1. Chapter 1: Arithmetic & Prealgebra
Section 1.5: Problem Solving
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

2. The Problem Solving Process
In his book, How to Solve It, George Polya outlined a step-by-step process for problem solving.
Understand the Problem
Devise a Plan
Carry Out the Plan
Look Back
A fifth step should also be considered.
Answer the Question
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

3. The Problem Solving Process
Understand the Problem
Do we understand all the words?
Can we restate the problem in our own words?
Do we know what is given?
Do we know what the goal is?
Is there enough information?
Is there extraneous information?
Is the problem similar to another problem we have solved?
Draw a picture of the problem statement, if applicable.
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

4. The Problem Solving Process
Devise a Plan - Using One of the Following Strategies:
Guess and Test
Look for a Pattern
Use a Variable
Make a List
Solve a Simpler Problem
Draw a Picture
Work Backwards
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

5. The Problem Solving Process
Carry Out the Plan
Implement the strategy or strategies we have chosen until the problem is solved or a new course of action is suggested.
Give ourselves a reasonable amount of time in which to solve the problem.
Don't be afraid of starting over. Often, a fresh start and a new strategy will lead to success.
Look Back
Is our solution correct?
Does our answer satisfy the statement of the problem?
Can we see an easier solution?
Can we use this solution to solve other problems?
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

6. The Problem Solving Process
Answer the Question
In written or verbal responses, we should always answer the asked question in a complete sentence.
Answering the question in a complete sentence forces you to examine the logistics of the situation. If you solve a problem and get an answer of 13.5, you may think you are done.
If, however, the question was, “How many cats are in the barn?” forcing yourself to write, “There are 13.5 cats in the barn.” may get you to realize 13.5 may not be correct.
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

7. Strategy: Guess and Test
Clues: The Guess and Test strategy may be appropriate when:
There are a limited number of possible answers.
We want to gain a better understanding of the problem.
We have a good idea of what the answer is.
We can systematically try possible answers.
There is no other obvious strategy to try.
Example: Fill in the blanks using some combination of the symbols +, −, ×, and ÷ to make a true statement.
7 __ 7 __ (7 __ 7) = 13
Possible answer: − 7 ÷7 =13
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

8. Strategy: Look for a Pattern
Clues: The Look for a Pattern strategy may be appropriate when:
A list of data is given.
Listing special cases helps you deal with complex problems.
We are asked to make a prediction or generalization.
Example: Determine the pattern and then generate the next two iterations for the following.
37 ✕ 3 = ✕ 6 = ✕ 9 = ✕ 12 = 444
Answer: The pattern is 37 ✕ 3n = nnn ✕ 15 = ✕ 18 = 666
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

9. Strategy: Use a Variable (Algebra)
Clues: The Use a Variable strategy may be appropriate when:
A phrase similar to "for any number" is present or implied.
A problem suggests an equation.
A proof or a general solution is required.
There is a large number of cases.
A proof is asked in a problem involving numbers.
We are trying to develop a general formula.
Example: A backyard fair charged $ admission for adults and $0.50 for children. The fair made $25 and sold 38 tickets. How many adult tickets were sold?
Answer: If x represents the number of adult tickets, then 38 – x is the number of children tickets. Then,
$1(x) + $0.50(38 – x) = $25 $1x + $19 - $0.50x = $25 $0.50x + $19 = $25 $0.50x = $6 x = $6/$0.50 = 12
12 adult tickets were sold.
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

10. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates
Strategy: Make a List
Clues: The Make a List strategy may be appropriate when:
The information can easily be organized and presented.
Data can be easily generated.
Asked "in how many ways" something can be done. 
Example: A person has 10 coins consisting of dimes and quarters. If the person has a total of $1.90, find the number of quarters.
Answer: Making a list…
10 quarters and no dimes is $ quarters and 1 dime is $ quarters and 2 dimes is $ quarters and 3 dimes is $ quarters and 4 dimes is $ …
The person has 6 quarters.
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

11. Strategy: Solve a Simpler Problem
Clues: The Solve a Simpler Problem strategy may be appropriate when:
The problem involves complicated computations.
The problem involves very large or small numbers.
We are asked to find the sum of a series of numbers.
A direct solution is too complex.
Example: A fruit stand is selling 8 apples for $1.25. At this rate, how much will 24 apples cost?
Answer: Think of them in bags of 8 apples. Three bags will give you 24 apples and will cost $3.75.
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

12. Strategy: Draw a Picture
Clues: The Draw a Picture strategy may be appropriate when:
A physical situation is involved.
Geometric figures or measurements are involved.
A visual representation of the problem is possible.
Example: A farmer is building a pen for his hogs. He will use the side of his barn, which is 30 feet long, as one side of the pen, and wants the width to be 12 feet. How many feet of fencing is needed?
Answer: Drawing the picture…
54 ft of fencing is needed.
BARN
12 ft
12 ft
30 ft
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

13. Strategy: Work Backwards
Clues: The Work Backwards strategy may be appropriate when:
We know the answer and need to find the process.
Example: Cindy was given her allowance on Monday. On Tuesday she spent $1.50 on candy. On Wednesday, Cindy found $1 on the ground. If Cindy now has $2, how much was her allowance?
Answer: Working backwards, since she now has $2, that means she had $1 before the money she found. Then, since she had $1 left after spending $1.50, she started with $2.50.
Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates