Table of Contents |
Word problems can be tricky. Often it takes a bit of practice to convert the English sentence into a mathematical sentence. This is what we will focus on here with some basic number problems, geometry problems, and parts problems. A few important phrases are described below that can give us clues on how to set up a problem.
Word to Math Phrases | Examples |
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A number (or unknown, a value, etc.) often becomes our variable | If you want to find the number of books you bought, you may use the variable b in your problem. |
"Is" (or other forms of "is": "was", "will be", "are", etc.) often represents equals (=). | "x is five" becomes |
"More than" often represents addition and is usually built backwards, writing the second part plus the first. | "Three more than a number" becomes |
"Less than" often represents subtraction and is usually built backwards as well, writing the second part minus the first. | "Four less than a number" becomes |
The product of two numbers often represents multiplication between two quantities. | "The product of two unknowns is 5" means that we have |
The quotient of two numbers often represents division between two quantities. | "The quotient of a number and 5 is 4" can be written as |
Using these key phrases, we can take a number problem and set up an equation and solve.
EXAMPLE
Suppose we were told Rachel earned $40,000 a year for some number of years and ended up making a total of $280,000 over that time period. For how many years did Rachel work? Can you determine an equation to represent the problem and solve the equation?Create an equation. Substitute given information, , | |
Solve for years, y, by dividing both sides by 40,000 | |
Our Solution |
Using these key phrases, we can take a number problem and set up an equation and solve.
EXAMPLE
If 28 less than five times a certain number is 232, what is the number?We know that we are multiplying a certain number by 5, so we'll start with 5x. We also know that we are taking 25 from that 5x, so subtract 25 from 5x | |
"Is" translates to "equals" so include | |
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Now we can solve this equation by first adding 28 to both sides |
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Divide both sides by 5 |
Our solution |
This same idea can be extended to a more involved problem as shown in the next example:
EXAMPLE
Seventeen more than three times a number is the same as ten less than 6 times the number. What is the number?We know that we are multiplying a certain number by 3, so we'll start with 3x. We also know that we are adding 17 more to this value, so add 17 to 3x | |
"Is" translates to "equals" so include an equal sign | |
Now find the second part. We know we are multiplying that same number by 6, so include a 6x | |
We also know that we are taking 10 from that 6x, so subtract 10 from 6x | |
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Now we can solve this equation by first subtracting 3x from both sides |
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Add 10 to both sides |
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Divide both sides by 3 |
Our solution |
When working with word problems, it is always a good idea to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consecutive numbers or geometry problems.
EXAMPLE
A sofa and a loveseat together cost $444. The sofa costs double the love seat. How much do they each cost?Together, the sofa and loveseat cost $444, which gives us this equation. With no other information about the loveseat, this becomes our variable, which we'll call x. | |
We know the sofa is double the loveseat, so we multiply x by 2 | |
Replace these values in the original equation | |
Combine like terms, x and 2x | |
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Divide both sides by 3 |
Our solution for x, which we gives us the cost of the loveseat | |
Now find the cost of the sofa by substituting the cost of the loveseat | |
Evaluate | |
The cost of the sofa |
Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License