 rectangles.

Let's evaluate each statement:

1. For every 2 rectangles there are 8 triangles.

   • If there are 2 rectangles, according to the original ratio, there would be \( \frac{6}{2} \times 2 = 6 \) triangles, not 8. False.

2. For every 6 triangles there are 8 rectangles.

   • If there are 6 triangles, according to the original ratio, there are \( \frac{2}{6} \times 6 = 2 \) rectangles, not 8. False.

3. For every 2 rectangles there are 4 triangles.

   • If there are 2 rectangles, according to the original ratio, there are \( \frac{6}{2} \times 2 = 6 \) triangles, not 4. False.

4. For every 6 triangles there are 2 rectangles.

   • This matches the original ratio. For every 6 triangles, there indeed are 2 rectangles as per the original count. True.

5. For every 6 triangles there are 2 rectangles.

   • This statement is the same as the one above and is also True.

The only true statement about the comparison of the shapes is: For every 6 triangles, there are 2 rectangles.