SAT Ratios & Proportions: Lesson, Practice Problems & Video Explanations

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By Glyn Caddell, BSME · Author of Digital SAT Math Secrets · Tutoring since 2008 — Updated August 18, 2026

Ratios and proportions questions ask students to analyze the relationship between quantities, determine how one value compares to another, and solve problems involving equivalent ratios. These questions often require students to simplify ratios, scale values up or down, and use cross-multiplication to find missing terms. They may appear in contexts such as comparing ingredients in a recipe, analyzing speed or density, or solving word problems involving part-to-whole or part-to-part comparisons. Ratio & proportion questions appear on approximately 5.7% of the Digital SAT. Ratio & proportion falls under Problem-Solving and Data Analysis according to the College Board.

Table of Contents

Ratios & Proportions Lesson
Tips
Example
- Video Explanation
- Text Explanation
Practice Problems with Video Explanations
Common Mistakes
FAQs
Related Topics

Ratios & Proportions Lesson

The lesson covers the types of ratios & proportions questions that show up on the Digital SAT. The lesson includes understanding the terms, solving proportions, ratios with a total, matching units, ratios in formulas, and ratios in shapes & similar figures.

What Do Ratios & Proportions Questions Look Like?

Here are some examples of Ratios & Proportions questions from the Digital SAT.

At this speed, what is the time, in seconds, that it would take for the object to travel 108 centimeters?
Which equation represents the total amount of paint $P$ , in gallons, needed to paint the walls of the room twice?
What is the area, in square miles, of this park?
If the width of the rectangular region increases by 7 units, how must the length change to maintain this ratio?
What is this rate, in miles per minute squared, rounded to the nearest tenth?
The area of circle B is how many times the area of circle A?

1. Understanding the Terms

An example of a rate is 25 miles/hr, which means the object will move 25 miles for every hour that passes.

An example of a ratio is 2 eggs:3 ounces of flour, which means the recipe calls for 2 eggs for every 3 ounces of flour used.

A statement that 2 rates or ratios are equal is a proportion.

For example, this is a proportion:

$\dfrac{2}{3}=\dfrac{8}{12}$

We can check if the proportion is correct by cross-multiplying

If we cross multiply, we get

$(2)(12)=(8)(3)$

which simplifies to

$24=24$

This proves that the proportion is correct.

2. How to Set Up & Solve a Proportion

When setting up a proportion, always write down how you will set it up.

For example, if we know that someone reads 13 pages every 4 minutes, we can find out how many pages he would read in 22 minutes.

Write down how it would be set up first.
$\dfrac{pages}{minute}$

Then insert the values.

$\dfrac{13}{4}=\dfrac{x}{22}$

Use cross-multiplication to solve

$\dfrac{13}{4}=\dfrac{x}{22}$

becomes

$(13)(22)=(x)(4)$

Simplify to get

$286=4x$

Divide both sides by 4 to get

$71.5=x$

That means based on the rate, the person would read 71.5 pages in 22 minutes.

3. How to Solve a Problem with a Ratio and a Total

A common type of proportion question on the SAT is a word problem in which a ratio is given. However, instead of giving you a value of one of the elements in the ratio, the question gives you the total of the elements in the ratio.

To solve for values when a ratio and a total is given, multiple each number in the ratio by x and then create an equation in which they all add up to the total. Then solve for x, and use that value of x to determine the other values.

Example

The ratio of boys to girls in a class is 2:3. If there are 45 students, how many boys are there?

Explanation & Solution

We know that the ratio of boys to girls is $2:3$ . That means the numbers could be 2 & 3, 4 & 6, 6 & 9. 8 & 12, etc.

What’s important is that whatever we multiply 2 by (for boys) we have to multiply 3 by (for girls). We can represent the number of boys as $2x$ and the number of girls as $3x$ .

We know that

(# of boys) + (# of girls) = (total # of students)

We can write the equation as

$2x+3x=45$

Combine like terms to get

$5x=45$

divide both sides by 5

$x=9$

This is not the answer. The question asks for the number of boys.

Number of boys is $2x$ , so $2(9) =18$ .

4. Use Matching Units in the Proportion

A way the SAT tests your knowledge of proportions is to mix up units on proportion questions.

Make sure to correct the units if they are different, before setting up the proportion.

Example

Tammy can read 10 pages in 4 minutes. How many pages can she read in 2 hours?

Explanation & Solution

Let’s set up the proportion as $\dfrac{pages}{minute}$

Now, let’s set it up.

If we write

$\dfrac{10}{4}=\dfrac{x}{2}$

This would be incorrect because in the denominator on the left the 4 is in minutes and in the denominator on the right the 2 is in hours. The units don’t match up.

