Translating Word Problems into Algebraic Expressions

1. Writing an Algebraic Expression

So far, most of the math we have covered has been arithmetic—or in other words, plain numbers and occasionally some variables. But we all know there are other types of math problems: word problems. Word problems may not be most people’s favorites. Why is that? Well, it’s because it can be difficult to understand what the word problem is asking you to do. Do you need to add? Subtract? How many steps, and if there are multiple steps needed, which step do you take first?!

Just breathe, it’s ok. First, there are some great general rules to remember when writing an expression to represent a situation:

1. Avoid skimming the problem. Read the entire problem to avoid missing or misinterpreting any information.
2. Determine what the question is asking you to find. This could mean you underline the important information or write it out yourself—whichever you prefer. These unknown quantities will help identify and define the variables in the expression.
3. Clearly state the letter that will be used for the variable and what that variable represents in the context of the problem. The letter x is commonly used as a variable, but any letter can be used to represent an unknown quantity.
4. Use the variables and other necessary given quantities from the situation to write an expression that represents the problem.
5. Review the expression to make sure it is logical and accurate.

So, now that we know how to approach word problems and how to identify important parts, let’s do some examples.

EXAMPLE

Lucy has 5 flowers. Max has 6 flowers. How many flowers do they have all together?

Let’s identify our important parts.

Lucy has 5 flowers. Max has 6 flowers. How many flowers do they have all together?

You may know the answer pretty quickly, but let’s break it down, just in case. The word problem used the term “all together.” According to our chart, what does that mean? It means you should add the values together.

5 + 6

So, they have 11 flowers.

EXAMPLE

Now, let’s try something a little more challenging.

You have a monthly cable bill. Each month, the cable bill is $50 and $2 for every movie you rented. How would you write this expression?

Step 1: Make sure you have read the entire problem to avoid missing or misinterpreting any information.

Step 2: Let’s identify the important parts.

You have a monthly cable bill. Each month, the cable bill is $50 and $2 for every movie you rented. How would you write this expression?

So this problem used the word “and.” According to our chart, this means that the operation we will use is addition.

Step 3: Identify and define the unknown variable.

We pay $2 for every movie rented. However, we don’t know how many movies you’re going to rent this month. As we have discussed earlier, a variable represents an unknown value—which is what we have here.

We have identified a variable and we’ll call it x for this example.

x = the amount of movies rented this month.

Because it is $2 for every movie and we don’t know how many we will watch, that would be $2 + $2 + $2 an unknown number of times. But repeated addition is the same as multiplication so it could also be written as 2 times x.

Step 4: Let’s add your variable to the base cost of your cable bill to make:

50 + 2x

Step 5: Review our expression. Does it make sense? A good way to test if the expression is accurate is to test it out with a value. Let’s say we rented 3 movies this month. Your bill would then be:

50 + 2(3)
50 + 6
$56

try it

Your plane has just landed and you need to take a taxi to your friend’s house. The taxi has a surcharge of $10 to pick you up at the airport plus charges an additional $1 for every mile. You’re not quite sure how far away your friend lives.

Write an expression that represents the total cost of a taxi ride.

Step 1: Make sure you have read the entire problem to avoid missing or misinterpreting any information.

Step 2: Identify the important parts

The taxi has a surcharge of $10 to pick you up at the airport plus charges an additional $1 for every mile.

The problem uses the word “plus”. From our chart, we know this represents addition.

Step 3: Identify and define the unknown variable.

As stated in the problem, we are not quite sure how far away or how many miles we will need to travel. This will be our unknown variable. Let’s call it y.

y = number of miles traveled in the taxi

Step 4: Let’s add your variable to the airport pick-up surcharge to make:

10 + 1y or we can just write 10 + y.

Step 5: Review our expression. Does it make sense?

A good way to test if the expression is accurate is to test it out with a value. Let’s say you traveled 20 miles. Your total cost would then be:

10 + 1(20)
10 + 20
$30