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Examples

• mathrm{Lauren’s:age:is:half:of:Joe’s:age.:Emma:is:four:years:older:than:Joe.:The:sum:of:Lauren,:Emma,:and:Joe’s:age:is:54.:How:old:is:Joe?}

• mathrm{Kira:went:for:a:drive:in:her:new:car.:She:drove:for:142.5:miles:at:a:speed:of:57:mph.:For:how:many:hours:did:she:drive?}

• mathrm{Bob’s:age:is:twice:that:of:Barry’s.:Five:years:ago,:Bob:was:three:times:older:than:Barry.:Find:the:age:of:both.}

• mathrm{Two:men:who:are:traveling:in:opposite:directions:at:the:rate:of:18:and:22:mph:respectively:started:at:the:same:time:at:the:same:place.:In:how:many:hours:will:they:be:250:apart?}

• mathrm{If:2:tacos:and:3:drinks:cost:12:and:3:tacos:and:2:drinks:cost:13:how:much:does:a:taco:cost?}

Frequently Asked Questions (FAQ)

• How do you solve word problems?

• To solve word problems start by reading the problem carefully and understanding what it’s asking. Try underlining or highlighting key information, such as numbers and key words that indicate what operation is needed to perform. Translate the problem into mathematical expressions or equations, and use the information and equations generated to solve for the answer.

• How do you identify word problems in math?

• Word problems in math can be identified by the use of language that describes a situation or scenario. Word problems often use words and phrases which indicate that performing calculations is needed to find a solution. Additionally, word problems will often include specific information such as numbers, measurements, and units that needed to be used to solve the problem.

• Is there a calculator that can solve word problems?

• Symbolab is the best calculator for solving a wide range of word problems, including age problems, distance problems, cost problems, investments problems, number problems, and percent problems.

• What is an age problem?

• An age problem is a type of word problem in math that involves calculating the age of one or more people at a specific point in time. These problems often use phrases such as «x years ago,» «in y years,» or «y years later,» which indicate that the problem is related to time and age.

word-problems-calculator

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We recommend reading any word problem at least three times. By the conclusion of the third reading, you should be magically whisked away to your home in Kansas.

Read it once…

…to get a general idea of what’s going on. Is the problem about money? Height and width? Distances? Money? Yeah, we already said money, but it’s important. Can you draw a picture in the margin that helps you visualize the problem?

Twice…

…to translate from English into math. Read the problem carefully this time, figuring out which pieces of information are important and which ones aren’t. Then ignore the unnecessary bits. This may sound a bit harsh, but they’re big boys. They’ll get over it.

Three times…

…a lady. Wait, that’s not it.

…to make sure you answered the right question. To be sure, read the question one last time before drawing a box around your final answer. Even if this method of double-checking saves you only once out of every 100 times, it’ll help you in the long run. One out of 100 is better than zero out of 100, but now we’re moving into some advanced mathematics.

Sample Problem

The local department store was having a sale. Holla! Gabe bought a pair of shoes for $21, although they would’ve been cheaper if he’d bought penny loafers. He also bought some shirts that were on sale for 25% off their normal retail price of $18 each. Gabe, always the bargain hunter, spent $75 total. How many shirts did he buy? Also, does he really think any of them will go with those shoes?

Read the problem:

Once…

…for a general idea of what’s going on: Gabe went shopping and spent money. Sounds like an old familiar story. Time for an intervention, friends of Gabe. Preferably before he maxes out his Diner’s Club card.

Now we need to figure out how many shirts he bought, so we read the problem:

Twice…

…to translate from English into math. Remember, we can translate a bit at a time, sort of like how Gabe pays for some of his major purchases when he has them on layaway. Sheesh, Gabe. Get a hold of yourself.

(total amount Gabe spent) = (amount he spent on shoes) + (amount he spent on shirts)

We know he spent $75 total, of which $21 was spent on shoes. This gives us the equation:

$75 = 21 + (amount he spent on shirts)

The problem has gotten smaller. It’s depressing when that happens with cake, but great when it happens to a word problem. Now all we need to do is come up with a symbolic expression for how much Gabe spent on shirts, or the cost per shirt times the number of shirts:

(amount he spent on shirts) = (cost per shirt)(number of shirts)

Since the shirts are 25% off their normal price of $18, they cost $18 – 0.25(18) = $13.50 each. What a deal, and real polyester, too!

We need to introduce a variable for the number of shirts; s will do the trick nicely. 

