Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.
Proportions & Ratios
This lesson is mostly dedicated to practice, but let's start with a quick review of ratios and proportions. A ratio is a comparison between two different quantities. For example, if you have 4 boys and 3 girls in a room, the ratio of boys to girls is 4 to 3.
Ratios can be expressed either with fractions or with a colon.
You can reduce ratios just like fractions. So, for example, the ratio of 4:3 is the same thing as the ratio of 16:12 or the ratio of 40:30.
Ratios come in two flavors: part-to-part or part-to-whole. An example of a part-to-part ratio is the ratio of boys to girls; an example of a part-to-whole ratio is the ratio of boys to all the students in the class.
A proportion is a statement that two ratios are equal. If you know three terms in a proportion, you can always solve for the fourth. So for example, if you know that 4/3 equals 16/x, you can solve to find that x equals 12.
When you set up proportions in SAT math problems, always be careful that the ratios you're comparing are actually the same. For example, you can't set up a proportion between a part-to-part and a part-to-whole ratio.
Now with that out of the way, let's look at a few examples. We'll start with one that's pretty simple.
In a certain kingdom, the ratio of dragons to princesses is 5:2. If there are 12 princesses in the kingdom, how many dragons are there?
First, we'll take the information in the problem to set up a proportion. 5/2 equals d/12.
Notice how we set up the ratios of dragons over princesses on both sides - remember that you have to keep the ratios the same on both sides of the proportion.
Now we'll cross-multiply to solve for d, or the number of dragons.
We learn that the number of dragons is 30. We can check this by plugging it back in to the original proportion to see if the ratio of dragons to princesses reduces to 5/2.
That checks out: 30/12 reduces to 5/2 if you divide both top and bottom by 6. So we know we've found the correct answer.
Ready for one that's a little tougher?
A bin of yarn contains red yarn and green yarn. If there are 3 balls of red yarn for every 7 balls of green yarn and the box contains 40 balls of yarn in total, how many balls of green yarn are there?
First, we'll take the information from the problem to set up our ratio. We know the ratio of red to green is 3:7. We also know that the total number of balls of yarn of both colors is 40.
Here we need to be very careful to avoid mixing up part-to-part and part-to-whole. The 3:7 ratio is a part-to-part comparison, but the number 40 is describing the whole. We cannot set up a proportion comparing unlike quantities!
In order to get something that we can compare to the 40, we need another ratio of part-to-whole.
This isn't hard to get, though.
If we look at the 3:7 ratio, we can see that together, the two parts form a group of 10 balls (3 + 7).
The whole in this case is just the sum of the red and the green. So if we want a ratio of green balls to all the balls, we'll just take 7:10.
Now we can set up a proportion with matching terms.
We'll cross-multiply and solve, just like before:
And we get that g equals 28.
Last problem: this one is a little challenging, but just stick with it.
Jim's goody bags contain candy bars, stickers, and toys to the ratio of 6:2:1. If each bag contains 8 stickers, how many total items does it contain?
This one might look harder because it's a part-to-part-to-part ratio, with three quantities being compared. To make this simpler, we'll start by breaking the big ratio up into two part-to-part ratios:
Now we can plug in 8 for the number of stickers in each of the two proportions and figure out the number of other items in each bag.
There are 24 candy bars and 4 toys.
Now we just add the numbers together: 24 + 8 + 4 = 36. So there are 36 total items in each bag.
In this lesson, you practiced using proportions and ratios to solve three problems:
- A pretty basic ratio setup
- A proportion with a part-to-whole twist
- A three-part ratio that you had to break into smaller groups
The key to these questions is to keep your work clear and organized. Also, make sure you're always keeping track of what you're comparing to what. Remember that you can only compare like quantities, so you can't set up a proportion between a part-to-part ratio and a part-to-whole ratio.
With the basic principles in mind, most SAT ratio problems should be pretty manageable. Ready to try your hand at a few of your own? Check out the quiz questions for some practice!
You will have the ability to do the following after watching this video lesson:
- Define ratio and proportion
- Describe what part-to-part and part-to-whole ratios are
- Solve basic and complex proportion problems
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