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### Course: Digital SAT Math>Unit 3

Lesson 1: Ratios, rates, and proportions: foundations

# Ratios, rates, and proportions | SAT lesson

A guide to ratios, rates, and proportions on the digital SAT

## What are ratios, rates, and proportions?

A ratio is a comparison of two quantities. The ratio of $a$ to $b$ can be expressed as $a:b$ or $\frac{a}{b}$.
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:
1. Learn to convert between part-to-part and part-to-whole ratios
2. Practice setting up proportions to solve for unknown quantities
3. Use rates to predict unknown values
You can learn anything. Let's do this!

## How do we identify and express ratios?

### Identifying a ratio

Part:whole ratiosSee video transcript

### 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 amounts 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 $5:10$ means the same thing as the ratio $1:2$.

### Try it!

Try: Identify parts and wholes
A high school randomly selected $50$ students to take a survey about extending their lunch period. Of students selected for the survey, $14$ were freshmen and $13$ were sophomores.
• $14:13$ is a
ratio.
• $13:50$ is a
ratio.
• $14:50$ is a
ratio, which could be reduced to
.

Try: find complementary ratios
A bag is filled with red marbles and blue marbles. There are $54$ total marbles in the bag, and $\frac{1}{3}$ 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?

### Writing proportions

Writing proportions exampleSee video transcript

### 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 $1$ cup of sugar for every $3$ cups of flour. What if we want to use $9$ cups of flour: how much sugar do we need?
• The ratio of sugar to flour must be $1:3$ to match the recipe.
• The ratio of sugar to flour in our batch can be written as $x:9$.
To determine how much sugar we need, we can set up the proportion $\frac{1}{3}=\frac{x}{9}$ and solve for $x$:
$\begin{array}{rl}\frac{1}{3}\cdot 9& =\frac{x}{9}\cdot 9\\ \\ 3& =x\end{array}$
We need $3$ 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:
1. Write an equation using equivalent ratios.
2. Plug in known values and use a variable to represent the unknown quantity.
3. Solve for the unknown quantity by isolating the variable.
Example: There are $340$ students at Du Bois Academy. If the student-to-teacher ratio is $17:2$, how many teachers are there?

### Try it!

Try: Set up a proportion
A local zoo houses $13$ penguins for every lion it houses. The zoo houses $78$ penguins.
Which proportion(s) would allow us to solve for $x$, the number of lions housed at the zoo?

## How do we use rates?

### Finding a per unit rate

Solving unit rate problemSee video transcript

### Applying a per unit rate

Rates are used to describe how quantities change. Common rates include speed ($\frac{\text{distance}}{\text{time}}$) and unit price ($\frac{\text{total price}}{\text{units purchased}}$).
For instance, if we know that a train traveled $120$ 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 $5$ hours:
Note: When working with rates on the SAT, you may need to do unit conversions. To learn more about unit conversions, see the Unit conversion lesson.

### Try it

Try: Calculate the unit price
Tony buys $6$ large pizzas for $\mathrm{}77.94$ before tax.
The price for a single large pizza is $\mathrm{}$
.
The price of $10$ large pizzas before tax would be $\mathrm{}$
.

Practice: Apply a ratio
There are two oxygen atoms and one carbon atom in one carbon dioxide molecule. How many oxygen atoms are in $78$ carbon dioxide molecules?

Practice: Solve a proportion
Building $A$ is $140$ feet tall, and Building $B$ is $85$ feet tall. The ratio of the heights of Building $A$ to Building $B$ is equal to the ratio of the heights of Building $C$ to Building $D$. If Building $C$ is $90$ feet tall, what is the height of Building $D$ to the nearest foot?

Practice: Use a rate
The $36$-inch tires on a pickup truck have a circumference of $9.42$ feet. To the nearest whole rotation, how many rotations must the tires make for the truck to travel $2$ miles in straight line? ()

## Want to join the conversation?

• I'm passing my exam on November 4th. Good luck everybody!
• Same here, good luckk :3
• Hello guys I’m going to give SAT on 26th august…I really haven’t touched maths and the exam is in 6 days. I’m going to complete everything in 6 days and become a math GOD 🗿. I’ll come back to share my results Lolll. My English score in mocks have been around 600-650 so I have to improve those too 🥹
• Put in the work and you'll get there. I'm in the same situation and I am grinding as much as I can. We'll get there mate!
• the advanced version of this lesson makes me just stare at my laptops screen clueless
• swrs, i cant even study the advanced math. but i'll try my possible best
• Hello! Fellow test takers, just like you all I am also preparing for my test, I wish you all ,and wish me too
Good luck out there test takers you've got this
• easy peasy lemon squeezy
• What I’m really goodat
• This unit is kinda hard... 🥲💔any tips?
• There are other ways that might help u better understand the concept, you could watch youtube videos or solve questions about the same concept until you understand.
• what's taught here doesn't seem to be enough for solving questions on the SAT practices
• This is the foundation level. It helps to set your base and also helps you to realise your weak points at an early level.
After completing foundations, move on to medium and then advanced level. It is also recommended for you to take a practice test in-between.
For example, after you complete advanced level (both eng and maths) take a practice test on the bluebook app. Understand your mistakes and then move onto medium level. Take practice test again after you finish medium level and then move onto advanced level.
Repeat the same with advanced level.