(amount he spent on shirts) = 13.5s

When we plug this into the earlier equation, we find that:

75 = 21 + 13.5s

Finally, we’ve reduced this thing to a super-simple-looking equation! Good riddance, vestiges of language! Begone, nouns and verbs!

Things look much nicer now, right? No worrying about shirts or shoes or prices or Gabe’s uncontrollable shopping addiction. For the moment, we can forget about the word problem and solve the equation. The answer is s = 4, by the way. In case you were interested.

Three times…

…to make sure we’re answering the right question. We want to know how many shirts Gabe bought. Is that the answer we arrived at? We found that s = 4, and s was the number of shirts Gabe bought, so we’re all done. Four new shirts for Gabe, and three of them feature a Hawaiian pattern. Gabe, if you insist on buying far more clothes than you need, can’t you at least have a decent fashion sense?

When translating from English into math, some information can be ignored. We don’t care that «the local department store was having a sale.» Gabe might, but we certainly don’t. We care about statements that tell us numbers, and statements that tell us what the question is. Any extraneous information has been placed there simply as a decoy. We’re not going to fall for that. Wait a second…two for one? We’ll grab our jacket and meet you there.

Some people find it helpful to underline the important pieces of information in a word problem. You’re like an actor highlighting in a script the lines that are important for him to remember. Unlike an actor, however, you can always look back at the original problem if you draw a blank. Also, you don’t need to wear any stage makeup.

In the problem we just did, the important bits might look something like this:

The local department store was having a sale. Gabe bought a pair of shoes for $21 and some shirts that were on sale for 25% off their normal retail price of $18 each. If Gabe spent $75 total, how many shirts did he buy?

As you practice, you’ll become better at figuring out which parts of the word problem you can ignore and which parts are important.

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Dynamically Created Word Problems

Here is a graphic preview for all of the word problems worksheets. You can select different variables to customize these word problems worksheets for your needs. The word problems worksheets are randomly created and will never repeat so you have an endless supply of quality word problems worksheets to use in the classroom or at home. Our word problems worksheets are free to download, easy to use, and very flexible.

These word problems worksheets are a great resource for children in 3rd Grade, 4th Grade, and 5th Grade.

Click here for a Detailed Description of all the Word Problems Worksheets.

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Words to Symbols Word Problems	Addition Word Problems Using 1 Digit with 2 Addends	Addition Word Problems Using 2 Digit with 2 Addends
Addition Word Problems Using 1 Digit with 3 Addends	Addition Word Problems Using 2 Digit with 3 Addends	Addition Word Problems 2 Digit Missing Addends
Subtraction Word Problems Using 1 Digit	Subtraction Word Problems Using 2 Digits	Addition and Subtraction Word Problems Using 1 Digit
Addition and Subtraction Word Problems Using 2 Digits	Addition and Subtraction Word Problems 2 Digits With No Regrouping	Addition and Subtraction Word Problems Using 3 Digits
Multi-Step Problems Addition and Subtraction	Multiplication Word Problems Using 1 Digit	Multiplication Word Problems Using Dozens
Multiplication Word Problems Using 2 Digits	Division Word Problems Using 1 Digit in Divisor	Division Word Problems Using Dozens in Divisor
Division Word Problems Using Multiple Digits in Divisor	Division Word Problems Dividing with Fractions	Multiplication and Division Word Problems Using 1 Digit
Mixed Operations Problems Using 1 or 2 Digits	One Step Equation Word Problems	Two Step Equation Word Problems
Multi-Step All Operations Word Problems	Fractions Word Problems Adding Two Fractions	Fractions Word Problems Adding Three Fractions
Fractions Word Problems Subtracting Fractions	Fractions Word Problems Adding Two Mixed Numbers	Fractions Word Problems Subtracting Two Mixed Numbers
U. S. Coins Word Problems Adding Coins Worksheets	Advanced Adding U. S. Coins Word Problems	U. S. Money Word Problems Adding Two Purchases
U. S. Money Word Problems Adding Three Purchases	U. S. Money Word Problems Change from a Purchase	Travel Time Word Problems
Reading a Calendar Word Problems	Elapsed Dates Word Problems	Ratios and Rate Word Problems
Percentage Word Problems	Mixed Operations with Key Phrases Word Problems	U.S. Money Change from a Purchase Multiplication Word Problems

Detailed Description for All Word Problems Worksheets

Words to Symbols Handout
This Word Problems Worksheet will produce a great handout to help students learn the symbols for different words and phrases in word problems.

