1.16: Problem-Solving Models
What if you were given a word problem like "A taxi cab charges $3 plus $0.75 per quarter mile. If you take a 3-mile cab ride, how much do you owe?" How could you devise a plan to solve this problem? After completing this Concept, you'll be able to compare alternative approaches to solving problems like this one.
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CK-12 Foundation: 0116S Problem Solving Plan
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The problem-solving plan used here is based on the ideas of George P\begin{align*}\acute{\text{o}}\end{align*}lya, who describes his useful problem-solving strategies in more detail in the book How to Solve It. Some of the techniques in the book can also be found on Wikipedia, in the entry http://en.wikipedia.org/wiki/How_to_Solve_It.
Guidance
We always think of mathematics as the subject in school where we solve lots of problems. Problem solving is necessary in all aspects of life. Buying a house, renting a car, or figuring out which is the better sale are just a few examples of situations where people use problem-solving techniques. In this book, you will learn different strategies and approaches to solving problems. In this section, we will introduce a problem-solving plan that will be useful throughout this book.
Read and Understand a Given Problem Situation
The first step to solving a word problem is to read and understand the problem. Here are a few questions that you should be asking yourself:
- What am I trying to find out?
- What information have I been given?
- Have I ever solved a similar problem?
This is also a good time to define any variables. When you identify your knowns and unknowns, it is often useful to assign them a letter to make notation and calculations easier.
Make a Plan to Solve the Problem
The next step in the problem-solving plan is to develop a strategy. How can the information you know assist you in figuring out the unknowns?
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, looking for patterns is a good strategy for most problems, and 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 in your study of algebra.
Solve the Problem and Check the Results
Once you develop a plan, you can implement it and solve the problem, carrying out all operations to arrive at the answer you are seeking.
The last step in solving any problem should always be to check and interpret the answer. Ask yourself:
- Does the answer make sense?
- If you plug the answer back into the problem, do all the numbers work out?
- Can you get the same answer through another method?
Compare Alternative Approaches to Solving the Problem
Sometimes one specific method is best for solving a problem. Most problems, however, 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, so we can demonstrate the strengths and weakness of different strategies for solving different types of problems.
Whichever 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
Read the problem carefully. Once the problem is read, list all the components and data that are involved. This is where you will be assigning your variables.
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 solve your problem solving plan.
Step 3:
Carry out the plan - Solve
This is where you solve the equation you developed in Step 2.
Step 4:
Look - Check and Interpret
Check to see if you used all your information. Then look to see if the answer makes sense.
The most difficult parts of problem-solving are most often the first two steps in our problem-solving plan. You need to read the problem and make sure you understand what you are being asked. Once you understand the problem, you can devise a strategy to solve it.
Let’s apply the first two steps to the following problem.
Example A
Six friends are buying pizza together and they are planning to split the check equally. After the pizza was ordered, one of the friends had to leave suddenly, before the pizza arrived. Everyone left had to pay $1 extra as a result. How much was the total bill?
Solution
Understand
We want to find how much the pizza cost.
We know that five people had to pay an extra $1 each when one of the original six friends had to leave.
Strategy
We can start by making a list of possible amounts for the total bill.
We divide the amount by six and then by five. The total divided by five should equal $1 more than the total divided by six.
Look for any patterns in the numbers that might lead you to the correct answer.
In the rest of this section you will learn how to make a table or look for a pattern to figure out a solution for this type of problem. After you finish reading the rest of the section, you can finish solving this problem for homework.
Develop and Use the Strategy: Make a Table
The method “Make a Table” is helpful when solving problems involving numerical relationships. When data is organized in a table, it is easier to recognize patterns and relationships between numbers. Let’s apply this strategy to the following example.
Example B
Josie takes up jogging. On the first week she jogs for 10 minutes per day, on the second week she jogs for 12 minutes per day. Each week, she wants to increase her jogging time by 2 minutes per day. If she jogs six days each week, what will be her total jogging time on the sixth week?
Solution
Understand
We know in the first week Josie jogs 10 minutes per day for six days.
We know in the second week Josie jogs 12 minutes per day for six days.
Each week, she increases her jogging time by 2 minutes per day and she jogs 6 days per week.
We want to find her total jogging time in week six.
Strategy
A good strategy is to list the data we have been given in a table and use the information we have been given to find new information.
We are told that Josie jogs 10 minutes per day for six days in the first week and 12 minutes per day for six days in the second week. We can enter this information in a table:
Week | Minutes per Day | Minutes per Week |
---|---|---|
1 | 10 | 60 |
2 | 12 | 72 |
You are told that each week Josie increases her jogging time by 2 minutes per day and jogs 6 times per week. We can use this information to continue filling in the table until we get to week six.
