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The Complete Guide to SAT Math Word Problems

Posted by Courtney Montgomery | Jun 1, 2018 8:00:00 AM

SAT Math

 

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About 25% of your total SAT Math section will be word problems, meaning you will have to create your own visuals and equations to solve for your answers. Though the actual math topics can vary, SAT word problems share a few commonalities, and we’re here to walk you through how to best solve them.

This post will be your complete guide to SAT Math word problems. We'll cover how to translate word problems into equations and diagrams, the different types of math word problems you’ll see on the test, and how to go about solving your word problems on test day.

Feature Image: Antonio Litterio/Wikimedia

 

What Are SAT Math Word Problems?

A word problem is any math problem based mostly or entirely on a written description. You will not be provided with an equation, diagram, or graph on a word problem and must instead use your reading skills to translate the words of the question into a workable math problem. Once you do this, you can then solve it.

You will be given word problems on the SAT Math section for a variety of reasons. For one, word problems test your reading comprehension and your ability to visualize information.

Secondly, these types of questions allow test makers to ask questions that'd be impossible to ask with just a diagram or an equation. For instance, if a math question asks you to fit as many small objects into a larger one as is possible, it'd be difficult to demonstrate and ask this with only a diagram.

 

Translating Math Word Problems Into Equations or Drawings

In order to translate your SAT word problems into actionable math equations you can solve, you’ll need to understand and know how to utilize some key math terms. Whenever you see these words, you can translate them into the proper mathematical action.

For instance, the word "sum" means the value when two or more items are added together. So if you need to find the sum of a and b, you’ll need to set up your equation like this: a+b.

Also, note that many mathematical actions have more than one term attached, which can be used interchangeably. 

Here is a chart with all the key terms and symbols you should know for SAT Math word problems:

Key Terms
Mathematical Action
Sum, increased by, added to, more than, total of
+
Difference, decreased by, less than, subtracted from
Product, times, __ times as much, __ times as many (a number, e.g., “three times as many”)
* or x
Divided by, per, __ as many, __ as much (a fraction, e.g., “one-third as much”)
/ or ÷
Equals, is, are, equivalent
=
Is less than
<
Is greater than
>
Is less than or equal to
Is greater than or equal to

 

Now, let's look at these math terms in action using a few official examples:

body_sat_math_sample_question_1
 
We can solve this problem by translating the information we're given into algebra. We know the individual price of each salad and drink, and the total revenue made from selling 209 salads and drinks combined. So let's write this out in algebraic form.
 
We'll say that the number of salads sold = S, and the number of drinks sold = D. The problem tells us that 209 salads and drinks have been sold, which we can think of as this:
 
S + D = 209
 
Finally, we've been told that a certain number of S and D have been sold and have brought in a total revenue of 836 dollars and 50 cents. We don't know the exact numbers of S and D, but we do know how much each unit costs. Therefore, we can write this equation:
 
6.50S + 2D = 836.5
 
We now have two equations with the same variables (S and D). Since we want to know how many salads were sold, we'll need to solve for D so that we can use this information to solve for S. The first equation tells us what S and D equal when added together, but we can rearrange this to tell us what just D equals in terms of S:
 
S + D = 209
 
Now, just subtract S from both sides to get what D equals:
 
D = 209  S
 
Finally, plug this expression in for D into our other equation, and then solve for S:
 
6.50S + 2(209 − S) = 836.5
6.50S + 418 − 2S = 836.5
6.50S − 2S = 418.5
4.5S = 418.5
S = 93
 

The correct answer choice is (B) 93.

 

body_sat_math_sample_question_2
 
This word problem asks us to solve for one possible solution (it asks for "a possible amount"), so we know right away that there will be multiple correct answers.
 
Wyatt can husk at least 12 dozen ears of corn and at most 18 dozen ears of corn per hour. If he husks 72 dozen at a rate of 12 dozen an hour, this is equal to 72 / 12 = 6 hours. You could therefore write 6 as your final answer.
 
If Wyatt husks 72 dozen at a rate of 18 dozen an hour (the highest rate possible he can do), this comes out to 72 / 18 = 4 hours. You could write 4 as your final answer.
 
Since the minimum time it takes Wyatt is 4 hours and the maximum time is 6 hours, any number from 4 to 6 would be correct.

 

body_Latin

Though the hardest SAT word problems might look like Latin to you right now, practice and study will soon have you translating them into workable questions.

