Math Skills: Fractions

Lesson 2: Lesson Two-Ratios and Proportions

Section Two: Proportions

Let's take the problem of males to females in the last problem.

A company needs to have a ratio of 3 males to 2 females in the workplace in order to be in Dept. of Labor compliance (btw, this is the way the government works, okay? I'm not trying to be "politically correct" here!). A workplace has 12 males and 6 females, which is NOT a 3 : 2 ratio, and the boss doesn't want to have to fire some of the males, so he has to hire more females. How many more?

To find out, you have to set up a proportion.

A proportion (and by that we mean a "true proportion") is a statement that looks like an equation because it has an equals sign. What it is is two equal ratios. So a proportion is two ratios with an equals sign in between them.

1/2 and 2/4 are ratios, but 1/2 = 2/4 is a proportion.

In the workplace problem, the desired ratio is 3/2 (3 males/2 females). We know that there are 12 males, and the boss wanted to know how many females would make the ratio 3/2. So, he sets up a proportion:

3 males/2 females = 12 males/? females. To find out what "?" is, the boss has to "solve" the proportion. The easiest way to solve a proportion is to multiply both outside terms together, then multiply the two inside terms together, and both answers must be equal.

The two outside terms and the 3 and the ?

The two inside terms are 2 and 12.

Thus (replacing the ? with a variable Y), 3 times Y = 3Y. 2 times 12 = 24. So we have 3Y = 24. Dividing both sides by 3, we get Y = 8.

Since 8 is the number of females the workplace must have in order to be in compliance with Dept. of Labor rules when the workplace has 12 males, then, since the workplace already has 6 females, the boss must hire two more females.

This problem is not meant to be an actual scenario involving the US Dept. of Labor.

The above problem can also be written: 3 12 --- = --- 2 Y

which is the way it would normally look on paper. But since in this online course everything has to be typed on a keyboard, what would normally appear as it does above looks like:

3/2 = 12/Y

Here's a more relevant practical problem (I mean, who cares what the US Dept. of Labor does?):

A cook has a recipe for 20 people, but she is only serving 15 people. The recipe calls for 4 tablespoons of baking powder in the recipe for 20. How many tablespoons will she need to serve 15?

Let's set this one up. NOTE: ALWAYS set up a proportion in such as way that units are the same, top and bottom.

The right way to set this one up is tablespoons-to-people for both ratios, or you can use people-to-tablespoons in both.

20 people/4 Tbsp = 15 people/Y Tbsp. Multiply the two outsides together and get 20 Y. Multiply the two insides together and get 4 x 15 = 60. So, 20Y = 60. Divide both sides by 20 and get Y = 3. Thus, to serve 15 people, use 3 tablespoons of baking powder.

Read more about proportions in Cambridge GED, pages 114, 115.

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