# Problem-Solving Models

## Use the Problem Solving plan to solve story problems

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Problem-Solving Models

Mario has a new baby sister. Mario's mom told him that his sister will gain weight pretty quickly at the beginning. When Mario's sister was one week old she weighed 8 pounds. A week later she weighed 8.5 pounds. A week later she weighed 9 pounds. Mario is impressed by how fast his sister is gaining weight and wonders when she will be 11 pounds if she keeps gaining weight in the same pattern. After how many more weeks will Mario's sister weigh 11 pounds?

In this concept, you will learn how to solve real world problems by making and implementing a plan.

### Problem-Solving Models

Solving real world problems can be challenging when you don't know how to start to solve them. A great strategy for starting the solving process is to ask yourself three key questions.

Key Questions:

1. What am I trying to find out?
2. What do I know?
3. How can I solve the problem?

Answering these questions helps you to slow down, understand the problem, and make a plan. You might have to read the problem a few times in order to answer the three questions and fully understand the problem.

Once you have a plan, your last two steps will be to implement your plan and then check to make sure your answer makes sense and is realistic.

Here is an example.

On Monday, Jake spent a total of 180 minutes on his math, history, and science homework. He spent 45 minutes on his math homework and 1 hour on his science homework. How many minutes did he spend on his history homework?

First, ask yourself “what am I trying to find out?”

The last sentence of the problem tells you that you are trying to find out how many minutes Jake spent on his history homework.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• Jake spent 180 minutes total on his math/history/science homework.
• Jake spent 45 minutes on math homework.
• Jake spent 60 minutes (1 hour) on science homework.

Then, ask yourself “how can I solve the problem?”

You know that the amount of time Jake spends on each of his three subjects combined should equal the total amount of time he spent on his homework. You can write this relationship as an equation and substitute what you know.

Now, implement your plan. Your unknown value is the amount of time Jake spent on history homework so you can let x=History Time\begin{align*}x = \text{History Time}\end{align*}.

Total Time180==Math Time+History Time+Science Time45+x+60\begin{align*}\begin{array}{rcl} \text{Total Time} & = & \text{Math Time} + \text{History Time} + \text{Science Time}\\ 180 & = & 45 + x + 60 \end{array}\end{align*}

Now, simplify the right side of the equation.

180=105+x\begin{align*}180 = 105 + x\end{align*}

Next, solve the equation using mental math. “180 equals 105 plus what number?” 105 plus 75 equals 180, so x\begin{align*}x\end{align*} must be equal to 75.

x=75\begin{align*}x=75\end{align*}

The answer is that Jake spent 75 minutes on his history homework.

Once you have an answer, make sure that you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how much time Jake spent on his history homework and that's exactly what you did. 75 minutes is a realistic amount of time to spend on a homework assignment so your answer makes sense.

### Examples

#### Example 1

Earlier, you were given a problem about Mario and his new baby sister.

She is gaining weight each week. When she was one week old she weighed 8 pounds. Then one week later she weighed 8.5 pounds. Then one week later she weighed 9 pounds. Mario wonders when she will weigh 11 pounds if she keeps gaining the same amount of weight each week.

First, ask yourself “what am I trying to find out?”

You are trying to figure out in how many weeks Mario's sister will weigh 11 pounds.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• At one week Mario's sister weighed 8 pounds, then one week later 8.5 pounds, then one week later 9 pounds.
• There is a pattern to how much Mario's sister weighs each week.

Then, ask yourself “how can I solve the problem?”

You should first figure out what the pattern is and see if you can extend it. Then, you can figure out when Mario's sister will hit 11 pounds.

Mario's sister's weights each week so far are

8, 8.5, 9, ...

The pattern is that she is gaining 0.5 pound each week. If she continues gaining at that rate her weights will look like

8, 8.5, 9, 9.5, 10, 10.5, 11, ...

If Mario's sister currently weighs 9 pounds, that means in 4 more weeks she will weigh 11 pounds.

The answer is that Mario's sister will weigh 11 pounds in 4 weeks.

Remember that once you have an answer, you want to check that you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out in how many weeks Mario's sister will weigh 11 pounds and that's what you did. 4 weeks is a realistic amount of time for Mario's sister to gain 2 pounds so your answer makes sense.

#### Example 2

Solve the following real world problem.

A sandwich shop sold 36 tuna fish sandwiches and 45 roast beef sandwiches. The shop sold three times as many turkey sandwiches as tuna sandwiches. How many turkey sandwiches did the shop sell?

First, ask yourself “what am I trying to find out?”

The last sentence of the problem tells you that you are trying to find out how many turkey sandwiches the shop sold.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• The shop sold 36 tuna fish sandwiches.
• The shop sold 45 roast beef sandwiches.
• The shop sold 3 times as many turkey sandwiches as tuna sandwiches.

Then, ask yourself “how can I solve the problem?”

You can use the fact that the shop sold 3 times as many turkey sandwiches as tuna sandwiches to determine the number of turkey sandwiches. Take the number of tuna sandwiches and multiply it by 3.

