1.5 Solving Word Problems

 Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student in the cartoon below?

When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.

Start with a fresh slate and begin to think positive thoughts like the student in the cartoon below. Read the positive thoughts and say them out loud.

If we take control and believe we can be successful, we will be able to master word problems.

Think of something that you can do now but couldn’t do three years ago. Whether it’s driving a car, snowboarding, cooking a gourmet meal, or speaking a new language, you have been able to learn and master a new skill. Word problems are no different. Even if you have struggled with word problems in the past, you have acquired many new math skills that will help you succeed now!

Use a Problem-Solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you’ve increased your math vocabulary as you learned about more algebraic procedures, and you’ve had more practice translating from words into algebra.

You have also translated word sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. You had to restate the situation in one sentence, assign a variable, and then write an equation to solve. This method works as long as the situation is familiar to you and the math is not too complicated.

Now we’ll develop a strategy you can use to solve any word problem. This strategy will help you become successful with word problems. We’ll demonstrate the strategy as we solve the following problem.

Try it

Translate the following verbal statement into an algebraic equation and then solve:

• Dion bought a bookcase on sale for $180, which was one-fourth of the original price.
• What was the original price of the bookcase?
• Use pp for your variable.

Answer

We list the steps we took to solve the previous example.

Problem-Solving Strategy

1. Read the word problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don’t understand, look them up in a dictionary or on the interne
2. Identify what you are looking for.
3. Name what you are looking for. Choose a variable to represent that quantity.
4. Translate into an equation. It may be helpful to first restate the problem in one sentence before translating.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem. Make sure it makes sense.
7. Answer the question with a complete sentence.

For a review of how to translate algebraic statements into words, watch the following video.^[1]

Try it

Translate the following scenario into an algebraic equation and then solve:

• Use j for your variable to represent journals.
• Pavati bought a total of 7 textbooks at the bookstore. She bought four more textbooks than journals.
• How many journals did she buy?

Answer

In the next example, we will apply our Problem-Solving Strategy to applications of percent.

Example

Nga’s car insurance premium increased by [latex]\text{\$60}[/latex], which was [latex]\text{8%}[/latex] of the original cost. What was the original cost of the premium?

Solution

Step 1. Read the problem. Remember, if there are words you don’t understand, look them up.
Step 2. Identify what you are looking for.	the original cost of the premium
Step 3. Name. Choose a variable to represent the original cost of premium.	Let [latex]c=\text{the original cost}[/latex]
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation.	[latex]60=0.08c[/latex]
Divide both sides by [latex]0.08[/latex].	[latex]\Large\frac{60}{\color{red}{0.08}}\normalsize =\Large\frac{0.08c}{\color{red}{0.08}}[/latex]
Simplify.	[latex]c=750[/latex]
Step 6. Check: Is our answer reasonable? Yes, a [latex]\text{\$750}[/latex] premium on auto insurance is reasonable. Now let’s check our algebra. Is 8% of 750 equal to [latex]60[/latex]?[latex]750=c[/latex] [latex]0.08(750)=60[/latex] [latex]60=60\quad\checkmark[/latex]
Step 7. Answer the question.	The original cost of Nga’s premium was [latex]\text{\$750}[/latex].

Try It

Translate the following verbal statement into an algebraic equation and then solve:

• Laila’s rent increased by 4%. The increase was $54.4.
• What was the original amount of Laila’s rent?
• Use r for your variable.