Final answer:
A proportion without variables can be represented as 1/2 = 2/4. The number of ways to rearrange the four numbers to keep equivalent ratios is 24 combinations or 4 factorial. Examples of proportions include 1/20 = 1/5.5 for meters and 1/48 = w/16 for feet, where w represents an unknown measure in feet.
Step-by-step explanation:
To write a proportion without using any variables, we can use specific numerical values. For instance, we can create a proportion like 1/2 = 2/4, where both sides represent the same relative sizes or ratios. The numerators and denominators can be rearranged to form equivalent ratios, proving the proportion remains true.
The number of different ways you can rearrange the four numbers in the ratios to keep the proportions equivalent is 24. This is calculated as 4 factorial (4!), which is 4 x 3 x 2 x 1. To systematically shuffle these numbers in an orderly manner, you could write out all 24 combinations. This helps to practice and comprehend the method of creating combinations. Consequently, you would discover that multiplying the numbers in descending order, as done in 4 factorial, is the correct method to count all unique arrangements.
An example of writing proportions using given units is to equate two ratios with the same unit. For instance, if the given unit is meters, we could write the proportion 1/20 = 1/5.5. Similarly, if the given unit is feet, we could write 1/48 = w/16, where w represents the unknown variable in feet.