Addition Word Problems Worksheets Using 1 Digit with 2 Addends
These addition word problems worksheets will produce 1 digit problems with two addends, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition Word Problems Worksheets Using 2 Digits with 2 Addends
These addition word problems worksheets will produce 2 digits problems with two addends, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition Word Problems Worksheets Using 1 Digit with 3 Addends
These addition word problems worksheets will produce 1 digit problems with three addends, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition Word Problems Worksheets Using 2 Digits with 3 Addends
These addition word problems worksheets will produce 2 digits problems with three addends, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition Word Problems Worksheets 2 Digits Missing Addends
These addition word problems worksheet will produce 2 digits problems with missing addends, with ten problems per worksheet. You may select between regrouping and non-regrouping type of problems. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Subtraction Word Problems Worksheets Using 1 Digit
These subtraction word problems worksheets will produce 1 digit problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Subtraction Word Problems Worksheets Using 2 Digits
These subtraction word problems worksheets will produce 2 digits problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition and Subtraction Word Problems Worksheets Using 1 Digit
These addition and subtraction word problems worksheets will produce 1 digit problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition and Subtraction Word Problems Worksheets Using 2 Digits
These addition and subtraction word problems worksheets will produce 2 digits problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition and Subtraction Word Problems Worksheets 2 Digits With No Regrouping
These addition and subtraction word problems worksheets will produce 2 digits problems with no regrouping and ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Addition and Subtraction Word Problems Worksheets Using 3 Digits
These addition and subtraction word problems worksheets will produce 3 digits problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Multi-Step Problems Addition and Subtraction
These multi-step addition and subtraction word problems worksheets will produce 10 problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Multiplication Word Problems Worksheets Using 1 Digit
These multiplication word problems worksheets will produce 1 digit problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Multiplication Word Problems Worksheets Using Dozens
These multiplication word problems worksheets will produce problems using dozens, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Multiplication Word Problems Worksheets Using 2 Digits
These multiplication word problems worksheets will produce 2 digits problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Division Word Problems Worksheets Using 1 Digit in Divisor
These division word problems worksheets will produce 1 digit divisor problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Division Word Problems Worksheets Using Dozens in Divisor
These division word problems worksheets will produce problems using dozens in the divisor, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Division Word Problems Worksheets Using Multiple Digits in Divisor
These division word problems worksheets will produce multiple digits in the divisor for the problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Dividing with Fractions Worksheets
These division Word Problems Worksheets will produce problems that focus on division with fractions. You have the option to select the range of denominators, as well as the types of fractions displayed. These word problems worksheets are appropriate for 4th Grade, 5th Grade, and 6th Grade.

Multiplication and Division Problems Using 1 Digit
These multiplication and division word problems worksheets will produce 1 digit problems, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Mixed Operations Word Problems Using 1 or 2 Digits
These mixed operations word problems worksheets will produce addition, multiplication, subtraction and division problems with 1 or 2 digit numbers. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

One Step Equation Word Problems
These equations worksheets will produce one step word problems. These worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Two Step Equation Word Problems
These equations worksheets will produce two step word problems. These worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Multi-Step All Operations Word Problems
These Word Problems worksheets will produce word problems involving all basic operations. You may choose the format of the answers. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Adding Two Fractions Word Problems
These fractions word problems worksheets will produce problems with the addition of two fractions. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Adding Three Fractions Word Problems
These fractions word problems worksheets will produce problems with the addition of three fractions. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Subtracting Fractions Word Problems
These fractions word problems worksheets will produce problems involving subtracting two fractions. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Adding Two Mixed Numbers Word Problems
These fractions word problems worksheets will produce problems involving adding two mixed numbers. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Subtracting Two Mixed Numbers Word Problems
These fractions word problems worksheets will produce problems involving subtracting two mixed numbers. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Adding U.S. Coins Word Problems
This U.S. coins word problems worksheet will produce coin addition problems. These worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Advanced Adding U.S. Coins Word Problems
These Word Problems Worksheets will produce word problems that focus on adding up different denominations of US coin currency. You have the option to select any combination of pennies, nickels, dimes, quarters, and half dollars for each worksheet. These worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

U.S. Money Adding Two Items Word Problems
These U.S. money word problems worksheest will produce purchase problems for adding two items. These worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

U.S. Money Adding Three Items Word Problems
These U.S. money word problems worksheets will produce purchase problems for adding three items. These worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

U.S. Money Change from a Purchase Word Problems
These U.S. money word problems worksheets will produce problems for calculating change from a purchase. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are a good resource for students in the 5th Grade through the 8th Grade.

