Proportions

According to A Maths Dictionary for Kids 2008 a proportion can be defined as:

-a part to whole comparison

-the equality of two ratios, written as an equation

How do we know when to use a proportion and when not to. Consider these two problems:

Memphis and Texas were shooting an equal number of free throws. Memphis shot first. When they had shot 9 free throws, Texas had shot 3 free throws. When Texas shot 15 free throws, how many had Memphis shot?

My in-laws just returned from Spain. The exchange rate for 3 U.S. dollars with the euro in Spain is 3 dollars for 2 euros. How many euros for 21 U.S. dollars?

Both would seem to be proportion-type problems. They each have 3 numbers and one piece of information that is missing. So you could set up a proportion in the a/b = c/d format for each problem. There is a difference though. In the first problem there is no multiplicative relationship between the numbers whereas in problem two there is a relationship. For every 3 dollars you can receive 2 euros (3:2, 3 to 2, or 3/2). If you are exchanging 21 U.S. dollars you would get 14 euros. You could set up a proportion such as:

proportion-2.jpg

In order to solve this problem you can treat it as an equivalent fraction/ratio type problem and see that 21 is 7 times bigger than 3, so x has to be 7 times bigger than 2 (which is 14). As well, you could cross multiply

proportion-cross-mult-1.jpg

and end up with 3x = 21 x 2.

3x = 42
3 3
x = 42/3
x = 14

So why wouldn’t the Memphis (who at half did actual shoot 9 free throws!) /Texas problem be solved proportionally? The number of free throws that Memphis shot compared to Texas can be explained with addition or subtraction. Memphis free throws equal Texas free throws + 6 or Texas free throws equal Memphis free throws -6. If Texas shot 15 free throws, then Memphis must have shot 21. The difference between the Memphis and Texas free throws will be + or – 6 (depending on which way you look at it).

In the in-law trip problem, they were exchanging 3 dollars for 2 euros. Therefore we could find any number of dollars for euros by multiplying the dollars by 2/3 (21 x 2/3= 42/3 which equals 14). And the opposite can be calculated as well, we can multiply the euros by 3/2 (3/2 x 14 = 42/2 which equals 21).

In order to understand proportions, one needs to understand ratios (comparing two things) and rates (a ratio that compares quantities of different units). Ratios can be look at as part-to-part or even part-to-whole. For example to make a banana smoothie you might use 1 cup of milk and 2 bananas (1:2). What if I wanted to make smoothies for myself and three friends? Since that is 4 people, keeping the rate the same I would need 4 cups of milk and 8 bananas.

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