Number of Turkey SandwichesNumber of Turkey Sandwiches==36×3108\begin{align*}\begin{array}{rcl} \text{Number of Turkey Sandwiches} & = & 36 \times 3\\ \text{Number of Turkey Sandwiches} & = & 108 \end{array}\end{align*}

The answer is that the shop sold 108 turkey sandwiches.

Notice that this problem contained extra information. You didn't need to know the number of roast beef sandwiches. Sometimes extra information is added to distract you.

Now that you have your answer, you want to make sure that you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how many turkey sandwiches the shop sold and that's what you did. 108 turkey sandwiches seems like a realistic number of sandwiches for a sandwich shop to sell, so your answer makes sense.

#### Example 3

When approaching a problem, what is the first thing you should do?

The answer is that the first thing you always have to do is read the problem carefully. Then, you should ask yourself “what am I trying to figure out?” as you start to understand the problem.

#### Example 4

Solve the following real world problem.

William began a fitness schedule. He ran 2 miles the first week, 2.5 miles the second week, and 3 miles the third week. If the pattern continues, how many miles will William have run after 5 weeks?

First, ask yourself “what am I trying to find out?”

The last sentence of the problem tells you that you are trying to find out how many miles William will have run total after 5 weeks.

Next, ask yourself “what do I know?”

You know the following pieces of information:

• William ran 2 miles the first week.
• William ran 2.5 miles the second week.
• William ran 3 miles the third week.
• There is a pattern to how many miles he runs each week.

Then, ask yourself “how can I solve the problem?”

You should first figure out what the pattern is and see if you can extend it. Then, you can determine how many miles William ran in weeks 4 and 5. Finally, you can add up how many miles he ran in each of the five weeks to find the total.

The number of miles he ran in the first three weeks is

2, 2.5, 3,...

The pattern is that he is running 0.5 mile more each week. If he continued the pattern it would look like

2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, ...

This means in the fourth week William ran 3.5 miles and in the fifth week William ran 4 miles.

Now, you can figure out how many miles William ran total in 5 weeks by adding up the number of miles from each of the 5 weeks.

2+2.5+3+3.5+4=15 miles\begin{align*}2 + 2.5 + 3 + 3.5 + 4 = 15 \ miles\end{align*}

The answer is that William ran 15 miles in 5 weeks.

Remember that once you have an answer, you want to make sure that you have actually answered the original question that was asked and that your answer seems realistic. You were trying to find out how many miles William ran in 5 weeks and that's what you did. 15 miles is a realistic number of miles to run in 5 weeks so your answer makes sense.

#### Example 5

True or false: Once you have an answer your work is finished.

The answer is this statement is false. It's always important to check your answer before moving on. You want to make sure you have answered the question that was asked and that your answer is realistic and makes sense.

### Review

Solve each real world problem by first asking yourself the key questions and making a plan. Be sure to check your answers.

1. Giovanni is working to improve his scores on his weekly science quizzes. The following were his scores for the first four weeks of school: 64, 70, 76, 82. If the pattern continues, in which week will Giovanni’s score be 100?
2. Lola’s Bakery uses 62 pounds of flour and 19 pounds of sugar each week. The bakery uses half as much butter as flour each week. How much butter will the bakery use in a month’s time?
3. A children’s pool holds 6 cubic meters of water. The length of the pool is three times the height and the width of the pool is twice the height. What is the height of the pool?
4. Martin went to the state fair with $30. He rode on 17 rides and came home with$4.50. How much did each ride cost?
5. Lionel arranged 24 photos in an album. The number of photos in each row is two more than the number of rows. How many rows of photos are there?
6. An aquarium has four fish tanks it wants to arrange on a shelf. The shelf has an area of 196 square feet. The area of second tank is twice the area of the first tank, and the area of the third tank is four times the area of the second tank. The area of the third tank is 56 square feet. If the aquarium puts all three tanks on the shelf, how much shelf area will be left over?
7. The Durands are driving 456 miles to the family reunion. If they split the drive equally over three days, how many miles will they drive each day?
8. At the end of a video game tournament, Raul and Martha had both scored twice as many points as Justice. If their total combined points is 285, how many points did each Justice score?
9. At the lunch counter, Ariana bought a sandwich and lemonade. The sandwich cost five times as much as the lemonade. She paid with $10 and got$2.50 in change. How much did the sandwich cost? How much did the lemonade cost?
10. The area of a playground is 40.5 m2\begin{align*}40.5 \ m^2\end{align*}. The width of the playground is 4.5 meters. What is the length of the playground?

Write five of your own problems. Be sure that one uses addition, one uses subtraction, one uses multiplication, one uses division and one uses a pattern that requires a table.

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Model A model is a mathematical expression or function used to describe a physical item or situation.
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.
Quotient The quotient is the result after two amounts have been divided.
Volume Volume is the amount of space inside the bounds of a three-dimensional object.