Week | Minutes per Day | Minutes per Week |
---|---|---|
1 | 10 | 60 |
2 | 12 | 72 |
3 | 14 | 84 |
4 | 16 | 96 |
5 | 18 | 108 |
6 | 20 | 120 |
Apply strategy/solve
To get the answer we read the entry for week six.
Answer: In week six Josie jogs a total of 120 minutes.
Check
Josie increases her jogging time by two minutes per day. She jogs six days per week. This means that she increases her jogging time by 12 minutes per week.
Josie starts at 60 minutes per week and she increases by 12 minutes per week for five weeks.
That means the total jogging time is \begin{align*}60 + 12 \times 5 = 120 \ minutes\end{align*}.
The answer checks out.
You can see that making a table helped us organize and clarify the information we were given, and helped guide us in the next steps of the problem. We solved this problem solely by making a table; in many situations, we would combine this strategy with others to get a solution.
Develop and Use the Strategy: Look for a Pattern
Looking for a pattern is another strategy that you can use to solve problems. The goal is to look for items or numbers that are repeated or a series of events that repeat. The following problem can be solved by finding a pattern.
Example C
You arrange tennis balls in triangular shapes as shown. How many balls will there be in a triangle that has 8 rows?
Solution
Understand
We know that we arrange tennis balls in triangles as shown.
We want to know how many balls there are in a triangle that has 8 rows.
Strategy
A good strategy is to make a table and list how many balls are in triangles of different rows.
One row: It is simple to see that a triangle with one row has only one ball.
Two rows: For a triangle with two rows, we add the balls from the top row to the balls from the bottom row. It is useful to make a sketch of the separate rows in the triangle.
\begin{align*}3 = 1 + 2\end{align*}
Three rows: We add the balls from the top triangle to the balls from the bottom row.
\begin{align*}6 = 3 + 3\end{align*}
Now we can fill in the first three rows of a table.
Number of Rows | Number of Balls |
---|---|
1 | 1 |
2 | 3 |
3 | 6 |
We can see a pattern.
To create the next triangle, we add a new bottom row to the existing triangle.
The new bottom row has the same number of balls as there are rows. (For example, a triangle with 3 rows has 3 balls in the bottom row.)
To get the total number of balls for the new triangle, we add the number of balls in the old triangle to the number of balls in the new bottom row.
Apply strategy/solve:
We can complete the table by following the pattern we discovered.
Number of balls = number of balls in previous triangle + number of rows in the new triangle
Number of Rows | Number of Balls |
---|---|
1 | 1 |
2 | 3 |
3 | 6 |
4 | \begin{align*}6 + 4 = 10\end{align*} |
5 | \begin{align*}10 + 5 = 15\end{align*} |
6 | \begin{align*}15 + 6 = 21\end{align*} |
7 | \begin{align*}21 + 7 = 28\end{align*} |
8 | \begin{align*}28 + 8 = 36 \end{align*} |
Answer There are 36 balls in a triangle arrangement with 8 rows.
Check
Each row of the triangle has one more ball than the previous one. In a triangle with 8 rows,
row 1 has 1 ball, row 2 has 2 balls, row 3 has 3 balls, row 4 has 4 balls, row 5 has 5 balls, row 6 has 6 balls, row 7 has 7 balls, row 8 has 8 balls.
When we add these we get: \begin{align*}1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 \ balls\end{align*}
The answer checks out.
Notice that in this example we made tables and drew diagrams to help us organize our information and find a pattern. Using several methods together is a very common practice and is very useful in solving word problems.
Watch this video for help with the Examples above.
CK-12 Foundation: Problem Solving Plan
Vocabulary
Whichever 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
Read the problem carefully. Once the problem is read, list all the components and data that are involved. This is where you will be assigning your variables.
- 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 solve your problem solving plan.
- Step 3:
Carry out the plan - Solve
This is where you solve the equation you developed in Step 2.
- Step 4:
Look - Check and Interpret
Check to see if you used all your information. Then look to see if the answer makes sense.
Guided Practice
Casey is twice as old as Marietta, who is two years younger than Jake. If Casey is 14, how old is Jake?
Solution:
Let \begin{align*}C\end{align*} be the age of Casey, \begin{align*}M\end{align*} be the age of Marietta and \begin{align*}J\end{align*} be the age of Jake. We can write the following equations:
\begin{align*}C=2M\end{align*}
and
\begin{align*}M=J-2.\end{align*}
We can substitute the second equation into the first, getting:
\begin{align*}C=2M=2(J-2)=2J-4\end{align*}.
