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## Digital SAT Math

### 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, colon, b or start fraction, a, divided by, b, end fraction.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

**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 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, colon, 10 means the same thing as the ratio 1, colon, 2.

### Try it!

## How do we use proportions?

### Writing 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 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, colon, 3 to match the recipe.
- The ratio of sugar to flour in our batch can be written as x, colon, 9.

To determine how much sugar we need, we can set up the proportion start fraction, 1, divided by, 3, end fraction, equals, start fraction, x, divided by, 9, end fraction and solve for x:

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:

- 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 340 students at Du Bois Academy. If the student-to-teacher ratio is 17, colon, 2, how many teachers are there?

### Try it!

## 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 (start fraction, start text, d, i, s, t, a, n, c, e, end text, divided by, start text, t, i, m, e, end text, end fraction) and unit price (start fraction, start text, t, o, t, a, l, space, p, r, i, c, e, end text, divided by, start text, u, n, i, t, s, space, p, u, r, c, h, a, s, e, d, end text, end fraction).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.

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## Your turn!

## Want to join the conversation?

- easy peasy lemon squeezy(23 votes)
- What I’m really goodat(1 vote)

- This unit is kinda hard... 🥲💔any tips?(7 votes)
- 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.(2 votes)

- Is anyone giving sat on june 3(4 votes)
- what's taught here doesn't seem to be enough for solving questions on the SAT practices(4 votes)
- you can use a calculator in this unit(3 votes)
- Where can I find the unit conversion lesson?(2 votes)
- how are we making the ratios equal if they are 2 different entities(1 vote)