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Create Equations from Word Problems

One of the algebra skills that students struggle with the most is writing equations from word problems. Ironically, translating real life problems into math is one of the key ways that algebra can become useful in real life (All those times you ask yourself, why do I need to learn this?  This is why you need to learn algebra!).

If you need to practice putting equations togther, visit the Creating Equations Lesson for all of the details.  But, if you can create equations when you are giving the numbers, you're ready to try it when you're given words.

Below is a quick review of the key words that signal what types of math operations that a word problem is asking for. We've included some word problem examples as well.

What operations do you use?
SymbolAddition $+$Subtraction $-$Multiplication $\times$Division $\div$
Key Words

in addition to
all together
in total
increased by

take away
left over
decreased by
reduced by

multiplied by


divided by

A number plus 6 is 8


If a ball costs \$6 and a bat costs \$12, what do they call all together?


The difference between 5 and x is 3


7 is 12 less than x


The product of 5 and x is 20

$5\times x=20$

He scored 3 times as many as Abe, who scored 14.


If you divide 6 apples among 2 kids, who many apples does each kid get?


Special notes:You can add numbers in any order (because of the communicative property of addition).Order matters!  It's often switched.  So, x less 7 is $x-7$ but 7 less than x is also $x-7$.  Think about it: You have \$10.  Your friend has three dollars less than you, would you do $3-10$ or $10-3$?  Always think about order when you write a subtraction problem. 

You can multiply numbers in any order (because of the communicative property of multiplication).

"Of" is often used when multiplying by decimals or fractions: one half of 20is ten or 45% of 100 is 45.  In both of these cases "of" means multiply.

When you write a division problem either with a division sign ($19\div4$) you enter numbers into a calculator in that order.  When you write a division problem as a fraction ($\dfrac{19}{3}$), you enter those numbers into a calculator from top to bottom ($19\div3$).  But, if you divide on scratch paper, using the "house," you write numbers in the opposite order: $3\vert\overline{19}$.


Once you have these key words in your head, you're ready to read the questions and start to break it down into an equation.

Example: The five members of a basketball team are getting new practice equipment.  Shoes cost \$129 a pair.  Balls cost \$15 each. Uniforms are \$80 a piece. What is the total cost of all of the new equipment for the team?

Sometimes the easiest thing to do is write out words and operations (don't worry about numbers and variables yet).

You are finding a total so you are going to want to add.

$\text{shoes} + \text{balls} + \text{uniforms} = \text{total}$

Great.  But, remember that there are five players on the team.

$5\times\text{shoes} + 5\times\text{balls} + 5\times \text{uniforms} = \text{total}$

That makes sense, right?  Now, just replace your words with the numbers provided in the word problem:

$5\times 129 + 5\times 15 + 5\times 80= \text{total}$

Now, you can solve:

$$\eqalign{5\times 129 + 5\times 15 + 5\times 80 &= \text{total}\\645 + 75 + 400 &=\text{total}\\1120 &=\text{total}}$$


Sometimes, a word problem will give you an answer and you have to write an equation in order to work backwards and solve for some other unknown number.

Example: Abraham purchased a gaming system for \$299 and several games at \$39.99 each.  If he paid \$418.97, how many games did he purchase?

Again, the easiest thing to do is write out words and operations (don't worry about numbers and variables yet).

You are finding a total so you are going to want to add. 

$\text{game system} + \text{games} = \text{total}$

But, remember that there are several games -- but we don't know how many, let's use a variable.

$\text{game system} + g\times\text{games}  = \text{total}$

Now, replace your words with the numbers provided in the word problem:

$299 + g\times39.99 = 418.97$

Now, you can solve:

$$\eqalign{$299 + g\times39.99 &= &&418.97\\299 + 39.99g &= &&418.97&&&\text{Rewrite in algebra format}\\-299& &&-299 &&&\text{Subtract 299 from each side}\\39.99g& = &&119.97\\\div 39.99& && \div 39.99&&&\text{divide each side by 39.99 to get } g \text{  alone}\\g&=&&3777&&&\text{He bought 3 games}}$$


The most complicated word problems have a lot of pieces, sometimes multiple variables, sometimes just a lot of terms. Move them them slowly and logically and you'll put the math problem together. 

ExampleA teacher has some number of cookies.  She is going to order more cookies so that she doubles her stock, plus an additional 30.  There are 24 kids in her class when she distributes all of the cookies, we know each student gets 3. Write an equation to show how many cookies the teacher started with.

There's a lot going on here.  First we need to figure out which quantity is unknown.  Then we have to add in the new order.  Then we have to divide the cookies by students. In the end, we set it equal to 3 (because each student gets three cookies).  Let's set this one up, step by step, and then solve it to see how we can actually figure out the answer to this problem.

$$\eqalign{&x \text{  cookies}&&\text{     Teacher starts with some number of cookies.  We use a variable. }\\&x\times 2&&\text{    Double the number of cookies}\\&2x + 30&&\text{     Simplify to 2x then add another 30 cookies}\\&\dfrac{2x+30}{24}=&&\text{     Divide total cookies by each student (24)}\\&\dfrac{2x+30}{24}=3&&\text{     Set expression equal to 3.  All cookies, divided by 24 students = 3 cookies each}}$$

Now that we have our equation, we can use algebra to solve for x and find out how many cookies the teacher started with. 

$$\eqalign{&\dfrac{2x+30}{24}&&=&&&3&&&&\text{Our equation.  Variable x represents number of cookies teacher started with}\\&\small{\times 24}&&\text{}&&&\small{\times 24}&&&&\text{     Multiply each side of the equation by 24 to clear the fraction}\\&2x+30&&=&&&72\\&\small{-30}&&\text{}&&&\small{-30}&&&&\text{     Add 30 to each side to clear the addition term}\\&2x&&=&&&102\\&\small{\div 2}&&\text{}&&&\small{\div 2}&&&&\text{     Divide each side by 2 to clear the 2 and isolate x}\\&x&&=&&&51\text{   cookies}}$$


Note that sometimes, a word problem does not give you an answer. You may not be able to solve this problem: you just need to write the expression that represents the word problem.

Example: Games4U has 35 video games in stock.  They then receive a shipment of 6y more games.  But, they sell 3 games.  What is the total number of games that they have in stock now?

The key word "total" signals that you're going to want to add.  But they also sell some games (so, will that be an addition or a reduction?).   Let's start from the beginning.

$$\eqalign{&35 \text{  games}&&\text{     They start with 35 games in stock}\\+&6y&&\text{     Add the new shipment}\\-&3&&\text{     Subtract the games that they sold}\\&35+6y-3=\text{total}&&\text{     Final expression}}$$


The keys to writing equations are:

  • Pay attention to all of the steps, 
  • Look for key words, and 
  • Use your logic.  
  • Equations are just math problems in which you do not know all of the numbers. Sometimes it helps to put real numbers in place of variables just to think about how you would solve the problem if you knew everything (so, for instance, if you knew that a teacher had 20 cookies, doubled that number, then added 30, what would you do?  You would do $20\times 2 + 30$.  To convert that same idea to an equation, just replace the $20$ with an $x$ and you're all set).  
  • The final key to equation success is practice. Translating thoughts into math is like learning a new language. The more you practice, the more natural it gets. Did you know that scientists, economists, and other people who work with math a lot often take notes in equation form rather than in word form? They are just so used to thinking in math that they find it easier and simpler than writing things down in words. 

So, don't be scared of these kinds of problems.  Do the practice, and keep doing more practice whenever you find it, and you'll soon be good at "thinking in math" and creating equations. 

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