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Geometry: High School15 chapters | 160 lessons

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how ratios and proportions are related. Also, learn how ratios and proportions are used in real life and how you can apply them to yours.

**Ratios** are used to compare values. They tell us how much of one thing there is compared to another. For example, ratios can be used to compare the number of girl puppies to boy puppies that were born. If we have a total of six puppies, where two are girls and four are boys, we can write that in ratio form as 2:4 (girls:boys). We can also write it in factor form as 2/4. To compare the number of boy puppies to girl puppies, we can simply rewrite our ratio with the number of boys first as 4:2 (boys:girls) or 4/2.

**Proportions** are related to ratios in that they tell you when two ratios are equal to each other. Let's see how proportions work for our puppies. Our first ratio of girls to boys is 2:4 for our litter of six. If our next litter had a ratio of 4:8 of girls to boys, it would be proportional to our first litter; because if we divide each of our ratios, we will find that they are equal: 2 / 4 = 0.5 and 4 / 8 = 0.5, as well. My ratios are proportional if they divide into the same number.

When things are proportional, they are also similar to each other, meaning that the only difference is the size. For our two litters of puppies, the ratio of girls to boys is the same. The only difference is that the second litter is twice as big as the first.

In the real world, ratios and proportions are used on a daily basis. Cooks use them when following recipes. I have a recipe for hummingbird food that calls for one part water to four parts sugar. In ratio form, the amount of water to sugar is 1:4. I can use one cup of water to four cups of sugar to make food for the hummingbirds. To make a bigger batch of hummingbird food, I use proportions to increase my batch. I can double it by doubling the ratio to 2:8. My two ratios, 1:4 and 2:8, are still the same since they both divide into the same number: 1 / 4 = 0.25 and 2 / 8 = 0.25.

Ratios and proportions are also used in business when dealing with money. For example, a business might have a ratio for the amount of profit earned per sale of a certain product such as $2.50:1, which says that the business gains $2.50 for each sale. The business can use proportions to figure out how much money they will earn if they sell more products. If the company sells ten products, for example, the proportional ratio is $25.00:10, which shows that for every ten products, the business will earn $25. These are proportional since both ratios divide into the same number: 2.50 / 1 = 2.5 and 25 / 10 = 2.5, also.

Just like these examples show, you can use ratios and proportions in a similar manner to help you solve problems. If a problem asks you to write the ratio for the number of apples to oranges in a certain gift basket, and it shows you that there are ten apples and 12 oranges in the basket, you would write the ratio as 10:12 (apples:oranges).

If the problem continues and asks you to make the gift basket three times bigger while maintaining the proportion of apples to oranges, you can do this by multiplying both numbers in the ratio by the amount you are increasing, in this case three. So, to triple our gift basket, we would multiply our 10 by three and our 12 by three to get 30:36 (apples:oranges). We can do this because we remember from algebra that multiplying a mathematical expression by the same number on both sides keeps the expression the same. We can check to see if our ratios are the same by dividing each of them: 10 / 12 = 0.833 and 30 / 36 = 0.833, which are equal. Because they are equal, it tells us that they are proportional.

What did we learn? We learned that **ratios** are value comparisons, and **proportions** are equal ratios. Ratios can be written with colons or as fractions. So, to compare the number of girls to boys in a litter of puppies, we can write 2:4 or 2/4 to say that there are two girls to four boys. If we double the litter size but the number of girls to boys changes to 4:8, we can say that both litters are in proportion since both ratios divide into the same number. And as we saw, ratios and proportions are used every day by cooks and business people, to name just a few.

Following this lesson, you should have the ability to:

- Define ratios and proportions and explain the relationship between them
- Identify two ways to write ratios
- Explain how to check whether two ratios are proportionate

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Geometry: High School15 chapters | 160 lessons

- Ratios and Proportions: Definition and Examples 5:17
- Angle Bisector Theorem: Definition and Example 4:58
- Solving Problems Involving Proportions: Definition and Examples 5:22
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- Constructing Similar Polygons 4:59
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- The Pythagorean Theorem: Converse and Special Cases 5:02
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