# 2.5.2: Problem-Solving Models

- Page ID
- 4367

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## Problem-Solving Models

Rima loves orangutans. She has learned about an organization that uses donations to care for and protect orangutans in rainforest sanctuaries around the world. Rima has decided to “adopt” an orangutan through this program. The program has three adoption levels: Bronze ($35 - $99), Silver ($100 - $499), and Gold ($500 - $1000). Rima earns money doing odd jobs for her family and neighbors, and she has calculated that with all her scheduled jobs for the first eight weeks of summer, she will earn $125 per week. How can Rima figure out how much money she will earn over the summer and what adoption level she will be able to afford?

In this concept, you will learn how to use a problem solving plan.

### Problem-Solving Plan

It is helpful to have a problem-solving plan to help you understand and solve problems. There are four parts to a problem solving plan.

**Problem-Solving Plan**

**Read and understand the problem****Make a plan to solve the problem****Solve the problem and check results****Compare alternative approaches to solving the problem**

**Read and understand the problem**

When looking at a problem, read it and underline all of the important information. Sometimes you will be given information that is not necessary to solving the problem.

**Make a plan to solve the problem**

Once you know what the problem is asking you to do, you will need to figure out the best method to do that in order to solve the problem.

**Solve the problem and check results**

After you have solved the problem, it is important to check your answer either against the given answer, or by using a different method to solve the problem.

**Compare alternative approaches to solving the problem**

Think about whether your approach was the best method in solving the problem, or if another method might have been easier, or gotten you to the answer in fewer steps.

**Examples**

Example 2.5.2.1

Earlier, you were given a problem about Rima and her furry adoption.

Rima needs to figure out how much money she will earn over the first eight weeks of summer if she earns $125 per week. She also needs to know which adoption level she will be able to afford: Bronze ($35-$99), Silver ($100-$499), and Gold ($500 -$1000).

**Solution**

First, identify the important information in the problem.

Rima earns $125 per week

Rima will be working for 8 weeks

Bronze adoption is $35-$99

Silver adoption is $100-$499

Gold adoption is $500-$1000

Next, make a plan.

You will need to write an expression to calculate the total **sum** of money Rima will earn, and you will need to compare this to the adoption level requirements.

Then, solve the problem.

Write an expression to calculate Rima’s earnings, then evaluate it.

125 x 8 = x

1,000=x

The answer is 1,000.

Compare this to the adoption level requirements.

$1,000 > $35-99

$1,000 > $100-499

$1,000 > $500 but = $1,000

The answer is Gold adoption

Rima will earn $1,000 over the summer and will be able to afford a Gold adoption of an orangutan.

Finally, consider alternative approaches to solving the problem.

Instead of writing an expression to calculate Rima’s earnings, you could have used repeated addition, but this would have been much slower - multiplication is a shortcut for repeated addition.

Example 2.5.2.2

A zebra can weigh between 770 and 990 pounds. What is the **difference** between the smallest possible weight and the largest possible weight for a zebra?

**Solution**

When you see the word "difference" in a problem, you know that you will need to use subtraction to solve the problem. In this case, you need to figure out the difference between the two weights.

Write this as an expression.

990−770=220

The answer is 220. So the difference between the two possible weights is 220 pounds

Example 2.5.2.3

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

**Solution**

First, read the problem to identify any action words and underline important information. In this problem, notice that you are given the weights 330 pounds and 500 pounds, and that the question indicates that you are looking for the * difference* between them.

Next, write a math expression for the problem you identified. Remember that a difference is the result of subtraction.

500−330

Finally, simplify the expression to learn the difference between the weights of the lions.

500−330=170

The difference between the weights of the lions is 170 pounds.

Example 2.5.2.4

If a large lion weighs in at 500 pounds, and there are four large lions in the habitat, how much do the lions weigh in all?

**Solution**

First, read the problem to identify any action words and important information. Here you should note the given weight of 500 lbs per lion, the number of lions, 4, and the action words "in all," which indicate addition or multiplication.

Next, write an expression for the problem. Since "in all" refers to four of the same weights, use multiplication instead of addition. Remember that multiplication is the shortcut for repeated addition.

500×4

Finally, simplify the expression to determine the total weight of the lions.

500×4=2000

The total weight of the four large lions is 2000 pounds.

Example 2.5.2.5

If a small lion weighs 330 pounds and there are five small lions in the habitat, what is the total weight of the small lions?

**Solution**

First, read the problem to identify any action words and important information. Here you should see that each small lion weighs 330 pounds, there are 5 small lions, and the action word "total" means you will need to add or multiply again.

Next, write an expression for the problem. Since "total" refers to five of the same weights, use multiplication instead of addition.

330×5Finally, simplify the expression to determine the total weight of the lions.

330×5=1650The total weight of the five small lions is 1650 pounds.

### Review

Use the four-part problem-solving plan to answer each question.

- Jana is working in the ticket booth at the Elephant ride. She earns $8.00 per hour. If she works for 7 hours, how much money will she earn in one day?
- If Jana makes this amount of money for one day, how much will she make after five days of work?
- If Jana works five days per week for 4 weeks, how much money will she make?
- 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?
- Jana has decided to purchase a bicycle with her summer earnings. She picks out a great mountain bicycle that costs $256.99. How much money does she have left after purchasing the bicycle?
- An 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?
- If an orangutan eats 12 kg every time it eats, and it eats three times per day, how many kilograms of food are consumed each day?
- If the orangutan eats 4 times per day, how many kilograms of food are consumed?
- If there are 12 orangutans in the habitat at the zoo, how many kilograms of food are consumed per feeding?
- Given this number, if all 12 orangutans eat three times per day, how many kilograms are consumed in one day?
- If all 12 eat four times per day, how many kilograms are consumed in one day?
- A giraffe can step 15 feet in one step. If a giraffe takes 9 steps, how many feet of ground did the giraffe cover?
- 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?
- How many inches closer does the giraffe need to move to be able to reach the tree?
- 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?
- If a male giraffe was to eat 98 pounds of food in one day, how many pounds would be consumed in one week?
- How much food would be consumed in one month?
- If a giraffe travels 15 feet with one step, how many steps would it take the giraffe to cover 120 feet?
- How many steps would it take for a giraffe to walk the length of a football field, which is 360 feet?
- If a lion can sleep 20 hours in one day, how many hours is a lion asleep over a period of three days?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.16.

### Vocabulary

Term | Definition |
---|---|

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. |

### Additional Resources

Video:

PLIX: Play, Learn, Interact, eXplore: **Problem-Solving Models: Subway vs. Taxi**

Activity: **Problem-Solving Models Discussion Questions**

Practice: **Problem-Solving Models**

Real World Application: **To the Moon!**