 

Typical SAT Word Problems

Word problems on the SAT can be grouped into three major categories: 

  • Word problems for which you must simply set up an equation
  • Word problems for which you must solve for a specific value
  • Word problems for which you must define the meaning of a value or variable

Below, we look at each world problem type and give you examples.

 

Word Problem Type 1: Setting Up an Equation

This is a fairly uncommon type of SAT word problem, but you’ll generally see it at least once on the Math section. You'll also most likely see it first on the section.

For these problems, you must use the information you’re given and then set up the equation. No need to solve for the missing variable—this is as far as you need to go.

Almost always, you’ll see this type of question in the first four questions on the SAT Math section, meaning that the College Board consider these questions easy. This is due to the fact that you only have to provide the setup and not the execution.

body_sat_math_sample_question_3

To solve this problem, we'll need to know both Armand's and Tyrone's situations, so let's look at them separately:

Armand: Armand sent m text messages each hour for 5 hours, so we can write this as 5m—the number of texts he sent per hour multiplied by the total number of hours he texted.

Tyrone: Tyrone sent p text messages each hour for 4 hours, so we can write this as 4p—the number of texts he sent per hour multiplied by the total number of hours he texted.

We now know that Armand's situation can be written algebraically as 5m, and Tyrone's can be written as 4p. Since we're being asked for the expression that represents the total number of texts sent by Armand and Tyrone, we must add together the two expressions:

5m + 4p

The correct answer is choice (C) 5m + 4p

 

Word Problem Type 2: Solving for a Missing Value

The vast majority of SAT Math word problem questions will fall into this category. For these questions, you must both set up your equation and solve for a specific piece of information.

Most (though not all) word problem questions of this type will be scenarios or stories covering all sorts of SAT Math topics, such as averages, single-variable equations, and ratios. You almost always must have a solid understanding of the math topic in question in order to solve the word problem on the topic.

body_sat_math_sample_question_4

Let's try to think about this problem in terms of x. If Type A trees produced 20% more pears than Type B did, we can write this as an expression:

x + 0.2x = 1.2x = # of pears produced by Type A

In this equation, x is the number of pears produced by Type B trees. If we add 20% of x (0.2x) to x, we get the number of pears produced by Type A trees. 

The problem tells us that Type A trees produced a total of 144 pears. Since we know that 1.2x is equal to the number of pears produced by Type A, we can write the following equation:

1.2x = 144

Now, all we have to do is divide both sides by 1.2 to find the number of pears produced by Type B trees:

x = 144 / 1.2

x = 120

The correct answer choice is (B) 120.

You might also get a geometry problem as a word problem, which might or might not be set up with a scenario, too. Geometry questions will be presented as word problems typically because the test makers felt the problem would be too easy to solve had you been given a diagram, or because the problem would be impossible to show with a diagram. (Note that geometry makes up a very small percentage of SAT Math.)

body_SAT_word_problem_5

This is a case of a problem that is difficult to show visually, since x is not a set degree value but rather a value greater than 55; thus, it must be presented as a word problem. 

Since we know that x must be an integer degree value greater than 55, let us assign it a value. In this case, let us call x 56°. (Why 56? There are other values x could be, but 56 is guaranteed to work since it's the smallest integer larger than 55. Basically, it's a safe bet!)

Now, because = 56, the next angle in the triangle—2x—must measure the following:

56*2 = 112

Let's make a rough (not to scale) sketch of what we know so far:

body_triangle_ex_1

Now, we know that there are 180° in a triangle, so we can find the value of y by saying this:

y = 180 − 112 − 56

y = 12

One possible value for y is 12. (Other possible values are 3, 6, and 9.)

 

Word Problem Type 3: Explaining the Meaning of a Variable or Value

This type of problem will show up at least once. It asks you to define part of an equation provided by the word problem—generally the meaning of a specific variable or number.
 
body_sat_math_sample_question_6
 
This question might sound tricky at first, but it's actually quite simple.
 
We know that P is the number of phones Kathy has left to fix, and d is the number of days she has worked in a week. If she's worked 0 days (i.e., if we plug 0 into the equation), here's what we get:
 
P = 108  23(0)
 
P = 108
 
This means that, without working any days of the week, Kathy has 108 phones to repair. The correct answer choice, therefore, is (B) Kathy starts each week with 108 phones to fix.
 
 
body_juggle

To help juggle all the various SAT word problems, let's look at the math strategies and tips at our disposal.

