Final answer:
To compare the ratios 4:5 and 19 to 30, we set up a proportion, but find they are not equivalent after cross-multiplying. For two squares where one has side lengths twice as long as the other, the area of the larger square is four times as large, because the ratio of the areas of similar figures is the square of the scale factor.
Step-by-step explanation:
To compare the two ratios 4:5 and 19 to 30, we need to express both ratios in the same form and then compare them by converting them to decimals or by cross-multiplying. The first ratio is already in the form of a: b, so we can transform the second ratio from 'to' form into a fraction to get 19/30.
Next, to find out if these two ratios are equivalent, we set up a proportion by writing an equation: 4/5 = 19/30. Once these are set equal, we can cross-multiply to check for equality: 4 x 30 = 5 x 19. Since 120 does not equal 95, the ratios are not equivalent.
As for the area comparison of two squares, if the side length of one square is 4 inches, and the side length of a similar square is twice that length (8 inches), then the area of the larger square is 82 or 64 square inches, and the smaller square's area would be 42 or 16 square inches. The ratio of the larger area to the smaller area is 64:16, which simplifies to 4:1. Therefore, the larger square's area is 4 times the smaller square's area. This illustrates the principle that when comparing areas of similar figures, the ratio of their areas is the square of the scale factor.