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Digital SAT Math

Course: Digital SAT Math > Unit 3

Lesson 1: Ratios, rates, and proportions: foundations
Test prep
Digital SAT Math
Foundations: Problem solving and data analysis
Ratios, rates, and proportions: foundations
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Ratios, rates, and proportions | SAT lesson

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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:
  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

Khan Academy video wrapper
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, colon, 10 means the same thing as the ratio 1, colon, 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, colon, 13 is a
    ratio.
  • 13, colon, 50 is a
    ratio.
  • 14, colon, 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 start fraction, 1, divided by, 3, end fraction 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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

How do we use proportions?

Writing proportions

Khan Academy video wrapper
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, 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:
13⋅9=x9⋅93=x\begin{aligned} \dfrac{1}{3}\purpleD{\cdot9} &= \dfrac{x}{9}\purpleD{\cdot9}\\\\ 3&=x \end{aligned}
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, colon, 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?
Choose all answers that apply:

How do we use rates?

Finding a per unit rate

Khan Academy video wrapper
Solving unit rate problemSee video transcript

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:
start fraction, 120, start text, space, m, i, l, e, s, end text, divided by, 2, start text, space, h, o, u, r, s, end text, end fraction, equals, 60, start text, space, m, i, l, e, s, space, p, e, r, space, h, o, u, r, end text
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:
start fraction, 60, start text, space, m, i, l, e, s, end text, divided by, 1, start cancel, start text, space, h, o, u, r, end text, end cancel, end fraction, dot, 5, start cancel, start text, space, h, o, u, r, s, end text, end cancel, equals, 300, start text, space, m, i, l, e, s, end text
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 dollar sign, 77, point, 94 before tax.
The price for a single large pizza is dollar sign
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.
The price of 10 large pizzas before tax would be dollar sign
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
.

Your turn!

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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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?
Choose 1 answer:

Practice: Use a rate
The 36-inch tires on a pickup truck have a circumference of 9, point, 42 feet. To the nearest whole rotation, how many rotations must the tires make for the truck to travel 2 miles in straight line? (1, start text, space, m, i, l, e, end text, equals, 5, comma, 280, start text, space, f, e, e, t, end text)
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

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