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# Problem-Solving Models

## Use the Problem Solving plan to solve story problems

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Practice Problem-Solving Models

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Understand the Problem-Solving Plan

Julien is designing flower gardens with his many nieces and nephews. He has to construct rectangular gardens where all the kids can plant their flower seeds. Each of the gardens is 6 feet by 12 feet. Each garden will have room for four kids to plant. How can Julien figure out how much space each of his nieces and nephews will have in the garden beds?

In this concept, you will learn to understand and use a problem-solving plan.

### Problem Solving Plan

When making a plan to solve a problem, you may choose one of several strategies, including:

• Act it Out
• Make a Model
• Look for a Pattern
• Guess and Check
• Make a Table
• Work Backwards
• Write an Equation
• Write a Proportion

When you read a problem, it is helpful to underline or highlight any important information. Important information can include words that identify an operation. You can also identify key words such as distance, time, speed, area or perimeter. All of these parts help you to identify a problem and what the problem is asking you to solve.

Let’s look at a situation.

Mollie is planning to meet a group of friends at the movies on Saturday night at 6:00 p.m. Mollie is incharge of driving a few friends to the movies. Mollie lives 15 minutes from Sara’s house. It takes 10 minutes to get from Sara’s house to Madison’s house. It is another 20 minutes to the movie theatre from Madison’s house. It takes Mollie one hour to get dressed and ready for the evening. At what time should Mollie begin to get ready for the evening?

First, what is the problem asking you to find out?

The problem is asking you to determine the time at which Mollie should begin getting ready for the evening if she is to beat the movies at 6:00 p.m. You must take into consideration the amount of travel time to each of her friends houses and the amount of time it takes for Mollie to get dressed.

This problem involves time. You need to backup the time that it takes for Molly to get to the movie theater.

Next, make a plan.

Mollie needs to beat the movies at 6:00 p.m. Work backward to determine the time Mollie should begin getting ready. If you work backwards, you will be able to figure out the time that she needs to get there.

Then, solve the problem.

Time of Movie – Time to Dress – Time to Movies – Time to Madison’s – Time to Sara’s

6:00 – 1:00 – 0:20 – 0:10 – 0:15

5:00 – 0:20 – 0:10 – 0:15

4:40 – 0:10 – 0:15

4:30 – 0.15

4:15

Therefore, with all that she has todo, Mollie should begin getting ready at 4:15 p.m.

Finally, check the results.

If Mollie begins getting dressed at 4:15 p.m. and takes 1 hour, she will be ready to leave at 5:15 p.m.

Since it takes Mollie 15 minutes to get to Sara’s house, she’ll arrive at 5:30 p.m.

10 minutes later at 5:40 Mollie will arrive at Madison’s house.

Since it takes 20 minutes to drive to the movies, Mollie will arrive at the movies promptly at 6:00 p.m.

Therefore, Mollie needs to begin getting ready at 4:15 p.m.

### Examples

#### Example 1

Earlier, you were given a problem about Julien and the kids’ garden space.

First, find the total area of one of the rectangular gardens. The formula for area is \begin{align*}A = l \times w\end{align*}.

You know that the length of each garden is 12 feet and the width of each garden is 6 feet. Substitute these values into the formula and solve for the area of the garden.

So the area of each garden is \begin{align*}72 \ ft^2\end{align*}.

Next, each garden needs to be divided into four equal spaces so that each child can have the same space.

\begin{align*}\frac{72}{4} = 18\end{align*}

Therefore, each child will have 18 square feet of garden to plant their flowers.

#### Example 2

The bowling alley charges $12 for one person to play,$10 for the second person to play, $8 for the third to person to play, and so on. What is the total cost for a family of five to bowl? How much money does the family save bowling together rather than separately? First, consider what the problem is asking. The problem is asking you to analyze the pattern to determine the cost for a family of five to bowl. After determining the cost of a family of five to bowl, determine the difference between the amount the family is spending and the amount it would cost for five individuals to play. Next, make a plan. To better observe the pattern, organize the information on a table. You know that every time you add a family member, the cost goes down by$2.

