# 1.15: Problem-Solving Models

**Basic**Created by: CK-12

**Practice**Problem-Solving Models

Suppose you're taking a standardized test to get into college and you encounter a type of problem that you've never seen before. What tools could you use to help solve the problem? Is there anything you should do before trying to solve the problem? Is there anything you should do afterwards?

### Problem Solving Strategies

Much of mathematics applies to real-world situations. To think critically and to problem solve are mathematical abilities. Although these capabilities may be the most challenging, they are also the most rewarding.

To be successful in applying mathematics in real-life situations, you must have a “toolbox” of strategies to assist you. Many algebra lessons are devoted to filling this toolbox so you become a better problem solver and can tackle mathematics in the real world.

#### Step 1: Read and Understand the Given Problem

Every problem you encounter gives you clues needed to solve it successfully. Here is a checklist you can use to help you understand the problem.

\begin{align*}\surd\end{align*} Read the problem carefully. Make sure you read all the sentences. Many mistakes have been made by failing to fully read the situation.

\begin{align*}\surd\end{align*} Underline or highlight key words. These include mathematical operations such as **sum***,* **difference**, and **product***,* and mathematical verbs such as **equal, more than, less than,** and **is***.* Key words also include the nouns the situation is describing, such as **time, distance, people,** etc.

\begin{align*}\surd\end{align*} Ask yourself if you have seen a problem like this before. Even though the nouns and verbs may be different, the general situation may be similar to something else you’ve seen.

\begin{align*}\surd\end{align*} What are you being asked to do? What is the question you are supposed to answer?

\begin{align*}\surd\end{align*} What facts are you given? These typically include numbers or other pieces of information.

Once you have discovered what the problem is about, the next step is to declare what variables will represent the nouns in the problem. Remember to use letters that make sense!

#### Step 2: Make a Plan to Solve the Problem

The next step in problem-solving is to **make a plan** or **develop a strategy.** How can the information you know assist you in figuring out the unknown quantities?

Here are some common strategies that you will learn.

- Drawing a diagram
- Making a table
- Looking for a pattern
- Using guess and check
- Working backwards
- Using a formula
- Reading and making graphs
- Writing equations
- Using linear models
- Using dimensional analysis
- Using the right type of function for the situation

In most problems, you will use a combination of strategies. For example, drawing a diagram and looking for patterns are good strategies for most problems. Also, making a table and drawing a graph are often used together. The “writing an equation” strategy is the one you will work with the most frequently in your study of algebra.

#### Step 3: Solve the Problem and Check the Results

Once you develop a plan, you can use it to **solve the problem.**

The last step in solving any problem should always be to **check and interpret** the answer. Here are some questions to help you to do that.

- Does the answer make sense?
- If you substitute the solution into the original problem, does it make the sentence true?
- Can you use another method to arrive at the same answer?

#### Step 4: Compare Alternative Approaches

Sometimes a certain problem is best solved by using a specific method. Most of the time, however, it can be solved by using several different strategies. When you are familiar with all of the problem-solving strategies, it is up to you to choose the methods that you are most comfortable with and that make sense to you. In this book, we will often use more than one method to solve a problem. This way we can demonstrate the strengths and weaknesses of different strategies when applied to different types of problems.

Regardless of the strategy you are using, you should always implement the problem-solving plan when you are solving word problems. Here is a summary of the problem-solving plan.

**Step 1:** Understand the problem.

**Step 2:** Devise a plan – Translate. Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart, or construct a table as a start to begin your problem-solving plan.

**Step 3:** Carry out the plan – Solve.

**Step 4:** Check and Interpret-Check to see if you have used all your information. Then look to see if the answer makes sense.

**Let's solve the following real-world problems using a plan:**

- Jeff is 10 years old. His younger brother, Ben, is 4 years old. How old will Jeff be when he is twice as old as Ben?

Begin by understanding the problem. Highlight the key words.

**Jeff** is **10** years old. His younger brother, **Ben**, is **4** years old. **How old** will Jeff be **when he is twice as old as Ben**?

The question we need to answer is. “What is Jeff’s age when he is twice as old as Ben?”

You could guess and check, use a formula, make a table, or look for a pattern.

The key is “twice as old.” This clue means two times, or double Ben’s age. Begin by doubling possible ages. Let’s look for a pattern.

\begin{align*}4 \times 2 = 8\end{align*}. Jeff is already older than 8.

\begin{align*}5 \times 2 = 10\end{align*}. This doesn’t make sense because Jeff is already 10.

\begin{align*}6 \times 2 = 12\end{align*}. In two years, Jeff will be 12 and Ben will be 6. Jeff will be twice as old.

Jeff will be 12 years old when he is twice as old as Ben.

Another way to solve this problem is to write an algebraic equation.

Let \begin{align*}x\end{align*} be the age of Ben. We want to know when Jeff will be twice as old as Ben, which can be expressed as \begin{align*}2x\end{align*}. We also know that since Jeff is 10 and Ben is 4, that Jeff is 6 years older than Ben. Jeff's age can be expressed as \begin{align*}x+6\end{align*}. We want to know when Jeff's age will be twice Ben's age, so putting these together, we get

\begin{align*}2x=x+6\end{align*}.

What value of \begin{align*}x\end{align*} would satisfy this equation? Solving this equation, we can find that \begin{align*}x=6\end{align*}. But \begin{align*}x=6\end{align*} is Ben's age, and Jeff is 6 years older so \begin{align*}x+6=6+6=12\end{align*}.

