# 1.16: Problem-Solving Models

**At Grade**Created by: CK-12

**Practice**Problem-Solving Models

Have you ever wondered how to solve a word problem or a story problem? Well, when you understand how to use a problem solving plan; it can make things a lot simpler.

Tyler loves to visit the orangutans at the city zoo. The orangutans are one of four living types of great apes. They are reddish-orange in color and swing and climb all around. Tyler thinks that they are very social as the orangutans often come up to the glass to peer at him when he visits. Tyler could stay at the orangutan exhibit for hours. In his last visit, Tyler saw a sign about orangutan adoption at the zoo. This piqued his interest, so he investigated more about it. At many zoos, including the city zoo in Tyler’s town, you can adopt a specific animal or species of animal. Any money donated goes directly to the care of this species of animal. You can adopt an animal for any amount from \begin{align*}\$35\end{align*} to \begin{align*}\$1000\end{align*}. Tyler has decided to use the money from his summer job to adopt an orangutan.

Tyler is working this summer doing yard work for his neighbors. Because of his excellent work ethic, he has many clients. Tyler figures out that he will make $125.00 per week on yard work.

There are different adoption pledge levels:

\begin{align*}&\text{Bronze} = \$35 - \$100 \\ &\text{Silver} = \$100 - \$500 \\ &\text{Gold} = \$500 - \$1000\end{align*}

If Tyler works for eight weeks, how much money will he collect? How much can he pledge to adopt the orangutan for at the end of eight weeks? What will Tyler’s pledge level be?

To help Tyler figure this out, we are going to use a problem-solving plan.

**In this Concept, you will learn all about a four-part problem-solving plan that can help you figure out how much money Tyler can use for his orangutan adoption.**

### Guidance

There are four parts to a problem solving plan.

- Four-Part Problem-Solving Plan
- Read and understand a given problem situation.
- Make a plan to solve the problem.
- Solve the problem and check the results
- Compare alternative approaches to solving the problem.

When looking at a problem, we first need to read it and underline all of the important information. Sometimes you will be given information that is not necessary to solving the problem. Next, you will need to figure out how you are going to solve the problem.

Now, let's go back and think about Tyler and his orangutan dilemma. Tyler loves to visit the orangutans at the city zoo. The orangutans are one of four living types of great apes. They are reddish-orange in color and swing and climb all around. Tyler thinks that they are very social as the orangutans often come up to the glass to peer at him when he visits. Tyler could stay at the orangutan exhibit for hours. In his last visit, Tyler saw a sign about orangutan adoption at the zoo. This piqued his interest and so he investigated more about it. At many zoos, including the city zoo in Tyler’s town, you can adopt a specific animal or species of animal. Any money donated goes directly to the care of this species of animal. You can adopt an animal for any amount from \begin{align*}\$35\end{align*} to \begin{align*}\$1000\end{align*}. Tyler has decided to use the money from his summer job to adopt an orangutan.

Tyler is working this summer doing yard work for his neighbors. Because of his excellent work ethic, he has many clients. Tyler figures out that he will make \begin{align*}\$125.00\end{align*} per week on yard work.

There are different adoption pledge levels:

\begin{align*}& \underline{\text{Bronze} = \$35 - \$100} \\ & \underline{\text{Silver} = \$100 - \$500} \\ & \underline{\text{Gold} = \$500 - \$1000}\end{align*}

If Tyler works for eight weeks, how much money will he collect? How much can he pledge to adopt the orangutan for at the end of eight weeks? What will Tyler’s pledge level be?

**There is a lot of information in this problem.**

**Some of it we need and some of it we don’t.**

**Notice that there are three questions at the end of the problem. These questions tell us what we need to solve for this problem.**

- If Tyler works for eight weeks, how much will he collect?
- How much can he pledge to adopt the orangutan?
- What will his pledge level be?

**The first thing that we will need to figure out is how much money Tyler will make in 8 weeks.** Here is the information that we have been given in the problem.

Tyler makes \begin{align*}\$125.00\end{align*} per week.

Tyler works for 8 weeks.

**Until we know how much he will make in 8 weeks, we can’t move on to answering any other questions.** **We need a plan to help us.** Each week, Tyler filled in the amount of money he made. He wrote this amount in on Friday. Tyler made $125.00 per week for 8 weeks.

**Once you have the given information, you will need to choose an operation to help you solve the problem.** **Which operation can help us to figure out Tyler’s total at the end of eight weeks?** **We could use addition or multiplication.** **Since multiplication is a short cut, let’s use multiplication.** **This is our plan for solving the problem.**

You can see how we are using a problem solving plan with this problem. Before we finish helping Tyler solve his problem, it is time for you to practice.

Try a few of these problems using a problem solving plan.

#### Example A

A small lion weighs in at 330 pounds. If a large lion weighs in at 500 pounds, what is the difference in weight between the two lions?

**Solution: 170 pounds**

#### Example B

If there are four large lions in the habitat, how much do the lions weigh in all?

**Solution: 2000 pounds**

#### Example C

If there are five small lions in the habitat, what is the total weight of the small lions?

**Solution: 1650 pounds**

Back to Tyler and the orangutan, now we can apply our plan.

To solve the problem and check our results, we are going to first write an equation. We use multiplication and our given information to write this equation.

$\begin{align*}125 \times 8 =\end{align*} total amount of money made

\begin{align*}\$1000 =\end{align*} the total amount of money made

**Our answer is that Tyler made $1000 in eight weeks.** **Next, we need to check our results.** **The best way to check our results is to think about other ways that we could have solved the problem.** **If we get the same answer using a different strategy, then we can be sure that our work is accurate.**

We chose to use multiplication to figure out the sum of Tyler’s money.

