- Ratios, rates, and proportions | Lesson
- Percents | Lesson
- Units | Lesson
- Table data | Lesson
- Scatterplots | Lesson
- Key features of graphs | Lesson
- Linear and exponential growth | Lesson
- Data inferences | Lesson
- Center, spread, and shape of distributions | Lesson
- Data collection and conclusions | Lesson
What are ratios, rates, and proportions, and how frequently do they appear on the test?
A ratio is a comparison of two quantities. The ratio of to can be expressed as or .
A proportion is an equality of two ratios. We write proportions to help us find equivalent ratios and solve for unknown quantities.
A rate is the quotient of a ratio where the quantities have different units.
In this lesson, we'll:
- Learn to convert between part-to-part and part-to-whole ratios
- Practice setting up proportions to solve for unknown quantities
- Use rates to predict unknown values
On your official SAT, you'll likely see 2 to 4 questions about ratios, rates, and proportions.
You can learn anything. Let's do this!
How do we identify and express ratios?
Identifying a ratio
Finding complementary ratios
Two common types of ratios we'll see are part-to-part and part-to-whole.
For example, if we're making lemonade:
- The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amount of two ingredients.
- The ratio of lemon juice to lemonade is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.
Since all the parts need to add up to the whole, part-to-part and part-to-whole ratios often imply each other. This means we can use the ratio(s) we're provided to find whichever ratio(s) we need to solve a problem!
Note: Just as fractions can be simplified, ratios can be reduced or expanded to find equivalent ratios. For example, the ratio means the same thing as the ratio .
Try: Identify parts and wholes
A high school randomly selected students to take a survey about extending their lunch period. Of students selected for the survey, were freshmen and were sophomores.
- is a ratio.
- is a ratio.
- is a ratio, which could be reduced to.
Try: find complementary ratios
A bag is filled with red marbles and blue marbles. There are total marbles in the bag, and of the marbles are blue.
The ratio of blue marbles to total marbles is
The ratio of red marbles to total marbles is
The ratio of red marbles to blue marbles is
How many red marbles are in the bag?
How do we use proportions?
Solving word problems using proportions
If we know a ratio and want to apply that ratio to a different scenario or population, we can use proportions to set up equivalent ratios and calculate any unknown quantities.
For example, say we're making cookies, and the recipe calls for cup of sugar for every cups of flour. What if we want to use cups of flour: how much sugar do we need?
- The ratio of sugar to flour must be to match the recipe.
- The ratio of sugar to flour in our batch can be written as .
To determine how much sugar we need, we can set up the proportion and solve for :
We need cups of sugar.
Note: There are multiple ways to set up a proportion. For a proportion to work, it must keep the same units either on the same side of the equation or on the same side of the divisor line.
To use a proportional relationship to find an unknown quantity:
- Write an equation using equivalent ratios.
- Plug in known values and use a variable to represent the unknown quantity.
- Solve for the unknown quantity by isolating the variable.
Example: There are students at Du Bois Academy. If the student-to-teacher ratio is , how many teachers are there?
Try: Set up a proportion
A local zoo houses penguins for every lion it houses. The zoo houses penguins.
Which proportion(s) would allow us to solve for , the number of lions housed at the zoo?
How do we use rates?
Finding a per unit rate
Applying a per unit rate
Rates are used to describe how quantities change. Common rates include speed () and unit price ().
For instance, if we know that a train traveled miles in two hours, we can calculate a rate that will tell us the train's average speed over those two hours:
We can then use that rate to predict other quantities, like how far that same train, traveling at the same rate, would travel in hours:
Note: When working with rates on the SAT, you may need to do unit conversions. To learn more about unit conversions, see the Units lesson.
Try: Calculate the unit price
Tony buys large pizzas for before tax.
The price for a single large pizza is
The price of large pizzas before tax would be
Practice: Apply a ratio
There are two oxygen atoms and one carbon atom in one carbon dioxide molecule. How many oxygen atoms are in carbon dioxide molecules?
Practice: Solve a proportion
Building is feet tall, and Building is feet tall. The ratio of the heights of Building to Building is equal to the ratio of the heights of Building to Building . If Building is feet tall, what is the height of Building to the nearest foot?
Practice: Use a rate
The -inch tires on a pickup truck have a circumference of feet. To the nearest whole rotation, how many rotations must the tires make for the truck to travel miles in straight line? ()
Want to join the conversation?
- For the last question, why is the answer 1121 and not 1122? After completing 1121 rotations, there's still .01911 rotations left. Shouldn't the answer be rounded up?(9 votes)
- If the answer needs to be rounded a different way than normal, then there will be something in the question that tells you to do so. For example, you could have: "What is the minimum number of boxes Usnavi needs to package all the treats?". Here, if you got a decimal, then you would have to round up regardless of whether it was greater or less than 0.5.
You don't have anything of the sort in this question. It simply asks for the number of rotations, rounded to the nearest whole. Therefore, you round as normal, which keeps the answer at 1121.(7 votes)
- So if 1/3 of the marbles are blue, how does that translate to 1:3 of the marbles being blue? 1/3 as a percent is 33.32%, but the ratio 1:3 means that for every one blue marble, there are 3 green marbles. So, as a percent, that would be 25%. Or is it different because it’s a proportion and not a fraction?(3 votes)
- Whenever you have a ratio, you have to make sure of what it's comparing. Here, the ratio we're asked to find is blue marbles to total marbles, instead of to green marbles. This means that it's just the same as the fraction. For every one blue marble, there are 3 marbles in total.(1 vote)