3.1 Use a Problem-Solving Strategy

Learning objectives

Learning objectives

• Approach word problems with a positive attitude

• Use a problem-solving strategy for word problems

• Solve number problems

Approach Word Problems with a Positive Attitude

“If you think you can… or think you can’t… you’re right.”—Henry Ford

The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?

How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does 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 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. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.3 and say them out loud.

Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.

Example 3.1

Solution

Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.

• In this problem, is it clear what is being discussed? Is every word familiar?

Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

• In this problem, the words “what was the original price of the purse” tell us what we need to find.

Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.

• Let $p=$ the original price of the purse.

Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.

Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.

Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.

Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that $p=36,$ which means “the original price” was $36.

• Does $36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.

Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”

• The answer to the question is: “The original price of the purse was $36.”

If this were a homework exercise, our work might look like this:

Pilar bought a purse on sale for $18, which is one-half the original price. What was the original price of the purse?

Let’s try this approach with another example.

Example 3.2

Solution

Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Example 3.3

Solution

Example 3.4

Solution

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example 3.5

Solution

Example 3.6

Solution

Example 3.7

Solution

Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

Notice that each number is one more than the number preceding it. So if we define the first integer as n, the next consecutive integer is $n+1.$ The one after that is one more than $n+1,$ so it is $n+1+1,$ which is $n+2.$

Example 3.8

Solution

Example 3.9

Solution

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

Notice each integer is 2 more than the number preceding it. If we call the first one n, then the next one is $n+2.$ The next one would be $n+2+2$ or $n+4.$

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.

Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

Example 3.10

Solution

Table

3.1

Example 3.11

Solution

Section 3.1 Exercises

Section 3.1 Exercises

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

Use a Problem-Solving Strategy for Word Problems

In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.

Solve Number Problems

In the following exercises, solve each number word problem.

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!

…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.

1. Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?

2. One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.

3. The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.

1. The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.

2. The sum of two numbers is $-18.$ One number is 40 more than the other. Find the numbers.

3. One number is eight more than twice another. Their sum is negative four. Find the numbers.

1. One number is three more than three times another. Their sum is $-5.$ Find the numbers.

2. The sum of two consecutive integers is $95.$ Find the numbers.

3. The sum of two consecutive integers is $-31.$ Find the numbers.

1. Find three consecutive integers whose sum is $-96.$

2. Find three consecutive integers whose sum is $-36.$

3. Find three consecutive even integers whose sum is 102.

1. Find three consecutive even integers whose sum is $-24.$

2. According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was$1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?

3. U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was$10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?

1. List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.

2. Philip pays $1,620 in rent every month. This amount is$120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?

3. Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is$8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?

1. Three times the sum of a number and nine is 12. Find the number.

2. One number is six more than the other. Their sum is 42. Find the numbers.

3. The sum of two numbers is 20. One number is four less than the other. Find the numbers.

1. The sum of two numbers is $-45.$ One number is nine more than the other. Find the numbers.

2. The sum of two numbers is $-316.$ One number is 94 less than the other. Find the numbers.

3. One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.

1. One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.

2. One number is one more than twice another. Their sum is $-5.$ Find the numbers.

3. The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.

1. The sum of two consecutive integers is 77. Find the integers.

2. The sum of two consecutive integers is $-23.$ Find the integers.

3. The sum of three consecutive integers is 78. Find the integers.

1. Find three consecutive integers whose sum is $-36.$

2. Find three consecutive even integers whose sum is 258.

3. Find three consecutive odd integers whose sum is 171.

1. Find three consecutive even integers whose sum is $-36.$

2. Find three consecutive odd integers whose sum is $-213.$

3. Sale Price Patty paid $35 for a purse on sale for$10 off the original price. What was the original price of the purse?

1. Buying in Bulk Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.

2. Price before Sales Tax Tom paid $1,166.40 for a new refrigerator, including$86.40 tax. What was the price of the refrigerator?

3. What has been your past experience solving word problems?

1. What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.