How to Solve Word Problems in Algebra

Co-authored by Daron Cam

Last Updated: December 19, 2022 Fact Checked

You can solve many real world problems with the help of math. In order to familiarize students with these kinds of problems, teachers include word problems in their math curriculum. However, word problems can present a real challenge if you don't know how to break them down and find the numbers underneath the story. Solving word problems is an art of transforming the words and sentences into mathematical expressions and then applying conventional algebraic techniques to solve the problem.

Part 1

Part 1 of 3:

Assessing the Problem

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Daron Cam
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Expert Interview. 29 May 2020.} A common setback when trying to solve algebra word problems is assuming what the question is asking before you read the entire problem. In order to be successful in solving a word problem, you need to read the whole problem in order to assess what information is provided, and what information is missing.^{[2]
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Determine what you are asked to find. In many problems, what you are asked to find is presented in the last sentence. This is not always true, however, so you need to read the entire problem carefully.^{[3]
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Research source} Write down what you need to find, or else underline it in the problem, so that you do not forget what your final answer means.^{[4]
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Research source} In an algebra word problem, you will likely be asked to find a certain value, or you may be asked to find an equation that represents a value.
- For example, you might have the following problem: Jane went to a book shop and bought a book. While at the store Jane found a second interesting book and bought it for $80. The price of the second book was $10 less than three times the price of he first book. What was the price of the first book?
- In this problem, you are asked to find the price of the first book Jane purchased.
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Summarize what you know, and what you need to know. Likely, the information you need to know is the same as what information you are asked to find. You also need to assess what information you already know. Again, underline or write out this information, so you can keep track of all the parts of the problem. For problems involving geometry, it is often helpful to draw a sketch at this point.^{[5]
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- For example, you know that Jane bought two books. You know that the second book was $80. You also know that the second book cost $10 less than 3 times the price of the first book. You don't know the price of the first book.
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Assign variables to the unknown quantities. If you are being asked to find a certain value, you will likely only have one variable. If, however, you are asked to find an equation, you will likely have multiple variables. No matter how many variables you have, you should list each one, and indicate what they are equal to.^{[6]
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- For example, assign the variable $x$ to the unknown in the problem, which is the price of the first book. Write $x={\text{the price of the first book}}$ .
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Look for keywords.^{[7]
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Daron Cam
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Expert Interview. 29 May 2020.} Word problems are full of keywords that give you clues about what operations to use. Locating and interpreting these keywords can help you translate the words into algebra.^{[8]
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- Multiplication keywords include times, of, and factor.^{[9]
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- Division keywords include per, out of, and percent.^{[10]
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- Addition keywords include some, more, and together.^{[11]
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- Subtraction keywords include difference, fewer, and decreased.^{[12]
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Part 2

Part 2 of 3:

Finding the Solution

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Write an equation. Use the information you learn from the problem, including keywords, to write an algebraic description of the story.^{[13]
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- For example, you know that the second book is $80, and you know what $80 equals in terms of the price of the first book ( $x$ ). So set 80 equal to $10 less ( $-10$ ) than 3 times the price of the first book ( $3x$ ). Putting everything together, you have $80=3x-10$ .
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Solve an equation for one variable. If you have only one unknown in your word problem, isolate the variable in your equation and find which number it is equal to. Use the normal rules of algebra to isolate the variable. Remember that you need to keep the equation balanced. This means that whatever you do to one side of the equation, you must also do to the other side.^{[14]
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- Use inverse operations to isolate a variable. For example, to isolate the variable in the equation $80=3x-10$ , you need to add 10 to both sides, then divide by 3:
  $80=3x-10$
  $80+10=3x-10+10$
  $90=3x$
  ${\frac {90}{3}}={\frac {3x}{3}}$
  $30=x$
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Solve an equation with multiple variables. If you have more than one unknown in your word problem, you need to make sure you combine like terms to simplify your equation.
- When combining like terms, remember that only terms with the same exponent and variable can be combined. For example, $4x$ and $2x$ can be combined, $3x^{2}$ and $5x^{2}$ can be combined, and $8xy$ and $4xy$ can be combined.
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Interpret your answer. Look back to your list of variables and unknown information. This will remind you what you were trying to solve. Write a statement indicating what your answer means.^{[15]
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- For example, since $x={\text{the price of the first book}}$ , and $30=x$ , you know that the price of the first book Jane bought was $30.

