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Ratios, rates, and proportions | 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 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
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

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 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 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:
139=x993=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:
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 Units 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:
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


Want to join the conversation?

  • stelly orange style avatar for user nnennao3
    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)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Hecretary Bird
      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)
  • blobby green style avatar for user mzann
    can I get more stuff that can help me with ratio need more help
    (3 votes)
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  • blobby green style avatar for user miladhmeda
    i really don't get it could you please explain i.
    (3 votes)
    Default Khan Academy avatar avatar for user
  • cacteye purple style avatar for user ishnita
    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)
    Default Khan Academy avatar avatar for user