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## SAT

### Course: SAT > Unit 6

Lesson 5: Problem Solving and Data Analysis: lessons by skill- 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

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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:

- 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 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 Units lesson.

### Try it

## Your turn!

## 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?(10 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)

- can I get more stuff that can help me with ratio need more help(3 votes)
- i really don't get it could you please explain i.(3 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)

- Can someone help me and explain on this question "A high school randomly selected

50

5050 students to take a survey about extending their lunch period. Of the students selected for the survey,

14

were freshmen and

13

were sophomores. at the end of this question, it says "14:50is a (my answer part to whole) ratio we also could reduce that to7:25which was my answer but why do we have to reduce it and how do we reduce it(0 votes)- We have to reduce this answer because all the multiple choice answer options on the SAT will be in their reduced, simplified form. To reduce a fraction or ratio to a simplified form, simply find a common factor in both the numerator and denominator and take that out. Here, 14 is 2*7 and 50 is 2*25. If we separate the 2 out, we get 7/25.

14/50 = (2*7) / (2*25) = (2/2) * (7/25) = 1 * (7/25) = 7/25(2 votes)