Let’s rewrite the 2 hours as 120 minutes, so both sides are using minutes.

$\dfrac{10}{4}=\dfrac{x}{120}$

Now, we can cross-multiply

$(10)(120)=(x)(4)$

Simplify

$1,200=4x$

Divide both sides by 4

$300=x$

5. Ratios in Formulas

On the SAT, you may be asked to compare the results of a formula when certain values are changed to an original value in the form of a ratio.

When given a formula, we can determine how one value will be affected by altering other values.

For example,

The weight of an object, $F_g$ , is found by multiplying the mass of an object, $m$ , by gravity, $g$ .
$F_g=mg$

If the mass, $m$ , is doubled, the weight, $F_g$ , will be doubled.

Just like when we solve equations, whatever we do to one side of the equation, we have to do to the other side of the equation.

Tip to Find the Multiple

Sometimes, it is not obvious how doubling/tripling/halving one variable will impact the overall value. It’s especially not obvious when one of the variables is raised to an exponent.

A trick to find the answer is to first substitute 1 in for all of the variables and evaluate the formula. Make a note of the value.

Now, you are going to find the new value with the variables that are supposed to be doubled/halved/tripled/etc.

Since all of the initial variables were all 1, any multiple done to a variable will just make it that number:

Substitute in a 2 for a variable that is supposed to be doubled.
Substitute in a 1/2 for a variable that is supposed to be halved.
Substitute in a 3 for a variable that is supposed to be tripled.
Substitute in a 1/3 for a variable that is supposed to be divided by 3
& so on.

After substituting in the new values for the variables and using 1 for variables that don’t change, calculate the new value.

Divide the new value by the original value to find the multiple.

Example

The elastic potential energy in a spring $U_s$ is dependent on its spring constant, $k$ , and the change in length of the spring, $x$ , due to stretching or compressing the spring. How is the value of $U_s$ affected by doubling $k$ and halving $x$ , in the formula $U_s=\dfrac{1}{2}kx^2$ ?

A) $U_s$ remains the same
B) $U_s$ is doubled
C) $U_s$ is halved
D) $U_s$ is quadrupled

Explanation

The answer is C.

Start with substituting 1 in for all variables

$U_s=\dfrac{1}{2}kx^2$

becomes

$U_s=\dfrac{1}{2}(1)(1)^2$

evaluate to get

$U_s=\dfrac{1}{2}$

Now, find the value of $U_s$ when $k$ is doubled and $x$ is halved.

Substitute 2 in for $k$ and $\dfrac{1}{2}$ for $x$ into $U_s=\dfrac{1}{2}kx^2$ to get

$U_s=\dfrac{1}{2}(2)(\dfrac{1}{2})^2$

evaluate to get

$U_s=\dfrac{1}{4}$

Therefore the value of $U_s$ changed from $\dfrac{1}{2}$ to $\dfrac{1}{4}$ , which means it was halved.

You can check by dividing the new value by the original value

$\dfrac{(1/4)}{(1/2)} = \dfrac{1}{2}$

6. Proportions in Similar Figures

Similar figures and shapes have proportionate side lengths.

A useful way to write down how we are setting up a proportion regarding geometric shapes is:

$\dfrac{Large}{Small}$

When dealing with proportionate shapes, make sure that you’re dealing with corresponding lengths. It is common for a question to ask for the length of a segment and not a full side.

Example

What is the value of $x$ in the figure below?

Explantation & Solution

Let’s set up the proportion as

$\dfrac{Large}{Small}$ ,

$x$ is not a length of the small or large triangle. 9 is the height of the small triangle. Let’s use $y$ to represent the height of the large triangle. After we solve for $y$ , we’ll be able to find the value of $x$ .

Now, we cans set up the proportion as

$\dfrac{Large}{Small}$ ,

$\dfrac{y}{9}=\dfrac{8}{6}$ ,

cross-multiply to get

$6y=72$ ,

divide both sides by 6

$y=12$

Remember than the question is asking for $x$ . $y$ is the total length.

To find $x$ we have to subtract 9 from $y$ .

$12-9=3$ ,

$x=3$

7. Converting Units of Area

A proportion can be used for converting measurements.

However, on the Digital SAT, it is common to get a question about converting from one unit of area measurement to another, but the given conversion is for lengths.

For example,

The area of a park is 34.5 square miles. What is the area of the park in square feet? (1 mile = 5,280 feet)

A common mistake is to set up the proportion like this:

$\dfrac{1}{5,280}=\dfrac{34.5}{x}$

The units don’t match up.

The ratio of lengths has to be converted to a ratio for areas. All you have to do is square the ratio.

