Algebraic Expressions and Word Problems Study Guide

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Updated on Oct 3, 2011

Introduction to Algebraic Expressions and Word Problems

"What's one and one and one and one and one and one and one and one and one and one?" "I don't know," said Alice. "I lost count."

—Lewis Carroll (1832–1898)

In this lesson, you'll see that translating word problems into algebra is important for both math tests and issues that arise every day. Specifically, you'll learn to tackle distance, mixture, and work problems.

When you are translating sentences and word problems into algebraic expressions and equations, it can seem like you are translating between two different languages.

There are some strategies, however, that will help you become fluent in both:

  • First, read the problem to determine what you are looking for.
  • Then, write the amount you are looking for in terms of x (or whatever letter you want to use). You can do this by writing "Let x =… " Write any other unknown amounts in terms of x, too.
  • Last, set up the algebraic expressions in an equation with an equals sign and solve for the variable.


When you are translating key words in an algebraic expression, the phrases less than and greater than do not translate in the same order as they are written in the sentence. For example, when you are translating the expression eight less than 15, the correct expression is 15 – 8, not 8 – 15.

Study the following to see how to translate word problems into mathematical statements and equations.

Equals Key Words is are has was were had

Addition Key Words Sum Together Total More Greater or Older than

Subtraction Key Words Difference Less Fewer or Younger than Remain Left Over

Multiplication Key Words Product Times Of

Division Key Words Per Evenly

Look at an example where knowing the key words is necessary:

Twenty less than five times a number is equal to the product of ten and the number. What is the number?

Let's let x equal the number we are trying to find. Now, translate the sentence piece by piece, and then solve the equation.

Twenty less than five times a number equals the product of 10 and x.

      5x– 20

The equation is 5x– 20 = 10x. Subtract 5x from both sides:

    5x– 5x– 20 = 10x– 5x

Now, divide both sides by 5:

    –4 = x

In this example, the key words less than tell you to subtract from the number and the key word product reminds you to multiply.

Problem Solving with Word Problems

There are a variety of different types of word problems you will encounter on tests or in your daily life. To help with these types of problems, always begin first by figuring out what you need to solve for and defining your variable(s) as what is unknown. Then write and solve an equation that matches the question asked.

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