E38: Teacher Certification Podcast | FTCE | General Knowledge | Mathematics | Ratios and Proportions
Ratios - Definition
A ratio is a comparison of two quantities. For example, if we have a bowl containing 3 red apples and 4 green apples, the ratio of red apples to green apples is 3 to 4. But how do we write that mathematically? Ratios are often expressed in colon form, such as 3 colon 4 or in fraction form which would be 3 over 4. Explanation: We have 3 red apples for every 4 green apples.
Mmmm… I love apples. Ok, where were we? Ah yes, Proportions!
Proportions - Definition
Now that we understand Ratios we can explain Proportions! Because basically proportions involve two ratios that are equal to each other.
What? Ok, Ok, get this. If we have two bowls with the same ratio of red to green apples, say 3 to 4, then they are in proportion. And we would express this in fraction form meaning 3 over 4 is equal to 3 over 4 signifying bowl one is equal to bowl two. Why is this important?
Let’s try to apply this to an example:
Let’s apply our knowledge of ratios and proportions to ingredients to a real-world problem like cooking. Let's say you have a recipe that serves 4 people, but are cooking for 8 people, sooo you’ve got to double the recipe, right? Well, how do you do that?
You can use proportions to scale up the ingredients. For example, If the original recipe calls for 2 cups of flour to serve 4 people thats 2 cups to 4 people. The question we are asking ourselves is how many cups for 8 people. We don’t know yet so we will let x equal the unknown amount of cups and 8 equal the total amount of people. Therefore, set up a proportion using the fraction form: 2 over 4 equals x over 8.
2/4 = x/8
Solving for x, you will cross multiply: 4 * x = 2 * 8. This simplifies to 4x = 16. Next solve for our unknown “x” by dividing both sides by 4, and you get x = 4. Therefore, you'll need 4 cups of flour to double the recipe.
Now there are few different ways to set that up, but what we did was keep it equal by having the cups to people equals cups to people proportion when solving for our unknown.
Stay with me now. You seem a little unsure so let’s try the example I was talking about earlier with the building and the statue from the FTCE Mathematics Practice test:
The problem states: A building 51 feet tall casts a shadow 48 feet long. Simultaneously, a nearby statue casts a shadow of 16 feet. How tall is the statue?
What? Where do we even start? Wait! I know, we can set this up using a proportion. So we have a building that is 51 feet tall with a 48 foot shadow. I’m going to go ahead and set up this ratio as 51 to 48 using the fraction form 51 over 48.
Next I have a statue that is “I dont’ know how tall” and a shadow of 16 feet. What do I do with an unknown? Yes! That’s right, I assign it as “x”. Now I can set up this ratio as x to 16 or in fraction form x over 16.
Let’s set these two ratios up using a proportion to solve for the height of the statue. This is going to look like building height over shadow length equals statue height over statue length. The numbers for this are 51 over 48 equals x over 16.
51/48 = x/16
Solving for x, you will cross multiply: 48 * x = 51 * 16. This simplifies to 48x = 816. Next solve for our unknown “x” by dividing both sides by 48, and you get x = 17. Therefore, the height of the statue is 17 feet, basically the statue is 17 feet tall.
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