(1 mile = 5,280 feet)

becomes

(1 square mile = 27,878,400 square feet)

The correct ratio is:

$\dfrac{1}{27,878,400}=\dfrac{34.5}{x}$

Cross multiple to get

$x=961,804,800$

Tips

Write down how you’re setting up the proportion before plugging numbers in.
Avoid picking numbers like 0, 1, or 2 when testing ratios in tricky problems.
Keep units consistent; don’t mix minutes with hours or inches with feet.

Digital SAT Example

A rectangular patio has an area of 384 square feet. What is the area, in square inches, of the patio? (

A) 4,608
B) 18,432
C) 55,296
D) 663,552

Video Explanation

Text Explanation

Answer: C

Since the question asks for an area conversion, square feet must be converted to square inches.

If:

then:

So 384 square feet is equal to:

$= 55, 296$

Therefore, the area of the patio is 55,296 square inches.

Common Mistakes

Three common mistakes students make when answering ratios & proportions questions are filling in the proportion incorrectly, solving for $x$ but not answering the questions, and using the conversion of lengths for the conversion of areas.

Filling in the Proportion Incorrectly

A common mistake for ratios & proportions questions is filling in the values into the proportion incorrectly. I recommend writing down how you will set up the proportion before creating the proportion. For example, write $\dfrac{pages}{minutes}$ , so you know to write the number of pages in the numerator (top) and the corresponding number of minutes in the denominator (bottom).

Solving for x but not Answering the Question

A common mistake for ratios & proportions questions that involve a total is solving for $x$ , but forgetting to use that to find the actual number.

For example, if the total number of books on a shelf is 25 and the ratio of fiction to nonfiction books is 4:1, then you can solve for the number of each type of book.

Set up your equation:

(# of fiction books) + (# of nonfiction books) =25

$4x+x=25$

Combine like-terms

$5x=25$

Divide both sides by 5

$x=5$

If the question asked for the number of fiction books, 5 would be the wrong answer because the number of fiction books is $4x$ .

You have to substitute 5 in for $x$ , to get the number of fiction books

# of fiction books = $4x$ = $4(5)$ = $20$

Using the Conversion of Lengths for the Conversion of Areas

A common mistake in ratios & proportions questions is using the conversion of lengths as the conversion of area. To get the conversion for areas, you have to square both terms in the conversion of lengths. If you think about it, it makes sense because lengths use regular units and areas use units squared.

For example,

1 $yard$ = 3 $feet$

therefore

1 $yard^2$ = 9 $feet^2$

Frequently Asked Questions

Here are three frequently asked questions that I get from students when reviewing ratios & proportions.

When can I use a proportion for finding missing sides in a triangle?

You can use a proportion for finding missing sides in a triangle if you have two similar triangles. It’s important to know that two triangles are similar if two pairs of angles are congruent.

Then you can make a proportion using sides you know and the missing side.

I recommend writing down how you will set up the proportion.

For example, I would write:

$\dfrac{SideOfBigTriangle}{SideOfSmall Triangle}$

Make sure you match up corresponding sides in the proportion, such as the biggest side of the big triangle is matched with the biggest side of the small triangle.

How do I find an equivalent area in different units, if I am only given the conversion for lengths?

You can find an equivalent area in different units if you are given the conversion for lengths by squaring the values in the conversion.

For example, if you are given that

1 $mile$ = 5,280 $feet$

Square both values and units to get

1 $mile^2$ = 27,878,400 $feet^2$

Can I use Desmos to solve proportions?

Yes, if your proportion is set up with only one variable, then you can use Desmos to solve it without cross-multiplying. Make sure to use $x$ as your variable. If the question uses a different variable such as $t$ or $p$ , just switch it to $x$ when entering it in Desmos.

SAT Ratios & Proportions: Lesson, Practice Problems & Video Explanations

Ratios & Proportions Lesson

What Do Ratios & Proportions Questions Look Like?

1. Understanding the Terms

2. How to Set Up & Solve a Proportion

3. How to Solve a Problem with a Ratio and a Total

Example

Explanation & Solution

4. Use Matching Units in the Proportion

Example

Explanation & Solution

5. Ratios in Formulas

Tip to Find the Multiple

Example

Explanation

6. Proportions in Similar Figures

Example

Explantation & Solution

7. Converting Units of Area

Tips

Digital SAT Example

Video Explanation

Text Explanation

Practice Problems with Video Explanations

Common Mistakes

Filling in the Proportion Incorrectly

Solving for x but not Answering the Question

Using the Conversion of Lengths for the Conversion of Areas

Frequently Asked Questions

When can I use a proportion for finding missing sides in a triangle?

How do I find an equivalent area in different units, if I am only given the conversion for lengths?

Can I use Desmos to solve proportions?

Related Topics

Angles, Polygons, & 3-D Shapes

Circles

All Math Topics