SAT / ACT Prep Online Guides and Tips

The Complete Guide to SAT Math Word Problems

Posted by Courtney Montgomery | Nov 21, 2015 9:21:43 PM

SAT Math

 

feature_words-1

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 answer. Though the math topics may 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 problemshow to translate your word problems into equations and diagrams, the different types of word problems you’ll see on the test, and how best to go about solving your word problems for test day.

Feature Image: Antonio Litterio/wikimedia

 

What Are Word Problems?

A word problem is any problem is based mostly or entirely on 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’ve done so, you can then solve for your information (if necessary).

You will be given word problems on the SAT for a variety of reasons. Word problems act to test your reading comprehension and your ability to visualize your information. These types of questions also allow the test-makers to ask questions that would be impossible to ask with a diagram or an equation. For instance, if the question asks you to fit as many small objects into a larger one as is possible, it would be difficult to demonstrate and ask this by using a diagram.

 

Translating Word Problems Into Equations or Drawings

In order to translate your word problems into actionable math equations you can then 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 with $a + b$.

Also take note that many mathematical actions have more than one term attached and they can be used interchangeably. 


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 terms in action in a few examples:

body_SAT_word_problem_13

We are told that the length of the rug is 2 feet more than the width. To have 2 feet "more than" means we must add 2 to our width in order for our width and length to be equal.

$l = w + 2$

Using that knowledge, you can solve the rest of the problem, since the length of the rug is 8 feet and the area of a rectangle is $lw$. 

[Note: the final answer is B, 48]

 

body_SAT_word_problem_14
 

Mr. Smith starts with an unknown amount of money on Monday, and he has one fourth the previous day's amount every day. We know that having a fraction like "one fourth the amount" means that we must divide his money by four every day to achieve the new value of his money. We also know we must perform this act five times and that we'll be left with one dollar at the very end. So $x$ must be divided by 4, then that amount must be divided by 4, then that amount would be divide by four, and so on two more times. This gives us:

${x/4}/4/4/4/4 = 1$

or

$(x/4)(1/4)(1/4)(1/4)(1/4) = 1$

[Note: the final answer is 1024]

 

body_Latin

Though the hardest SAT word problems may look like Latin 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 where you must simply set up an equation, word problems in which you must solve for a specific value, and word problem logic puzzles. We will look at each type with 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. You'll also most likely see it first on the test.

For these problems, you must use the information you’re given in the problem 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 an SAT math section, meaning that the test-makers consider these questions “easy.” This is due to the fact that you only have to provide the set-up and not the execution.

body_SAT_word_problem_1

Let us make an organized list of who is driving what number of miles based on what we're told.

Sam: $m$ miles

Now, if Kara drove "twice as many" miles as Sam, she is multiplying Sam's mileage by 2. 

Kara: $2m$ miles

And if Darin drove "20 miles fewer" than Kara, we must take her mileage and subtract 20 to find Darin's mileage. 

Darin: $2m - 20$

Our final answer is B, $2m - 20$

 

Word Problem Type 2: Solving for Your Missing Value

The vast majority of your SAT 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, including averages, single variable equations, and ratios, among others. 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_word_problem_3

Because we are asked for a guarantee, we must imagine that we are working with the worst odds possible. So what we have here is a part probability question, part counting question. We need 3 scarves of a single color (doesn't matter which color) and we are working with 4 different colors total. So if we imagine that we are choosing scarves at random, the worst scenario would be to get a different color scarf on each selection. So:

Pick 1: red

Pick 2: brown

Pick 3: yellow

Pick 4: gray

Well that certainly doesn't get us 3 scarves of the same color, so let's do it again. 

Pick 5: red

Pick 6: brown

Pick 7: yellow

Pick 8: gray

Again, the worst case scenario was that we picked a different colored scarf each time. Now we have 2 scarves of each color. But wait! That means that the next pick will give us 3 scarves of the same color, no matter which color we pick!

Pick 9: ?

No matter what color we pick on our 9th selection, we will now have 3 scarves that are the same color, whether it be red, brown, yellow, or gray. This means that it takes 9 selections to guarantee that we will get 3 scarves of the same color, no matter which color it is. 

Our final answer is E, 9.

 

You might also be given a geometry problem as a word problem, which may or may not be set up with a scenario as well. 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.

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 is instead simply a value greater than 55. Thus, it must be presented to us as a word problem. 

But since we know that $x$ must be an integer degree value greater than 55, let us simply 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 is the smallest integer larger than 55. Basically, it's a safe bet.)

Now, since $x = 56$, the next angle in the triangle--$2x$--must measure:

$56 * 2$

$112$

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 degrees in a triangle, so we can find the value of y by saying:

$180 - 112 - 56$

$y = 12$

One possible value for y is 12. 