When Jeff is 12, he will be twice Ben's age, since 12 is twice the age of 6.

- Matthew is planning to harvest his corn crop this fall. The field has 660 rows of corn with 300 ears per row. Matthew estimates his crew will have the crop harvested in 20 hours. How many ears of corn will his crew harvest per hour?

Begin by highlighting the key information.

Matthew is planning to harvest his corn crop this fall. The field has **660 rows** of corn with **300 ears per row**. Matthew estimates his crew will have the **crop harvested in 20 hours. How many ears of corn** will his crew **harvest per hour**?

You could draw a picture (it may take a while), write an equation, look for a pattern, or make a table. Let’s try to use reasoning.

We need to figure out how many ears of corn are in the field: \begin{align*}660(300) = 198,000\end{align*}. There are 198,000 ears in the field. It will take 20 hours to harvest the entire field, so we need to divide 198,000 by 20 to get the number of ears picked per hour.

\begin{align*}\frac{198,000}{20} = 9,900\end{align*}

The crew can harvest 9,900 ears per hour.

### Examples

#### Example 1

Earlier, you were asked what tools you have to solve a problem you haven't seen before.

As described in this concept, there are 4 important steps:

Step 1: Understand the problem.

**Step 2: **Devise a plan – Come up with a way to solve the problem. Set up an equation, draw a diagram, make a chart, or construct a table to help.

**Step 3: **Carry out the plan.

**Step 4: **Check and interpret – Check to see if your answer makes sense and you have used the information provided.

#### Example 2

The sum of angles in a triangle is 180 degrees. If the second angle is twice the size of the first angle and the third angle is three times the size of the first angle, what are the measures of the angles in the triangle?

Step 1 is to read and determine what the problem is asking us. After reading, we can see that we need to determine the measure of each angle in the triangle. We will use the information given to figure this out.

Step 2 tells us to devise a plan. Since we are given a lot of information about how the different pieces are related, it looks like we can write some algebraic expressions and equations in order to solve this problem. Let \begin{align*}a\end{align*} be the measure of the first angle. The second angle is twice the first, so think about how you can express that algebraically. The correct expression is \begin{align*}2a\end{align*}. Also, the third angle is three times the size of the first so that would be \begin{align*}3a\end{align*}. Now the other piece of information given to us is that all three angles must add up to 180 degrees. From this we will write an equation, adding together the expressions of the three angles and setting them equal to 180.

\begin{align*}a+2a+3a=180\end{align*}

Step 3 is to solve the problem. Simplifying this we get

\begin{align*}6a=180\end{align*}

\begin{align*}a=30\end{align*}

Now we know that the first angle is 30 degrees, which means that the second angle is 60 degrees and the third is 90 degrees. Let's check whether these three angles add up to 180 degrees.

\begin{align*}30+60+90=180\end{align*}

The three angles do add up to 180 degrees.

Step 4 is to consider other possible methods. We could have used guess and check and possibly found the correct answer. However, there are many choices we could have made. What would have been our first guess? There are so many possibilities for where to start with guess and check that solving this problem algebraically was the simplest way.

### Review

- What are the four steps to solving a problem?
- Name three strategies you can use to help make a plan. Which one(s) are you most familiar with already?
- Which types of strategies work well together? Why?
- Suppose Matthew’s crew takes 36 hours to harvest the field. How many ears per hour will they harvest?
- Why is it difficult to solve Ben and Jeff’s age problem by drawing a diagram?
- How do you check a solution to a problem? What is the purpose of checking the solution?
- There were 12 people on a jury, with four more women than men. How many women were there?
- A rope 14 feet long is cut into two pieces. One piece is 2.25 feet longer than the other. What are the lengths of the two pieces?
- A sweatshirt costs $35. Find the total cost if the sales tax is 7.75%.
- This year you got a 5% raise. If your new salary is $45,000, what was your salary before the raise?
- It costs $250 to carpet a room that is \begin{align*}14 \ ft \times 18 \ ft\end{align*}. How much does it cost to carpet a room that is \begin{align*}9 \ ft \times 10 \ ft\end{align*}?
- A department store has a 15% discount for employees. Suppose an employee has a coupon worth $10 off any item and she wants to buy a $65 purse. What is the final cost of the purse if the employee discount is applied before the coupon is subtracted?
- To host a dance at a hotel, you must pay $250 plus $20 per guest. How much money would you have to pay for 25 guests?
- It costs $12 to get into the San Diego County Fair and $1.50 per ride. If Rena spent $24 in total, how many rides did she go on?
- An ice cream shop sells a small cone for $2.92, a medium cone for $3.50, and a large cone for $4.25. Last Saturday, the shop sold 22 small cones, 26 medium cones, and 15 large cones. How much money did the store take in?

**Mixed Review**

- Choose an appropriate variable for the following situation: It takes Lily 45 minutes to bathe and groom a dog. How many dogs can she groom in an 9-hour day?
- Translate the following into an algebraic inequality: Fourteen less than twice a number is greater than or equal to 16.
- Write the pattern of the table below in words and using an algebraic equation.

\begin{align*}&& x && -2 && -1 && 0 && 1\\ && y && -8 && -4 && 0 && 4\end{align*}

- Check that \begin{align*}m=4\end{align*} is a solution to \begin{align*}3y-11 \ge -3\end{align*}.
- What is the domain and range of the graph shown?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.15.

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### Image Attributions

Here you'll be exposed to many different methods that you can use to solve a problem and the way that these methods should fit into your overall problem-solving plan.

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