SAT Math Word Problems

Most students tremble with fear when they think about preparing for the Math portion of the SAT.  After all, there are 20+ topics tested, and the test requires mastery of topics ranging from arithmetic to trigonometry. Some topics cover basic math questions from algebra class, such as solving linear equations or recognizing graphs of parabolas. Solving questions such as these is tough enough, but the SAT adds a layer of difficulty by presenting word problems from a wide range of disciplines that require you to use your algebra skills to solve them.

To tackle these word problems, you must first understand the context of the word problem and then create the equations that you’ll need to solve, based on the information provided in the question.

In this article, I’ll first cover two critical algebra skills you need to master. Then I’ll provide a number of strategies for efficiently solving hard SAT word problems, as well as examples, so we can see these strategies in action and avoid common mistakes.

Here are the topics we’ll cover:

• Basic Algebra – Solving Single-Variable Linear Equations 
  • Example 1: Linear Equation with One Variable
• Sales Tax Questions
  • Sales Tax: Example 1
• Linear Equations with Two Variables/Systems of Linear Equations
  • Using the Substitution Method to Solve a System of Linear Equations
    • Substitution Method Practice 1
    • Substitution Method Practice 2
• When Dealing With Word Problems, Translation Is Key
• You Must Make Sure Your Equations are Balanced
  • Balancing Equations: Example 1
  • Balancing Equations: Example 2
  • Balancing Equations: Example 3
• “Number of Items” Questions
  • “Number of Items”: Example 1
  • “Number of Items”: Example 2
• Age Problems
  • Age Problem: Example 1
• Money Problems
  • Money Problem: Example 1
• In Conclusion: SAT Math Word Problems
• What’s Next?

We’ll start with solving basic linear equations, then we’ll review the substitution method of solving SAT algebra problems. Finally, we’ll use these two techniques to solve math word problem examples.

Basic Algebra – Solving Single-Variable Linear Equations 

The simplest algebra questions on the SAT are single-variable linear equations. Your goal is to solve for the value of the variable:

• 2x + 5 = x

• 4 = 5n

• 2v + 5 = 3v – 7

Note that solving single-variable equations generally requires that we combine all terms containing the variable on one side of the equation and all constant terms on the other side.

Let’s practice simplifying and solving an equation with one variable.

If 8q – 2 = 10q + 14, then what is the value of q?

First, we combine our constant terms by adding 2 to both sides:

8q – 2 + 2 = 10q + 14 + 2

8q = 10q + 16

Next, we combine the variable terms by subtracting 10q from both sides of the equation:

8q – 10q = 10q + 16 – 10q

-2q = 16

Finally, we divide both sides of the equation by -2 to isolate the variable:

-2q / -2 = 16 / -2

q = -8

KEY FACT:

When solving for a single variable, we combine like terms and isolate the variable we are solving for.

Now, let’s practice with an example.

Now let’s turn our attention to applying algebra to solve a straightforward word problem involving sales tax.

Sales Tax Questions

A common single-variable word problem that you might be asked on the SAT involves the calculation of sales tax. When you buy an item, you not only pay for the item itself but you also generally pay a surcharge called sales tax, typically stated as a percentage of the cost of the item. We add the sales tax to the cost of the item to determine the actual amount we will pay at the register.

The equation for purchasing an item in an area where sales tax is charged is:

Cost of Item + Sales Tax = Total

Let’s look at a basic example of the calculation of sales tax.

Laura buys a blender for $40.00 in a town where the sales tax rate is 4.5%. How much will she have to pay, including sales tax?

We see that the cost of the item is $40.00, and the sales tax rate is 4.5%. Thus, the sales tax amount is 4.5% of $40.00, or 40 x 0.045. Let’s put these into the equation:

Cost of Item + Sales Tax = Total

40 + (40)(0.045) = Total

40 + 1.8 = Total

41.8 = Total

Thus, Laura will pay a total of $41.80: the item cost of $40.00 and the sales tax of $1.80.

TTP PRO TIP:

Know the sales tax formula: Cost of Item + Sales Tax = Total

Let’s look at a sample SAT question about sales tax.

Next, let’s discuss linear equations with two variables.

Linear Equations with Two Variables/Systems of Linear Equations

Two-variable equations have not just one but two different variables:

• v – u = 12

• 2x + y = 10

• 5z – 3y = 42

These are called two-variable equations because they contain two different variables. If we have a second equation that contains one or both of the variables, the two equations are called a system of linear equations. One common way to solve for the values of the variables is by using the substitution method. Let’s discuss the substitution method now.

Using the Substitution Method to Solve a System of Linear Equations

When we’re working with SAT systems of equations word problems, the essence of the substitution method is that we first isolate one variable in one of the equations. Then we substitute whatever that variable is equal to into the other equation. Let’s practice with an example.

Substitution Method Practice 1

In the system of linear equations below, determine the value of a.

b = 3a (equation 1)

a + b = 12 (equation 2)

Solution:

Looking at our two equations, we see that b is already isolated in equation 1. So, we can substitute 3a (from equation one) for b (in equation two). This gives us:

a + 3a = 12

4a = 12

a = 3

Substitution Method Practice 2

For the system of linear equations below, what is the value of x?