Travel Time Word Problems Worksheets
These time word problems worksheets produce problems for finding the duration, start, and end times of trips. Users may select the units of time to use in the problems. These word problems worksheets will produce ten problems per worksheet. These word problems worksheets are appropriate for students in the 5th Grade through the 8th Grade.

Reading a Calendar Word Problems Worksheets
These calendar worksheets will produce word problems revolving around reading and understanding a monthly calendar. You may select the month and the year within a range of the years 1800 to 3999. These word problems worksheets will produce nine problems per worksheet. These word problems worksheets are appropriate for students in the 5th Grade through the 8th Grade.

Elapsed Dates Word Problems Worksheets
These time word problems worksheets will produce questions with elapsed days, weeks, months, and years, with ten problems per worksheet. These word problems worksheets are appropriate for 3rd Grade, 4th Grade, and 5th Grade.

Ratios and Rates Word Problems Worksheets
These ratio word problems worksheets will produce eight ratio and rates word problems for the students to solve. These ratio word problems worksheets are appropriate for 3rd Grade, 4th Grade, 5th Grade, 6th Grade, and 7th Grade.

Percentage Word Problems Worksheets
These Percentage Word Problems Worksheets will produce problems that focus on finding and working with percentages. You have the option to select the types of numbers, as well as the types of problem you want. These percentage word problems worksheets are appropriate for 3rd Grade, 4th Grade, 5th Grade, 6th Grade, and 7th Grade.

Mixed Word Problems with Key Phrases Worksheets
These Word Problems Worksheets will produce addition, multiplication, subtraction and division problems using clear key phrases to give the student a clue as to which type of operation to use. These word problems worksheets are appropriate for 4th Grade, 5th Grade, 6th Grade, and 7th Grade.

U.S. Money Change from a Purchase Multiplication Word Problems
These Word Problems Worksheets will produce problems that ask students to use multiplication to calculate the monetary value of a purchase and then find how much change is given from the purchase.

If you need help figuring out how old someone is (or how old you are) given a birthday, you can use this great Age
Calculator. to answer those questions and many more!

Eight interesting and fun least common multiple word problems you can give to your students to tease them. If your students can solve these problems, they can probably solve any word problems about the least common multiple.

Word problem #1

Today, both the soccer team and the basketball team had games. The soccer team plays every 3 days and the basketball team plays every 5 days. When will both teams have games on the same day again?

Word problem #2

A manager at a restaurant can buy hamburger buns in packages of 8 and hamburger patties in packages of 6. Suppose that the manager cannot buy part of a package. What is the least number of packages of each product he can buy to have an equal number of hamburger patties and buns? 

Word problem #3

A man smiles at his beautiful wife every 3 seconds while the wife smiles back at him every 6 seconds. When will both husband and wife smile at each other at the same time?

Word problem #4

Steve can save 9 dollars every day while Maria can save 12 dollars every day. What is the least number of days it will take each person to save the same amount of money?

More Interesting and fun least common multiple word problems

Word problem #5

A least common multiple word problem about barking dogs

Five dogs in a neighborhood were barking consistently last night. The names of the dogs are Lucy, Max, Murphy, Daisy, and Sam. The dogs started barking at 10 P.M.
Then, Lucy barked every 5 minutes, Max barked every 3 minutes, Murphy barked every 4 minutes, Daisy barked every 3 minutes, and Sam barked every minute. Why did Mr. Smith suddenly awaken at 11 P.M.?

Ratio Word Problems

Ratio Word Problems

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Math-Salamanders.com

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What are word problems?

A word problem is a math problem written out as a short story or scenario. Basically, it describes a realistic problem and asks you to imagine how you would solve it using math. If you’ve ever taken a math class, you’ve probably solved a word problem. For instance, does this sound familiar?

Johnny has 12 apples. If he gives four to Susie, how many will he have left?

You could solve this problem by looking at the numbers and figuring out what the problem is asking you to do. In this case, you’re supposed to find out how many apples Johnny has left at the end of the problem. By reading the problem, you know Johnny starts out with 12 apples. By the end, he has 4 less because he gave them away. You could write this as:

12 — 4

12 — 4 = 8, so you know Johnny has 8 apples left.