This gives us \begin{align*}C=2J-4.\end{align*}
What are possible ages for Jake that would make Casey's age 14? We can make a table based on the equation:
\begin{align*} J \qquad 2 \qquad 3 \qquad 4 \qquad 5 \qquad 6 \qquad 7 \qquad 8 \qquad 9\\ C \qquad 0 \qquad 2 \qquad 4 \qquad 6 \qquad 8 \qquad 10\qquad 12\qquad 14 \end{align*}
Looking at the table, when Casey's age is 14, Jake's age is 9.
To check the answer, evaluate the equation for \begin{align*}J=9\end{align*}:
\begin{align*}C&=2J-4 \qquad \text{Start with the equation.}\\ C&=2(9)-4 \qquad \text{Substitute in J=9 .}\\ C&=18-4 \qquad \text{Simplify.}\\ C&=14 \qquad \text{It's correct!}\\ \end{align*}
Practice
- 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?
- Mariana deposits $500 in a savings account that pays 3% simple interest per year. How much will be in her account after three years?
- It costs $250 to carpet a room that is 14 ft by 18 ft. How much does it cost to carpet a room that is 9 ft by 10 ft?
- 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?
- Yusef’s phone plan costs $10 a month plus $0.05 per minute. If his phone bill for last month was $25.80, how many minutes did he spend on the phone?
- 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?
- How much would she have spent in total if she had gone on five more rides?
- An ice cream shop sells a small cone for $2.95, 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 earn?
- In Lise’s chemistry class, there are two midterm exams, each worth 30% of her total grade, and a final exam worth 40%. If Lise scores 90% on both midterms and 80% on the final exam, what is her overall score in the class?
- The sum of the 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?
- A television that normally costs $120 goes on sale for 20% off. What is the new price?
- A cake recipe calls for \begin{align*}1 \frac{3}{4}\end{align*} cup of flour. Jeremy wants to make four cakes. How many cups of flour will he need?
- Kylie is mowing lawns to earn money for a new bike. After mowing four lawns, she still needs $40 more to pay for the bike. After mowing three more lawns, she has $5 more than she needs to pay for the bike.
- How much does she earn per lawn?
- What is the cost of the bike?
- Jared goes trick-or-treating with his brother and sister. At the first house they stop at, they collect three pieces of candy each; at the next three houses, they collect two pieces of candy each. Then they split up and go down different blocks, where Jared collects 12 pieces of candy and his brother and sister collect 14 each.
- How many pieces of candy does Jared end up with?
- How many pieces of candy do all three of them together end up with?
- Marco’s daughter Elena has four boxes of toy blocks, with 50 blocks in each one. One day she dumps them all out on the floor, and some of them get lost. When Marco tries to put them away again, he ends up with 45 blocks in one box, 53 in another, 46 in a third, and 51 in the fourth. How many blocks are missing?
- A certain hour-long TV show usually includes 16 minutes of commercials. If the season finale is two and a half hours long, how many minutes of commercials should it include to keep the same ratio of commercial time to show time?
- Karen and Chase bet on a baseball game: if the home team wins, Karen owes Chase fifty cents for every run scored by both teams, and Chase owes Karen the same amount if the visiting team wins. The game runs nine innings, and the home team scores one run in every odd-numbered inning, while the visiting team scores two runs in the third inning and two in the sixth. Who owes whom how much?
- Kelly, Chris, and Morgan are playing a card game. In this game, the first player to empty their hand scores points for all the cards left in the other players’ hands as follows: aces are worth one point, face cards ten points, and all other cards are face value. When Kelly empties her hand, Morgan is holding two aces, a king, and a three; Chris is holding a five, a seven, and a queen. How many points does Kelly score?
- A local club rents out a social hall to host an event. The hall rents for $350, and they hope to make back the rental price by charging $15 admission per person. How many people need to attend for the club to break even?
- You plan to host a barbecue, and you expect 10 friends, 8 neighbors, and 7relatives to show up.
- If you expect each person (including yourself) to eat about two ounces of potato salad, how many half-pound containers of potato salad should you buy?
- If hot dogs come in ten-packs that cost $4.80 apiece and hot dog buns come in eight-packs that cost $2.80 apiece, how much will you need to spend to have hot dogs and buns for everyone?
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Here you'll learn a variety of problem-solving strategies you can use to solve a problem situation. Once you've devised a plan, you'll then solve problems and check your results.