3x + y = 11 (equation 1)

2x + 5y = 3 (equation 2)

Solution:

First, let’s isolate y in equation 1 by subtracting 3x from both sides of the equation:

3x + y = 11 (equation 1)

y = 11 – 3x

Now, we can substitute 11 – 3x for y in equation 2. This gives us:

2x + 5(11 – 3x) = 3

2x + 55 – 15x = 3

-13x = -52

x = 4

KEY FACT:

We can use the substitution method to solve a system of linear equations.

Next, let’s discuss the many topics on the SAT in which we use algebra to determine the solution.

When Dealing With Word Problems, Translation Is Key

Here are a couple of sample SAT math word problems that you might encounter:

Luca buys apples, which cost $2 each, and bananas, which cost $3 each, at the market. If he buys twice as many bananas as apples and spends $48 at the market, how many bananas does he buy?

Tyrone and Greg have a total of 40 marbles. If Tyrone has 4 times the number of marbles that Greg has, how many marbles does Greg have?

The bottom line is that a word problem presents a scenario requiring us to translate the given information into an algebraic equation that we then solve.

Word problems are not just about solving equations; they are also about translating words into equations! Let’s look at some common translations:

• “Is” translates to equals (=)

Kendra is the same age as Carla

Kendra’s age = Carla’s age

• “More” translates to addition (+)

Arita has 6 more marbles than Pablo

Arita = Pablo + 6

• “Less/fewer” translates to subtraction (-)

Sammy has 3 fewer coins than Rati

Sammy = Rati – 3

• “Times as many” translates to multiplication (✖)

Harold has 5 times as many newspapers as Phoebe

Harold = Phoebe ✖ 5

TTP PRO TIP:

Know the common translations of words to algebraic equations.

Before jumping into word problem practice questions, let’s discuss one point of confusion students have when translating words into equations: properly balancing the equations.

You Must Make Sure Your Equations are Balanced

A basic principle for translating words into equations is that the equations must be balanced correctly. If they are not, you might obtain reversed values for your variables’ values, or you might get a negative answer. Let’s look at a few correct and incorrect ways to balance equations, and we’ll explain why the equations are balanced correctly.

Balancing Equations: Example 1

Sherry has 30 more dollars than Melanie.

Solution:

When we are setting up this equation, we must understand that Sherry has more money than Melanie. For example, if Melanie has 20 dollars, then Sherry has 50 dollars.

So, to properly balance the equation, we must add 30 dollars to Melanie’s amount and set it equal to Sherry’s amount.

If we let S = Sherry’s money and M = Melanie’s money, we have:

S = 30 + M

Balancing Equations: Example 2

Nala has 4 fewer toys than Frank.

Solution:

When considering this equation, we must understand that Nala has fewer toys than Frank.

In other words, Frank has more toys than Nala.

So, to set up a balanced equation, we must subtract 4 from Frank’s amount and set that equal to Nala’s.

If we let N = the number of Nala’s toys and F = the number of Frank’s toys, we have:

N – 4 = F

Balancing Equations: Example 3

There are 5 times as many baseballs as tennis balls.

Solution:

When considering this equation, we must understand that there are more baseballs than tennis balls.

So, to set up a correct equation, we multiply the number of tennis balls by 5 and set that amount equal to the number of baseballs.

If we let B = the number of baseballs and T = the number of tennis balls, we have:

B = 5T

Now that we are familiar with how to translate, balance, and solve equations, let’s jump into the various types of general word problems you will encounter on the SAT, and how to solve them.

“Number of Items” Questions

We have already encountered some simple examples of “number of items” questions in this article. Now, let’s attack a full-fledged problem like those you might encounter on test day.

Let’s practice one more. This time, we will add a twist.

Next, let’s discuss another common type of SAT word problem: age problems.

Age Problems

Age problems are a common type of SAT problem with a unique spin. Instead of having a straightforward translation, as we practiced above, an age problem will usually make us translate words into equations based on an age in the past, an age in the future, or even both.

For example, we might need to consider Ann’s age 10 years from now. To do this, we let Ann’s age today = A, so her age in 10 years will be A + 10.

Similarly, Ann’s age 12 years ago would be expressed as A – 12.

KEY FACT:

Age problems generally require us to compare ages in the past or future.

Now, let’s practice with an example.

Next, let’s discuss money problems.

Money Problems

In general, money problems on the SAT involve two different commodities, such as adult tickets and child tickets, and the revenue earned from selling them. We can create two equations, one relating the number of items and the second relating the revenue earned from the sale of the items.

We will see that the substitution method is extremely useful for solving the two equations that are created from the given information.

TTP PRO TIP:

Be prepared to use the substitution method when dealing with money problems.

In Conclusion: SAT Math Word Problems

Of the 20+ major math topics on the SAT, one of the most challenging is word problems. In order to be successful with this type of problem, you need to be skilled both at translating words into equations and at solving algebra equations. The ability to solve SAT word problems is a must for scoring well on the exam.

The two most common skills for solving math word problems are knowing how to solve a linear equation and using the substitution method for solving systems of linear equations. You must translate the question into an equation or equations first, then use the appropriate algebraic method to arrive at the answer.

Some of the common types of word problems that we have covered here are sales tax questions, “number of items” questions, age questions, and money questions.

What’s Next?

In this article, we have covered how to solve word problems on the SAT. You have a good start in doing well on this particular topic. But don’t lose your perspective: word problems are only one of many topics you must master in order to get a great SAT score.  Read our article on how to improve your SAT score for more strategies, tips, and tricks.

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