Word problems in algebra

If you were able to solve this problem, you should also be able to solve algebra word problems. Yes, they involve more complicated math, but they use the same basic problem-solving skills as simpler word problems.

You can tackle any word problem by following these five steps:

1. Read through the problem carefully, and figure out what it’s about.
2. Represent unknown numbers with variables.
3. Translate the rest of the problem into a mathematical expression.
4. Solve the problem.
5. Check your work.

We’ll work through an algebra word problem using these steps. Here’s a typical problem:

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took two days, and the van cost $360. How many miles did she drive?

It might seem complicated at first glance, but we already have all of the information we need to solve it. Let’s go through it step by step.

Step 1: Read through the problem carefully.

With any problem, start by reading through the problem. As you’re reading, consider:

• What question is the problem asking?
• What information do you already have?

Let’s take a look at our problem again. What question is the problem asking? In other words, what are you trying to find out?

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?

There’s only one question here. We’re trying to find out how many miles Jada drove. Now we need to locate any information that will help us answer this question.

There are a few important things we know that will help us figure out the total mileage Jada drove:

• The van cost $30 per day.
• In addition to paying a daily charge, Jada paid $0.50 per mile.
• Jada had the van for 2 days.
• The total cost was $360.

Step 2: Represent unknown numbers with variables.

In algebra, you represent unknown numbers with letters called variables. (To learn more about variables, see our lesson on reading algebraic expressions.) You can use a variable in the place of any amount you don’t know. Looking at our problem, do you see a quantity we should represent with a variable? It’s often the number we’re trying to find out.

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?

Since we’re trying to find the total number of miles Jada drove, we’ll represent that amount with a variable—at least until we know it. We’ll use the variable m for miles. Of course, we could use any variable, but m should be easy to remember.

Step 3: Translate the rest of the problem.

Let’s take another look at the problem, with the facts we’ll use to solve it highlighted.

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?

We know the total cost of the van, and we know that it includes a fee for the number of days, plus another fee for the number of miles. It’s $30 per day, and $0.50 per mile. A simpler way to say this would be:

$30 per day plus $0.50 per mile is $360.

If you look at this sentence and the original problem, you can see that they basically say the same thing: It cost Jada $30 per day and $0.50 per mile, and her total cost was $360. The shorter version will be easier to translate into a mathematical expression.

Let’s start by translating $30 per day. To calculate the cost of something that costs a certain amount per day, you’d multiply the per-day cost by the number of days—in other words, 30 per day could be written as 30 ⋅days, or 30 times the number of days. (Not sure why you’d translate it this way? Check out our lesson on writing algebraic expressions.)

$30 per day and $.50 per mile is $360

$30 ⋅ day + $.50 ⋅ mile = $360

As you can see, there were a few other words we could translate into operators, so and $.50 became + $.50, $.50 per mile became $.50 ⋅ mile, and is became =.

Next, we’ll add in the numbers and variables we already know. We already know the number of days Jada drove, 2, so we can replace that. We’ve also already said we’ll use m to represent the number of miles, so we can replace that too. We should also take the dollar signs off of the money amounts to make them consistent with the other numbers.

$30 ⋅ day + $.50 ⋅ mile = $360

30 ⋅ 2 + .5 ⋅ m = 360

Now we have our expression. All that’s left to do is solve it.

Step 4: Solve the problem.

This problem will take a few steps to solve. (If you’re not sure how to do the math in this section, you might want to review our lesson on simplifying expressions.) First, let’s simplify the expression as much as possible. We can multiply 30 and 2, so let’s go ahead and do that. We can also write .5 ⋅ m as 0.5m.

30 ⋅ 2 + .5 ⋅ m = 360

60 + .5m = 360

Next, we need to do what we can to get the m alone on the left side of the equals sign. Once we do that, we’ll know what m is equal to—in other words, it will let us know the number of miles in our word problem.

We can start by getting rid of the 60 on the left side by subtracting it from both sides.

The only thing left to get rid of is .5. Since it’s being multiplied with m, we’ll do the reverse and divide both sides of the equation with it.

.5m / .5 is m and 300 / 0.50 is 600, so m = 600. In other words, the answer to our problem is 600—we now know Jada drove 600 miles.

Step 5: Check the problem.

To make sure we solved the problem correctly, we should check our work. To do this, we can use the answer we just got—600—and calculate backward to find another of the quantities in our problem. In other words, if our answer for Jada’s distance is correct, we should be able to use it to work backward and find another value, like the total cost. Let’s take another look at the problem.