**Is there another way that we could have solved the problem?** **We could have used repeated addition to solve the problem.** **Let’s do this and then see if we get the same answer that we did when we multiplied.**

\begin{align*}& \quad \ ^21^4 25 \\ & \qquad 125 \\ & \qquad 125 \\ & \qquad 125 \\ & \qquad 125 \\ & \qquad 125 \\ & \qquad 125 \\ & \ \underline{+ \quad 125} \\ & \quad \ \ 1000\end{align*}

**Our answer is** \begin{align*}\$1000\end{align*}.

**By solving this problem using another method, we can be sure that our work is correct.**

Now we can sum up our results.

**If Tyler works for eight weeks and makes** \begin{align*}\$125.00\end{align*} **per week, he will have** \begin{align*}\$1000\end{align*} **to adopt an orangutan.** **Given the pledge levels, Tyler will be at the highest pledge level.** **Tyler's purchase will be a GOLD level adoption.** *Tyler lets us know that everyone can make a difference. You can investigate adopting an animal by visiting your local zoo or animal shelter.*

### Vocabulary

Here are the vocabulary words that are found in this Concept.

- Product
- the answer to a multiplication problem

- Quotient
- the answer to a division problem

- Word Problem
- A problem that uses verbal language to explain a mathematical situation.

- Sum
- the answer in an addition problem

- Difference
- the answer in a subtraction problem

### Guided Practice

Here is one for you to try on your own.

Zebras are interesting animals. There are two types of zebras categorized by their scientific names. We can nickname the two types as Grevy’s and Burchell’s. A Grevy’s Zebra can weigh between 770 and 990 pounds. What is the difference between the smallest Grevy’s zebra and the largest Grevy’s zebra?

**Answer**

Anytime you see the word "difference" in a problem, you should know that you will need to use subtraction to solve the problem. In this practice problem, you need to figure out the difference between the two weights for the zebras.

Let's set it up.

\begin{align*}990 - 770 = 220\end{align*}

**There is a 220 pound difference between the two types of zebras.**

### Video Review

Here's a video to help you review this concept.

Khan Academy: Word Problem Solving Plan 1

### Practice

Directions: Use what you have learned about the four-part problem-solving plan to answer each question.

1. Jana is working in the ticket booth at the Elephant ride. She earns \begin{align*}\$8.00\end{align*} per hour. If she works for 7 hours, how much will she make in one day?

2. If Jana makes this amount of money for one day, how much will she make after five days of work?

3. If Jana works five days per week for 4 weeks, how much money will she make?

4. If Jana keeps up this schedule for the ten weeks of summer vacation, how much money will she have at the end of the summer?

5. Jana has decided to purchase a bicycle with her summer earnings. She picks out a great mountain bicycle that costs \begin{align*}\$256.99\end{align*}. How much money does she have left after purchasing the bicycle?

6. Zoey goes with Tyler to see the orangutans. She is really interested in how much an orangutan eats in one day. Zoey asks the zookeeper for this information. The zookeeper says that each orangutan will eat about 12 kg of fruit and vegetables every time it eats. They also eat every 6-8 hours. If an orangutan eats every 6-8 hours, how many times does one eat in a 24 hour period?

7. If an orangutan eats 12 kg every time it eats, and it eats three times per day, how many kilograms of food is consumed each day?

8. If the orangutan eats 4 times per day, how many kilograms of food is consumed?

9. If there are 12 orangutans in the habitat at the zoo, how many kilograms of food is consumed per feeding?

10. Given this number, if all 12 eat three times per day, how many kilograms are consumed in one day?

11. If all 12 eat four times per day, how many kilograms are consumed in one day?

12. A giraffe can step 15 feet in one step. If a giraffe takes 9 steps, how many feet of ground did the giraffe cover?

13. If a giraffe’s tongue is 27 inches long, and a tree is 3 feet away from where he is standing, can the giraffe reach the tree with its tongue?

14. How many inches closer does the giraffe need to move to be able to reach the tree?

15. A male giraffe can eat up to 100 pounds of food in a day. If a female giraffe eats about half of what a male eats, how many pounds does the female consume in one day?

16. If a male giraffe were to eat 98 pounds of food in one day, how many pounds would be consumed in one week?

17. How much food would be consumed in one month?

18. If a giraffe travels 15 feet with one step, how many steps would it take the giraffe to cover 120 feet?

19. How many steps would it take for a giraffe to walk the length of a football field, which is 360 feet?

20. If a lion can sleep 20 hours in one day, how many hours is a lion asleep over a period of three days?

Difference

The result of a subtraction operation is called a difference.Model

A model is a mathematical expression or function used to describe a physical item or situation.Product

The product is the result after two amounts have been multiplied.Proportion

A proportion is an equation that shows two equivalent ratios.Quotient

The quotient is the result after two amounts have been divided.Sum

The sum is the result after two or more amounts have been added together.Volume

Volume is the amount of space inside the bounds of a three-dimensional object.Word Problem

A word problem is a problem that uses verbal language to explain a mathematical situation.### Image Attributions

Here you'll learn how to use a problem solving plan.

## Concept Nodes:

Difference

The result of a subtraction operation is called a difference.Model

A model is a mathematical expression or function used to describe a physical item or situation.Product

The product is the result after two amounts have been multiplied.Proportion

A proportion is an equation that shows two equivalent ratios.Quotient

The quotient is the result after two amounts have been divided.Sum

The sum is the result after two or more amounts have been added together.Volume

Volume is the amount of space inside the bounds of a three-dimensional object.Word Problem

A word problem is a problem that uses verbal language to explain a mathematical situation.