 

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SAT Math Strategies for Word Problems

Though you’ll see word problems on the SAT Math section on a variety of math topics, there are still a few techniques you can apply to solve word problems as a whole.

 

#1: Draw It Out

Whether your problem is a geometry problem or an algebra problem, sometimes making a quick sketch of the scene can help you understand what exactly you're working with. For instance, let's look at how a picture can help you solve a word problem about a circle (specifically, a pizza):

body_sat_math_sample_question_7_2

If you often have trouble visualizing problems such as these, draw it out. We know that we're dealing with a circle since our focus is a pizza. We also know that the pizza weighs 3 pounds.

Because we'll need to solve the weight of each slice in ounces, let's first convert the total weight of our pizza from pounds into ounces. We're given the conversion (1 pound = 16 ounces), so all we have to do is multiply our 3-pound pizza by 16 to get our answer:

3 * 16 = 48 ounces (for whole pizza)

Now, let's draw a picture. First, the pizza is divided in half (not drawn to scale):

body_sat_math_sample_question_7_diagram_1

We now have two equal-sized pieces. Let's continue drawing. The problem then says that we divide each half into three equal pieces (again, not drawn to scale):

body_sat_math_sample_question_7_diagram_2

This gives us a total of six equal-sized pieces. Since we know the total weight of the pizza is 48 ounces, all we have to do is divide by 6 (the number of pieces) to get the weight (in ounces) per piece of pizza:

48 / 6 = 8 ounces per piece

The correct answer choice is (C) 8.

As for geometry problems, remember that you might get a geometry word problem written as a word problem. In this case, make your own drawing of the scene. Even a rough sketch can help you visualize the math problem and keep all your information in order.

 

#2: Memorize Key Terms

If you’re not used to translating English words and descriptions into mathematical equations, then SAT word problems might be difficult to wrap your head around at first. Look at the chart we gave you above so you can learn how to translate keywords into their math equivalents. This way, you can understand exactly what a problem is asking you to find and how you’re supposed to find it.

There are free SAT Math questions available online, so memorize your terms and then practice on realistic SAT word problems to make sure you’ve got your definitions down and can apply them to the actual test.

 

#3: Underline and/or Write Out Important Information

The key to solving a word problem is to bring together all the key pieces of given information and put them in the right places. Make sure you write out all these givens on the diagram you’ve drawn (if the problem calls for a diagram) so that all your moving pieces are in order.

One of the best ways to keep all your pieces straight is to underline your key information in the problem, and then write them out yourself before you set up your equation. So take a moment to perform this step before you zero in on solving the question.

 

#4: Pay Close Attention to What's Being Asked

It can be infuriating to find yourself solving for the wrong variable or writing in your given values in the wrong places. And yet this is entirely too easy to do when working with math word problems.

Make sure you pay strict attention to exactly what you’re meant to be solving for and exactly what pieces of information go where. Are you looking for the area or the perimeter? The value of x, 2x, or y?

It’s always better to double-check what you’re supposed to find before you start than to realize two minutes down the line that you have to begin solving the problem all over again.

 

#5: Brush Up on Any Specific Math Topic You Feel Weak In

You're likely to see both a diagram/equation problem and a word problem for almost every SAT Math topic on the test. This is why there are so many different types of word problems and why you’ll need to know the ins and outs of every SAT Math topic in order to be able to solve a word problem about it.

For example, if you don’t know how to find an average given a set of numbers, you certainly won’t know how to solve a word problem that deals with averages!

Understand that solving an SAT Math word problem is a two-step process: it requires you to both understand how word problems work and to understand the math topic in question. If you have any areas of mathematical weakness, now's a good time to brush up on them—or else SAT word problems might be trickier than you were expecting!

 

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All set? Let's go!
 

