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I have been wrestling my way through the RP domain and the creation of units and professional development. This has come up and I was wondering what other people thought…
I was recently told that “Two equivalent ratios are a proportion; however, they are not proportional. Only quantities can be proportional.”
My question is… “If ratios are the comparison of two quantities and ratios represent those two quantities then why can’t they be proportional? Also — If numbers are quantities then why aren’t two numbers that are in a relationship and form a ratio considered to be a quantity?”
Any clarification would be greatly appreciated.
In the second half of your statement “If ratios are the comparison of two quantities and ratios represent those two quantities”, you say that ratios represent the quantities. But they don’t. A ratio represents the relationship between the quantities. The quantities are what they are. A ratio associates them.
Then your second question essentially asks why ratios aren’t quantities by asserting that numbers are quantities (?) and numbers in a relationship form a ratio. I guess I wouldn’t say that numbers form ratios. Ratios are associations that describe the relationship between two or more quantities – cups of apple juice to cups of grape juice, or meters walked to seconds elapsed.
Ratios can be equivalent if they have the same value. The value of a ratio is the quotient A/B. A and B are the measurements of the quantities described in the ratio. 2 cups of apple juice to 3 cups of grape juice is equivalent to 6 cups of apple juice to 9 cups of grape juice because 2/3 = 6/9.
If we collect a bunch of those pairs of numbers that are in equivalent ratios, we have a proportional relationship: (2,3), (6,9), (1,2/3), (10,30), … which we can describe with an equation y = kx.
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