 Group size Cost to play 1st person $12 2nd person$10 3rd person $8 Then, solve the problem. Continue the pattern for the fourth and fifth family members. Add the cost to play for each family member to determine the total cost.  Group size Cost to play 1st person$12 2nd person $10 3rd person$8 4th person $6 5th person$4 Total $40 Then, solve for the amount you save by playing as a family. To determine the amount saved playing as a family, subtract the total cost for a family of five from the total cost for five individuals. To play as an individual it costs$12. Therefore the cost for five individuals is \begin{align*}\12 \times 5\end{align*} or $60. To play as a family, it costs$40.

\begin{align*}\begin{array}{rcl} \text{Cost to Play Individually} - \text{Cost to Play as a Family} & = & \text{Amount Saved}\\ \60 - \40 & = & \20 \end{array}\end{align*}

The answer is $20. Therefore the cost for a family of five to bowl is$40. A family of five saves 20 playing together rather than individually. Use this situation to answer the following example questions. For every day that Jesse harvested vegetables in the garden, he collected 4 pounds of vegetables. #### Example 3 If Jesse continued this for 45 days, how many total pounds of vegetables will he have collected? First, what is the problem asking you to find out? The problem is asking you to determine the number of pounds Jesse collects in 45 days. Next, make a plan. Jesse can collect four pounds of vegetables a day. Set up an equation to solve for the number of pounds he can collect in 45 days. \begin{align*}\# \ pounds = \frac{4 \ pounds}{day} \times 45 \ days\end{align*} Then, solve the problem. \begin{align*}\begin{array}{rcl} \# \ pounds & = & \frac{4 \ pounds}{day} \times 45 \ days\\ \# \ pounds & = & 180 \end{array}\end{align*} The answer is 180. Therefore, Jesse can collect 180 pounds of vegetables. #### Example 4 If Jesse collects 20 pounds a week, how many days does he work a week? First, what is the problem asking you to find out? The problem is asking you to determine the number of days Jesse works a week. Next, make a plan. Use the equation from the previous example to solve for the number of days Jesse works in a week. \begin{align*}\# \ 20 \ pounds = \frac{4 \ pounds}{day} \times x \ days\end{align*} Then, solve the problem. \begin{align*}\begin{array}{rcl} \# \ 20 \ pounds & = & \frac{4 \ pounds}{day} \times x \ days\\ x \ days & = & 20 \ pounds \times \frac{1 \ day}{4 \ pounds}\\ x \ days & = & 5 \ days \end{array}\end{align*} The answer is 15. Therefore, Jesse works 5 days a week. #### Example 5 If Jesse collected vegetables for 90 days, how many pounds would he gather at this same rate? First, what is the problem asking you to find out? The problem is asking you to determine the number of pounds Jesse collects in 90 days. Next, make a plan. Jesse can collect four pounds of vegetables a day. Set up an equation to solve for the number of pounds he can collect in 90 days. \begin{align*}\# \ pounds = \frac{4 \ pounds}{day} \times 90 \ days\end{align*} Then, solve the problem. \begin{align*}\begin{array}{rcl} \# \ pounds & = & \frac{4 \ pounds}{day} \times 90 \ days\\ \# \ pounds & = & 360 \end{array}\end{align*} The answer is 360. Therefore, Jesse can collect 360 pounds of vegetables. ### Review Read each problem and then answer the questions following each problem. At the end of a phone call home, Brad had0.85. The initial cost of the phone call was $0.75 plus$0.12 per minute. If Brad spoke on the phone for 20 minutes, how much money did he have before making the phone call home?

1. Should you use working backwards or writing a proportion for this problem?

2. Why wouldn’t you use a proportion for this problem?

3. You will need to multiply one part of this problem, which part?

4. What equation could you write to solve this problem?

5. How much money did Brad have before making the phone call?

Suppose you have $75 in your savings account. You plan to save an additional$25 per week. After how many weeks will you have saved $500? 6. Which strategy makes the most sense look for a pattern or working backwards? 7. What is the unknown quantity that you are trying to figure out? 8. What equation can you use to solve this problem? 9. How many weeks will it take to save$500.00?

An online music store charges $1.90 to download 2 songs. Determine the cost of downloading 13 songs. 10. Which strategy would you use look for a pattern or use a proportion? 11. Why would you use that method? 12. What is the cost for the 13 songs? 13. What would be the cost for double the songs? 14. If the cost had been$2.25 per song, how much would 2 songs cost?

15. What would be the cost for 4 songs?

16. If six friends each downloaded four songs, how much would the total cost be?

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

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Color Highlighted Text Notes

### Vocabulary Language: English

Area

Area is the space within the perimeter of a two-dimensional figure.

Model

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

Perimeter

Perimeter is the distance around a two-dimensional figure.

Problem Solving

Problem solving is using key words and operations to solve mathematical dilemmas written in verbal language.

Proportion

A proportion is an equation that shows two equivalent ratios.

Volume

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