{Note: the other possible values for y are: 3, 6, or 9]

 

Word Problem Type 3: Logic Puzzle

This type of question will never show up more than once per test and is essentially a math brainteaserFor this kind of problem, you will be given one or two pieces of information, and from there you must find what does or does not fit with the pattern.

body_SAT_word_problem_2

The one piece of information we have is that every number in set P is also in set Q. We are then asked to choose the one option that does NOT fit with this knowledge. So let us cross off every answer choice that does match what we know.

Answer choice A says that 4 is in both P and Q. Because all the numbers in P must also be in Q, this could be correct. We can eliminate answer choice A. 

Answer choice B says that 5 is neither in P nor Q. This could very well be true, since all we know about these number sets is that Q has every number P has (and perhaps even more). We do not know which numbers are actually included in each set, so this is potentially correct. We can eliminate answer choice B. 

Answer choice C says that 6 is in P, but not Q. This CANNOT possibly be correct, since all we know is that each and every number that is in set P must also be in set Q. Therefore, if 6 is in P, 6 must also be in Q. Answer choice C is likely our final answer, but let us see if we can eliminate D and E as well, just to make sure. 

Answer choice D says that 7 is in set Q, but not P. This could very well be true, since number set Q might have numbers that set P does not. All we know is that Q includes (but is not limited to) every number in set P. We can eliminate answer choice D. 

Finally, answer choice E gives us a hypothetical which says that if 8 is not in set Q, it is must not be in set P either. This would be true if set Q and set P were equal and set Q had no additional numbers. Because it might be true, we cannot eliminate this answer choice. 

We are left with answer choice C, which is the only answer that is always and irrefutably UNTRUE, based on the information we were given. 

Our final answer is C. 

 
body_juggle

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

 

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 your 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 ratio problem (especially if ratios give you any hesitation or pause):

body_SAT_word_problems_12

Because we're working with three different values that all sound similar, let's draw ourselves a picture of the scene. We know that, by weight, we have 2 pineapples, 3 pears, and 5 peaches, so:

body_fruit_diagram

Now, we can clearly see what's going on with the problem and can simply count the number of pineapples (2) and the number of fruits total (10). This gives us the fraction of weight by pineapple ($2/10$, or $1/5$). 

Our final answer is A, $1/5$.

As for geometry problems, rememberyou’re often given a geometry word problem as a word problem because the question would be too simple to solve had you had a diagram to work with from the beginning. So make your own drawing of the scene. Even a rough sketch can help you visualize the problem and keep all your information in order.

 

#2: Memorize Key Terms

If you’re not used to translating English words into mathematical equations, then SAT word problems might be difficult to wrap your head around at first. Look to the chart and learn how to translate keywords into their math equivalents so that you can understand exactly what the 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 real SAT word problems to make sure you’ve got your definitions down and can apply them to real SAT problems.

 

#3: Underline and/or Write Out Important Information

The key to solving a word problem is bringing together all the key pieces of given information and putting them in the right places. Make sure you write out all your 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 start in on solving the question.

 

#4: Pay Close Attention to Exactly What Is Being Asked of You

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 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 $x$ or $2x$ 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 in Which You Feel Weak

You are likely to see both a diagram/equation problem and a word problem for most 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 any particular math topic in order to solve its corresponding word problem. 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 on averages.

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

 
body_ready-1
All set? Let's go!
 

Test Your Knowledge

Now it's time to test your word problem know-how against real SAT math problems. 

1)

body_SAT_word_problem_11


2)

body_SAT_word_problem_7


3)

body_SAT_word_problem_15

4)

body_SAT_word_problem_9


Answers: A, E, E, D

 

Answer Explanations:

1) Here, we have three people who all made different numbers of sandwiches in relation to one another. Because we are told that Ben made "three times" as many sandwiches as Ali, and Carla made "twice" as many as Ben, we know we are dealing with multiplication. 

So let us say that Ali made $x$ number of sandwiches. Ben would, therefore, have made $3x$ sandwiches. And finally, Carla would have made $2(3x)$ or $6x$ sandwiches. Now we can add these values together to get to the total number of sandwiches, 20, to find out how many Ali made. 

$x + 3x + 6x = 20$

$10x = 20$

$x = 2$

Ali made two sandwiches. 

Our final answer is A, two. 

Alternatively, and as usual, we can make a sketch of this scene by starting with the assumption that Ali made one sandwich. From there, we can figure out how many Ben and Carla made. 

body_sandwiches

If we count all the sandwiches up, we can see that there are 10 total. We know that the three of them actually made 20 sandwiches total, so all we need to do is multiply each of their sandwiches by $20/10 = 2$. 

If we multiply Ali's sandwiches by 2, we get: $1 * 2 = 2$. So we know that Ali made 2 sandwiches total.

Our final answer is, again, A, two. 

 

2) All we know for sure is what we are told is true--some of the integers in set X are even. From here, we can eliminate all answers that are not necessarily true. 

Answer choice A says that if a number is even, it must be in set X. This is not necessarily true, as we were only told that some values in set X are even, NOT that set X contains all even numbers. We can eliminate answer choice A. 