The rate to rent a small moving van is $30 per day, plus $0.50 per mile. Jada rented a van to drive to her new home. It took 2 days, and the van cost $360. How many miles did she drive?

According to the problem, the van costs $30 per day and $0.50 per mile. If Jada really did drive 600 miles in 2 days, she could calculate the cost like this:

$30 per day and $0.50 per mile

30 ⋅ day + .5 ⋅ mile

30 ⋅ 2 + .5 ⋅ 600

60 + 300

360

According to our math, the van would cost $360, which is exactly what the problem says. This means our solution was correct. We’re done!

While some word problems will be more complicated than others, you can use these basic steps to approach any word problem. On the next page, you can try it for yourself.

Practice!

Let’s practice with a couple more problems. You can solve these problems the same way we solved the first one—just follow the problem-solving steps we covered earlier. For your reference, these steps are:

1. Read through the problem carefully, and figure out what it’s about.
2. Represent unknown numbers with variables.
3. Translate the rest of the problem into a mathematical expression.
4. Solve the problem.
5. Check your work.

If you get stuck, you might want to review the problem on page 1. You can also take a look at our lesson on writing algebraic expressions for some tips on translating written words into math.

Problem 1

Try completing this problem on your own. When you’re done, move on to the next page to check your answer and see an explanation of the steps.

A single ticket to the fair costs $8. A family pass costs $25 more than half of that. How much does a family pass cost?

Problem 2

Here’s another problem to do on your own. As with the last problem, you can find the answer and explanation to this one on the next page.

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?

Problem 1 Answer

Here’s Problem 1:

A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?

Answer: $29

Let’s solve this problem step by step. We’ll solve it the same way we solved the problem on page 1.

Step 1: Read through the problem carefully

The first in solving any word problem is to find out what question the problem is asking you to solve and identify the information that will help you solve it. Let’s look at the problem again. The question is right there in plain sight:

A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?

So is the information we’ll need to answer the question:

• A single ticket costs $8.
• The family pass costs $25 more than half the price of the single ticket.

Step 2: Represent the unknown numbers with variables

The unknown number in this problem is the cost of the family pass. We’ll represent it with the variable f.

Step 3: Translate the rest of the problem

Let’s look at the problem again. This time, the important facts are highlighted.

A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?

In other words, we could say that the cost of a family pass equals half of $8, plus $25. To turn this into a problem we can solve, we’ll have to translate it into math. Here’s how:

1. First, replace the cost of a family pass with our variable f.

f equals half of $8 plus $25

2. Next, take out the dollar signs and replace words like plus and equals with operators.

f = half of 8 + 25

3. Finally, translate the rest of the problem. Half of can be written as 1/2 times, or 1/2 ⋅ :

f = 1/2 ⋅ 8 + 25

Step 4: Solve the problem

Now all we have to do is solve our problem. Like with any problem, we can solve this one by following the order of operations.

1. f is already alone on the left side of the equation, so all we have to do is calculate the right side.

f = 1/2 ⋅ 8 + 25

2. First, multiply 1/2 by 8. 1/2 ⋅ 8 is 4.

f = 4 + 25

3. Next, add 4 and 25. 4 + 25 equals 29 .

f = 29

That’s it! f is equal to 29. In other words, the cost of a family pass is $29.

Step 5: Check your work

Finally, let’s check our work by working backward from our answer. In this case, we should be able to correctly calculate the cost of a single ticket by using the cost we calculated for the family pass. Let’s look at the original problem again.

A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?

We calculated that a family pass costs $29. Our problem says the pass costs $25 more than half the cost of a single ticket. In other words, half the cost of a single ticket will be $25 less than $29.

1. We could translate this into this equation, with s standing for the cost of a single ticket.

1/2s = 29 — 25

2. Let’s work on the right side first. 29 — 25 is 4.

1/2s = 4

3. To find the value of s, we have to get it alone on the left side of the equation. This means getting rid of 1/2. To do this, we’ll multiply each side by the inverse of 1/2: 2.

s = 8

According to our math, s = 8. In other words, if the family pass costs $29, the single ticket will cost $8. Looking at our original problem, that’s correct!

A single ticket to the fair costs $8. A family pass costs $25 more than half that. How much does a family pass cost?

So now we’re sure about the answer to our problem: The cost of a family pass is $29.

Problem 2 Answer

Here’s Problem 2:

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?

Answer: $70

Let’s go through this problem one step at a time.

