Algebra Topics

Introduction to Word Problems

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#### Lesson 9: Introduction to Word Problems

### What are word problems?

### Word problems in algebra

#### Step 1: Read through the problem carefully.

#### Step 2: Represent unknown numbers with variables.

#### Step 3: Translate the rest of the problem.

#### Step 4: Solve the problem.

#### Step 5: Check the problem.

### Practice!

#### Problem 1

#### Problem 2

### Problem 1 Answer

#### Step 1: Read through the problem carefully

#### Step 2: Represent the unknown numbers with variables

#### Step 3: Translate the rest of the problem

#### Step 4: Solve the problem

#### Step 5: Check your work

### Problem 2 Answer

#### Step 1: Read through the problem carefully

#### Step 2: Represent the unknown numbers with variables

#### Step 3: Translate the rest of the problem

#### Step 4: Solve the problem

#### Step 5: Check your work

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?

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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.

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:

**Read**through the problem carefully, and figure out what it's about.**Represent**unknown numbers with variables.**Translate**the rest of the problem into a mathematical expression.**Solve**the problem.**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.

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**.

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.

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.

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 0.5*m*.

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**.

60 | + .5m = | 360 |

-60 | -60 |

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 | = | 300 |

.5 | .5 |

**.5 m / .5 **is

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.

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:

**Read**through the problem carefully, and figure out what it's about.**Represent**unknown numbers with variables.**Translate**the rest of the problem into a mathematical expression.**Solve**the problem.**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.

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?

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?

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.

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.

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

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:

- First, replace
**the cost of a family pass**with our variable*f*. - Next, take out the dollar signs and replace words like
**plus**and**equals**with operators. - Finally, translate the rest of the problem.
**Half of**can be written as 1/2 times, or 1/2 ⋅ :

f equals half of $8 plus $25

f = half of 8 + 25

f = 1/2 ⋅ 8 + 25

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.

*f*is already alone on the left side of the equation, so all we have to do is calculate the right side.- First, multiply 1/2 by 8.
**1/2 ⋅ 8**is 4. - Next, add 4 and 25.
**4 + 25**equals 29 .

f = 1/2 ⋅ 8 + 25

f = 4 + 25

f = 29

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

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.

- We could translate this into this equation, with
*s*standing for the cost of a single ticket. - Let's work on the right side first.
**29 - 25**is 4. - 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.

1/2s = 29 - 25

1/2s = 4

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.

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.

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**

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

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:

- 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*. - Next, we can put in
**mathematical operators**in place of certain words. We'll also take out the dollar sign. - Finally, let's write
**three times**mathematically.**Three times**can also be written as*m**3*⋅*m*, or just 3*m*.

m plus three times m equals $280

m + three times m = 280

m + 3m = 280

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

- To get the correct answer, we'll have to get
*m*alone on one side of the equation. - To start, let's add
*m*and 3*m*. That's 4*m*. - We can get rid of the 4 next to the
*m*by dividing**both sides**by 4.**4**is*m*/ 4*m*, and**280 / 4**is 70.

m + 3m = 280

4m = 280

m = 70.

We've got our answer: ** m = 70**. In other words,

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

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

- We can write our new equation like this:
- The order of operations calls for us to multiply first.
**3 ⋅ 70**is 210. - The last step is to add 70 and 210.
**70**plus**210**equals 280.

70 + 3 ⋅ 70 = 280

70 + 210 = 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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