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Calculations with Ratios and Proportions

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  • 0:05 Let's Compare
  • 0:41 Definitions
  • 1:27 Ratio Problems
  • 2:46 Proportion Problems
  • 5:07 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

Even if only 99 out of 100 people need to know how to work with ratios, the odds that they'll be useful to you are very high. In this lesson, we'll practice performing calculations with ratios and proportions.

Let's Compare

Comparisons can get kind of a bad rap. I might say my favorite football team is better than yours. Or, I could say that I'm better-looking than you. Or, maybe I tell you that my dog is smarter than yours.

First of all, my dog is sweet, but I don't think he's winning any IQ contests. And, aside from their possible lack of accuracy, these kinds of comparisons are kind of mean.

But comparisons can serve useful purposes - math purposes. And, I don't mean abstract, make-you-nervous, pop quiz math purposes; I mean legitimately useful math purposes. Let's look at how this works.

Definitions

First, let's get some definitions on the table to give us a framework. When we're talking about comparisons, we mean ratios and proportions.

A ratio is a comparison between two things. For example, a soccer team may need 5 soccer balls for every 10 players during practice. The ratio of soccer balls to players if 5:10.

A proportion is a pair of ratios that are equal to each other. It's like this: a/b = c/d or a:b = c:d. For example, let's say you save $5 out of every $100 you earn, so you save at a ratio of 5:100. If you earn $500, you'll save $25. The proportion can be written as 5/100 = 25/500.

Ratio Problems

Okay, now let's try some math. Let's start with ratio problems.

Here's a bowl of fruit:

The ratio of apples to oranges is 3:4.
image of bowl of fruit for ratio example

What is the ratio of apples to oranges? There are 3 apples and 4 oranges, so the ratio is 3:4. Now, I like apples way more than oranges, so this ratio just won't do. I'm more of an all-apples-to-no-oranges ratio kind of guy.

Let's talk about milk. You can buy milk in quarts or gallons or other sizes not relevant to this question. What's the ratio of quarts to gallons? This question is really asking how many quarts are in a gallon. If you know your milk, you know it's 4. So, the ratio of quarts to gallons is 4:1. Incidentally, this ratio holds true for chocolate milk; it's just tastier that way.

Also, always pay attention to the order of the ratio. 4:1 is not the same as 1:4.

How about this? In a town with 400 commuters, 120 people drive to work. The rest ride their bikes. What is the ratio of drivers to cyclists? We just need to know the number of each. We know there are 120 drivers. And, if there are a total of 400 commuters, then the cyclists must be 400 - 120, or 280. So, the ratio of drivers to cyclists is 120:280. This is one bike-loving town.

Proportion Problems

Let's get more complicated. What if Angie's Car Batteries and Decorative Garden Gnomes sells car batteries and gnomes in a ratio of 35:2? By the time Angie has sold 280 car batteries, how many gnomes did she sell?

We just need to set this up as a proportion. Again, that's two ratios that are equal to each other. Our first ratio is 35:2. Our second ratio is 280:g, where g is the number of gnomes. Just cross-multiply, and we find that Angie has sold 16 gnomes. We also learned that Angie should probably stick to car batteries, as those gnomes just aren't moving very well.

What about this? In the annals of garage rock band history, guitar players outnumber drummers 13:3. In Rockville, which is your typical garage rock-loving town, there are 12 drummers. How many guitar players are there?

We set this up as 13:3 = g:12, where 12 is our Ringos and g is our number of aspiring Eddie Van Halens. Just solve for g. We get 3g = 13*12. So, 3g = 156, and g = 52. So, Rockville has 52 guitarists. Some of those people are really going to need to learn bass.

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