Step 1: Read through the problem carefully

Start by asking what question the problem is asking you to solve and identifying the information that will help you solve it. What’s the question here?

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?

To solve the problem, you’ll have to find out how much money Mo gave to charity. All the important information you need is in the problem:

• The amount Flor donated is three times as much the amount Mo donated
• Flor and Mo’s donations add up to $280 total

Step 2: Represent the unknown numbers with variables

The unknown number we’re trying to identify in this problem is Mo’s donation. We’ll represent it with the variable m.

Step 3: Translate the rest of the problem

Here’s the problem again. This time, the important facts are highlighted.

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?

The important facts of the problem could also be expressed this way:

Mo’s donation plus Flor’s donation equals $280

Because we know that Flor’s donation is three times as much as Mo’s donation, we could go even further and say:

Mo’s donation plus three times Mo’s donation equals $280

We can translate this into a math problem in only a few steps. Here’s how:

1. Because we’ve already said we’ll represent the amount of Mo’s donation with the variable m, let’s start by replacing Mo’s donation with m.

m plus three times m equals $280

2. Next, we can put in mathematical operators in place of certain words. We’ll also take out the dollar sign.

m + three times m = 280

3. Finally, let’s write three times mathematically. Three times m can also be written as 3 ⋅ m, or just 3m.

m + 3m = 280

Step 4: Solve the problem

It will only take a few steps to solve this problem.

1. To get the correct answer, we’ll have to get m alone on one side of the equation.

m + 3m = 280

2. To start, let’s add m and 3m. That’s 4m.

4m = 280

3. We can get rid of the 4 next to the m by dividing both sides by 4. 4m / 4 is m, and 280 / 4 is 70.

m = 70.

We’ve got our answer: m = 70. In other words, Mo donated $70.

Step 5: Check your work

The answer to our problem is $70, but we should check just to be sure. Let’s look at our problem again.

Flor and Mo both donated money to the same charity. Flor gave three times as much as Mo. Between the two of them, they donated $280. How much money did Mo give?

If our answer is correct, $70 and three times $70 should add up to $280.

1. We can write our new equation like this:

70 + 3 ⋅ 70 = 280

2. The order of operations calls for us to multiply first. 3 ⋅ 70 is 210.

70 + 210 = 280

3. The last step is to add 70 and 210. 70 plus 210 equals 280.

280 = 280

280 is the combined cost of the tickets in our original problem. Our answer is correct: Mo gave $70 to charity.

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Though the majority of ACT math problems use diagrams or simply ask you to solve given mathematical equations, you will also see approximately 15-18 word problems on any given ACT (between 25% and 30% of the total math section). This means that knowing how best to deal with word problems will help you significantly when taking the test. Though there are many different types of ACT word problems, most of them are not nearly as difficult or cumbersome as they may appear. 

This post will be your complete guide to ACT word problems: how to translate your word problems into equations and diagrams, the different types of word problems you’ll see on the test, and how best to go about solving your word problems for test day.

What Are Word Problems?

A word problem is any problem that is based mostly or entirely on written description and does not provide you with an equation, diagram, or graph. You must use your reading skills to translate the words of the question into a workable math problem and then solve for your information.

Word problems will show up on the test for a variety of reasons. Most of the time, these types of questions act to test your reading and visualization skills, as well act as a medium to deliver questions that would otherwise be untestable. For instance, if you must determine the number of sides of an unknown polygon based on given information, a diagram would certainly give the game away!

Translating Word Problems Into Equations or Drawings

In order to translate your word problems into actionable math equations that you can solve, you’ll need to know and utilize some key math terms. Whenever you see these words, you can translate them into the proper action. For instance, the word “product” means “the value of two or more values that have been multiplied together,” so if you need to find “the product of a and b,” you’ll need to set up your equation with $a * b$. 

Key Terms	Mathematical Action
Sum, increased by, added to, total of	+
Difference, decreased by, subtracted from	—
Product, times	* or x
Divided by	/ or ÷
Equals, is, are, equivalent, same	=
Is less than	<
Is greater than	>
Is less than or equal to	≤
Is greater than or equal to	≥

Let’s take a look at this in action with an example problem:

Learning the language of ACT word problems will help you to unravel much of the mystery of these types of questions. 

Typical ACT Word Problems

ACT word problems can be grouped into two major categories: word problems where you must simply set up an equation and word problems in which you must solve for a specific piece of information.

