Final answer:
The ratios 2:3, 4:6, and 8:12 are all equivalent, as they can be simplified to the same fundamental ratio of 2:3. Proportions formed by these ratios confirm their equivalency, and understanding scale factors allows converting scale measurements to actual dimensions.
Step-by-step explanation:
The ratios 2:3, 4:6, and 8:12 are equivalent because they can all be simplified to the same fundamental ratio. To show this, we need to divide the numbers in each ratio by their greatest common divisor (GCD). For 2:3, the GCD is 1, so it remains the same. For 4:6, dividing both terms by their GCD of 2, we get 2:3. For 8:12, dividing both terms by their GCD of 4, we again get 2:3.
If we express these ratios in fraction form, 2/3, 4/6, and 8/12, it becomes even clearer. Simplifying 4/6 and 8/12, we get 2/3 in both cases, proving they are equivalent ratios. Furthermore, when we create proportions by setting them equal to each other, such as 2/3 = 4/6 and 4/6 = 8/12, we see that the cross products are equal, confirming the proportions are true.
Using scale factor to find actual dimensions is a practical application of understanding equivalent ratios. For example, a scale factor of 2":3' means that 2 inches on the model represent 3 feet in real life. If the scale measurement is 6 inches, we set up the proportion 2/3 = 6/x, where x is the actual dimension in feet. Solving this proportion, we find that x equals 9 feet.