Test Your SAT Math Word Problem Knowledge

Finally, it's time to test your word problem know-how against real SAT Math problems:

 

Word Problems

1. No Calculator

body_sat_math_test_question_1

 

2. Calculator OK

body_sat_math_test_question_2

 

3. Calculator OK

body_sat_math_test_question_3

 

4. Calculator OK

body_sat_math_test_question_4

 


Answers: C, B, A, 1160

 

Answer Explanations

1. For this problem, we have to use the information we're given to set up an equation.

We know that Ken spent x dollars, and Paul spent 1 dollar more than Ken did. Therefore, we can write the following equation for Paul:

x + 1

Ken and Paul split the bill evenly. This means that we'll have to solve for the total amount of both their sandwiches and then divide it by 2. Since Ken's sandwich cost x dollars and Paul's cost x + 1, here's what our equation looks like when we combine the two expressions:

x + x + 1

2x + 1

Now, we can divide this expression by 2 to get the price each person paid:

(2x + 1) / 2

x + 0.5

But we're not finished yet. We know that both Ken and Paul also paid a 20% tip on their bills. As a result, we have to multiply the total cost of one bill by 0.2, and then add this tip to the bill. Algebraically, this looks like this:

(x + 0.5) + 0.2(x + 0.5)

x + 0.5 + 0.2x + 0.1

1.2x + 0.6

The correct answer choice is (C) 1.2x + 0.6

 

2. You'll have to be familiar with statistics in order to understand what this question is asking.

Since Nick surveyed a random sample of his freshman class, we can say that this sample will accurately reflect the opinion (and thus the same percentages) as the entire freshman class.

Of the 90 freshmen sampled, 25.6% said that they wanted the Fall Festival held in October. All we have to do now is find this percentage of the entire freshmen class (which consists of 225 students) to determine how many total freshmen would prefer an October festival:

225 * 0.256 = 57.6

Since the question is asking "about how many students"—and since we obviously can't have a fraction of a person!—we'll have to round this number to the nearest answer choice available, which is 60, or answer choice (B).

 

3. This is one of those problems that is asking you to define a value in the equation given. It might look confusing, but don't be scared—it's actually not as difficult as it appears!

First off, we know that t represents the number of seconds passed after an object is launched upward. But what if no time has passed yet? This would mean that t = 0. Here's what happens to the equation when we plug in 0 for t:

h(0) = -16(0)2 + 110(0) + 72

h(0) = 0 + 0 + 72

h(0) = 72

As we can see, before the object is even launched, it has a height of 72 feet. This means that 72 must represent the initial height, in feet, of the object, or answer choice (A).

 

4. You might be tempted to draw a diagram for this problem since it's talking about a pool (rectangle), but it's actually quicker to just look at the numbers given and work from there.

We know that the pool currently holds 600 gallons of water and that water has been hosed into it at a rate of 8 gallons a minute for a total of 70 minutes.

To find the amount of water in the pool now, we'll have to first solve for the amount of water added to the pool by hose. We know that 8 gallons were added each minute for 70 minutes, so all we have to do is multiply 8 by 70:

8 * 70 = 560 gallons

This tells us that 560 gallons of water were added to our already-filled, 600-gallon pool. To find the total amount of water, then, we simply add these two numbers together:

560 + 600 = 1160

The correct answer is 1160.

 

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Aaaaaaaaaaand time for a nap.

 

Key Takeaways: Making Sense of SAT Math Word Problems

Word problems make up a significant portion of the SAT Math section, so it’s a good idea to understand how they work and how to translate the words on the page into a proper expression or equation. But this is still only half the battle. 

Though you won’t know how to solve a word problem if you don’t know what a product is or how to draw a right triangle, you also won’t know how to solve a word problem about ratios if you don’t know how ratios work. 

Therefore, be sure to learn not only how to approach math word problems as a whole, but also how to narrow your focus on any SAT Math topics you need help with. You can find links to all of our SAT Math topic guides here to help you in your studies.

 

What’s Next?

Want to brush up on SAT Math topics? Check out our individual math guides to get an overview of each and every topic on SAT Math. From polygons and slopes to probabilities and sequences, we've got you covered!

Running out of time on the SAT Math section? We have the know-how to help you beat the clock and maximize your score

Been procrastinating on your SAT studying? Learn how you can overcome your desire to procrastinate and make a well-balanced prep plan.

Trying to get a perfect SAT score? Take a look at our guide to getting a perfect 800 on SAT Math, written by a perfect scorer. 

 

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Courtney Montgomery
About the Author

Courtney scored in the 99th percentile on the SAT in high school and went on to graduate from Stanford University with a degree in Cultural and Social Anthropology. She is passionate about bringing education and the tools to succeed to students from all backgrounds and walks of life, as she believes open education is one of the great societal equalizers. She has years of tutoring experience and writes creative works in her free time.



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