Word Problem Type 1: Setting Up an Equation

This is the less common type of word problem on the test, but you’ll generally see it at least once or twice. You’ll also usually see this type of word problem first. For this type of question, you must use the given information to set up the equation, even though you don’t need to solve for the missing variable.

Almost always, you’ll see this type of question in the first ten questions on the test, meaning that the ACT test-makers consider them fairly “easy.” This is due to the fact that you only have to provide the set-up and not the execution.

 

Word Problem Type 2: Solving for Your Information

Other than the few set-up word questions you’ll see, the rest of your ACT word problem questions will fall into this category. For these questions, you must both set up your equation and solve for a specific piece of information.

Most (though not all) word problem questions of this type will be scenarios or stories covering all sorts of ACT math topics, including averages, single variable equations, and probabilities, among others. You almost always must have a solid understanding of the math topic in question in order to solve the word problem on the topic.

You might also be given a geometry problem as a word problem, which may or may not be set up with a scenario as well.

Geometry questions will be presented as word problems typically because the test-makers felt the problem would be too easy to solve had you been given a diagram.

Now, how do we put our knowledge to its best effect? Let’s take a look.

ACT Math Strategies for Your Word Problems

Though you’ll see word problems on a myriad of different types of ACT math topics, there are still a few techniques you can apply to solve your word problems as a whole.

#1: Draw It Out

Whether your problem is a geometry problem or an algebra problem, sometimes making a quick sketch of the scene can help you understand what, exactly, you’re working with. For instance, let’s look at how a picture can help you solve a ratio/division problem:

As for geometry problems, remember—you’re often given a word problem as a word problem because it would be too simple to solve had you had a diagram to work with from the get-go. So take back the advantage and draw the picture yourself. Even a quick and dirty sketch can help you visualize the problem much easier than you can in your head and help keep all your information clear.

#2: Memorize Important Terms

If you’re not used to translating English words into mathematical equations, then ACT word problems can sound like so much nonsense and leave you floundering to set up the proper equation. Look to the chart and learn how to translate your keywords into their math equivalents. Doing so will help you to understand exactly what the problem is asking you to find.

There are free ACT math questions available online, so memorize your terms and then practice on real ACT word problems to make sure you’ve got your definitions down and can apply them to real problems.

#3: Underline and Write Out the Key Information

The key to solving a word problem is bringing together all the relevant pieces of given information and putting them in the right places. Make sure you write out all your givens on the diagram you’ve drawn (if the problem calls for a diagram) and that all your moving pieces are in order.

One of the best ways to keep all your pieces straight is to underline them in the problem and then write them out yourself before you set up your equation, so take a moment to perform this step.

#4: Pay Close Attention to Exactly What Is Being Asked of You

Little is more frustrating than solving for the wrong variable or writing in your given values in the wrong places. And yet this is entirely too easy to do when working with word problems.

Make sure you pay strict attention to exactly what you’re meant to be solving for and exactly what pieces of information go where. Are you looking for the area or the perimeter? The value of $x$ or $x + y$? Better to make sure before you start what you’re supposed to find than realize two minutes down the line that you have to solve the problem all over again.

#5: Brush Up on Any Specific Math Topic in Which You Feel Weak

You are likely to see both diagram/equation problems and word problems for any given ACT math topic on the test. Many of the topics can swing either way, which is why there are so many different types of word problems and why you’ll need to know the ins and outs of any particular math topic in order to solve its corresponding word problem. For example, if you don’t know how to properly set up a system of equations problem, you will have a difficult time of it when presented with a word problem on the topic.

So understand that solving a word problem is a two-step process: it requires you to both understand how word problems themselves work and to understand the math topic in question. If you have any areas of mathematical weakness, now is a good time to brush up on them, or else the word problem might be trickier than you were expecting.

All set? Time to shine!

Test Your Knowledge

Now to put your word problem know-how to the test with real ACT math problems. 

Answers: K, C, A, E

Answer Explanations:

A round of applause to your success!
Picture: John Morris/Flickr

The Take-Aways

Word problems comprise a significant portion of the ACT, so it’s a good idea to understand how they work and how to translate the words into a proper equation. But remember that translating your word problems is still only half the battle.

You must also supplement this knowledge of how to solve word problems with a solid understanding of the math topic in question. For example, it won’t do a lot of good if you can translate a probability word problem if you don’t understand exactly how probabilities work. So be sure to not only learn how to approach your word problems, but also hone your focus on any math topics you feel you need to improve upon. You can find links to all of our ACT math topic guides here to help your studies.

What’s Next? 

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