Answer choice B says that if an integer is odd, it must be in set X. Much like answer choice A, this could be true, but is not necessarily true. We don't know that there are an infinite number of odd numbers in set X. We can therefore eliminate answer choice B. 

Answer choice C says that all the numbers in set X are even. All we know is that "some" integers in set X are even, so this is not necessarily true. We can eliminate answer choice C. 

Answer choice D says that all the numbers in set X are odd. We know for certain that this is NOT true, since we were told that some numbers in set X were even. We can eliminate answer choice D. 

Finally, answer choice E says that not all the numbers in set X are odd. We know that this is true, since we were told that some of the numbers are even. Answer choice E is correct. 

Our final answer is E. 


3) We are given a set of numbers and some rules as to what this set looks like. So let's make a sketch of the scene from what we know. 

First, we know that there are a total of 7 numbers in the set and that the first and last numbers are 2 and 20, respectively.

body_set_1

We also know that the median (middle) number is 6.

body_numbers_1_ex.png

Finally, we know that the number 3 occurs the most often. Because there are only two slots available, it must mean that the number 3 occurs twice and no other number repeats

body_numbers_2_ex.png

 

Now, the missing slots can be any two real numbers between the values of 6 and 20 that are NOT equal to one another and that do NOT equal 6 or 20. (Why not? Because we were told that 3 was the number that occurs the most, which means that no other number in the set can possibly repeat). 

In order to find an average, we must add up our set and divide by the total number of numbers in the set. So let's find the sum of the numbers in our set:

$2 + 3 + 3 + 6 + x + y + 20$

$34 + x + y$

Now let's experiment by trial and error which averages fit with our information. 

Answer option I gives us an average of 7, so let's see if this is possible. 

$7 = {34 + y + y}/7$

$49 = 34 + x + y$

$15 = x + y$

Answer option I is possible, since $x$ and $y$ could fit with our given criteria and add up to equal 15. For instance, $x$ could equal 7 and $y$ could equal 8, and $7 + 8 = 15$. 

Because option I is possible, we can eliminate answer choice D, II and III only. 

Now let us test out answer option II. If we test our answer option II, we get:

$8.5 = {34 + y + y}/7$

$59.5 = 34 + x + y$

$25.5 = x + y$

It is entirely possibly for the sum of $x$ and $y$ to equal 25.5 with real numbers between 6 and 20. For instance, $x = 10$ and $y = 15.5$ would give us a sum of 25.5. 

This means we can eliminate answer choices A and C as well. 

Finally, let us test answer option III. Answer option III gives us:

$10 = {34 + y + y}/7$

$70 = 34 + x + y$

$36 = x + y$

Again, it is entirely possible for the sum of $x$ and $y$ to equal 36 with real numbers between 6 and 20. For instance, $x = 17$ and $y = 19$ would give us a sum of 36. 

We have seen here that all three answer options work, so only answer E can be correct. 

Our final answer is E, I, II, and III only. 


4) Notebooks cost 2 dollars each, which means that the cost of some number of notebooks will be a multiple of 2. So $n$ notebooks will cost $2n$. 

Backpacks cost 32 dollars each and so whatever multiple of backpacks we have will be multiplied by 32. This means that $b$ backpacks will cost $32b$. 

If we put these together, we will have:

$2n + 32b$

Our final answer is D, $2n + 32b$. 

 

body_sleepy-1

Aaaaaaaaaaand time for a nap.

 

The Take-Aways

Word problems make up a significant portion of the SAT, so it’s a good idea to understand how they work and how to translate the words on the page into a proper 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 your own right triangle, you also won’t know how to solve a word problem on ratios if you don’t know anything about how ratios work. So be sure to not only learn how to approach your word problems as a whole, but also hone your focus on any math topics you feel you need to improve upon. 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 any of your other math topics? Check out our individual math guides to get the walk-through on each and every topic on the SAT math test. From polygons to slopes, probabilities to 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 studying? Learn how you can overcome your desire to procrastinate and make a well-balanced study plan. 

Trying to get a perfect score? Check out our guide to getting a perfect 800 on SAT math, written by a perfect-scorer. 

 

Want to improve your SAT score by 240 points? 

Check out our best-in-class online SAT prep program. We guarantee your money back if you don't improve your SAT score by 240 points or more.

Our program is entirely online, and it customizes what you study to your strengths and weaknesses. If you liked this Math strategy guide, you'll love our program. Along with more detailed lessons, you'll get thousands of practice problems organized by individual skills so you learn most effectively. We'll also give you a step-by-step program to follow so you'll never be confused about what to study next.

Check out our 5-day free trial:

SAT Free Signup

 

Have friends who also need help with test prep? Share this article!
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.



Get Free Guides to Boost Your SAT/ACT
100% Privacy. No spam ever.

You should definitely follow us on social media. You'll get updates on our latest articles right on your feed. Follow us on all 3 of our social networks:

Twitter and Google+



Ask a Question Below

Have any questions about this article or other topics? Ask below and we'll reply!