Part 3

Part 3 of 3:

Completing a Sample Problem

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Solve the following problem. This problem has more than one unknown value, so its equation will have multiple variables. This means you cannot solve for a specific numerical value of a variable. Instead, you will solve to find an equation that describes a variable.
- Robyn and Billy run a lemonade stand. They are giving all the money that they make to a cat shelter. They will combine their profits from selling lemonade with their tips. They sell cups of lemonade for 75 cents. Their mom and dad have agreed to double whatever amount they receive in tips. Write an equation that describes the amount of money Robyn and Billy will give to the shelter.
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Daron Cam
Academic Tutor

Expert Interview. 29 May 2020.} You are asked to find how much money Robyn and Billy will give to the cat shelter.
You know that Robyn and Billy will make money from selling cups of lemonade and from getting tips. You know that they will sell each cup for 75 cents. You also know that their mom and dad will double the amount they make in tips. You don't know how many cups of lemonade they sell, or how much tip money they get.
Since you have three unknowns, you will have three variables. Let $x$ equal the amount of money they will give to the shelter. Let $c$ equal the number of cups they sell. Let $t$ equal the number of dollars they make in tips.
Since they will “combine” their profits and tips, you know addition will be involved. Since their mom and dad will “double” their tips, you know you need to multiply their tips by a factor of 2.
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Write an equation. Since you are writing an equation that describes the amount of money they will give to the shelter, the variable $x$ $x$ will be alone on one side of the equation.
- Since you are combining their profits and tips, you will be adding two terms. So, x = __ + __.
- The first term will be equal to their profits. Since they make $0.75 for every cup of lemonade they sell, their profits are equal to $.75c$ . So, $x=.75c\;+\;??$ .
- The second term will be equal to their tips. Since their parents are doubling their tips, their tips will be equal to $2t$ . So, $x=.75c+2t$ . Since the variable you are describing is already isolated, and all like terms are combined, you have arrived at your final answer.
The variable $x$ equals the amount of money Robyn and Billy will donate to the cat shelter. So, the amount they donate can be found by multiplying the number of cups of lemonade they sell by .75, and adding this product to the product of their tip money and 2.

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How do you solve an algebra word problem?

Daron Cam
Academic Tutor
Daron Cam is an Academic Tutor and the Founder of Bay Area Tutors, Inc., a San Francisco Bay Area-based tutoring service that provides tutoring in mathematics, science, and overall academic confidence building. Daron has over eight years of teaching math in classrooms and over nine years of one-on-one tutoring experience. He teaches all levels of math including calculus, pre-algebra, algebra I, geometry, and SAT/ACT math prep. Daron holds a BA from the University of California, Berkeley and a math teaching credential from St. Mary's College.

Daron Cam

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Expert Answer

Carefully read the problem and figure out what information you're given and what that information should be used for. Once you know what you need to do with the values they've given you, the problem should be a lot easier to solve.

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If Deborah and Colin have $150 between them, and Deborah has $27 more than Colin, how much money does Deborah have?

Donagan

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Let x = Deborah's money. Then (x - 27) = Colin's money. That means that (x) + (x - 27) = 150. Combining terms: 2x - 27 = 150. Adding 27 to both sides: 2x = 177. So x = 88.50, and (x - 27) = 61.50. Deborah has $88.50, and Colin has $61.50, which together add up to $150.

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Karl is twice as old Bob. Nine years ago, Karl was three times as old as Bob. How old is each now?

Donagan

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Let x be Bob's current age. Then Karl's current age is 2x. Nine years ago Bob's age was x-9, and Karl's age was 2x-9. We're told that nine years ago Karl's age (2x-9) was three times Bob's age (x-9). Therefore, 2x-9 = 3(x-9) = 3x-27. Subtract 2x from both sides, and add 27 to both sides: 18 = x. So Bob's current age is 18, and Karl's current age is 36, twice Bob's current age. (Nine years ago Bob would have been 9, and Karl would have been 27, or three times Bob's age then.)

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Word problems can have more than one unknown and more the one variable.

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The number of variables is always equal to the number of unknowns.

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While solving word problems you should always read every sentence carefully and try to extract all the numerical information.

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