The document provides examples and explanations of how to use Polya's four-step problem solving procedure. It begins by introducing Polya and his general four-step approach of understanding the problem, devising a plan, carrying out the plan, and checking the results. It then gives detailed guidelines for each step. Several math word problems are presented and solved using this method to illustrate the process.

1. Lesson 3:
Problem Solving

2. Polya’s Procedure
George Polya (1887-1985), a mathematician
who was educated in Europe and taught at
Stanford, developed a general procedure for
solving problems.
1.3-2

3. Guidelines for Problem Solving
1. Understand the Problem.
2. Devise a Plan to Solve the Problem.
3. Carry Out the Plan.
4. Check the Results.
1.3-3

4. 1. Understand the Problem.
• Read the problem carefully at least twice.
• Try to make a sketch of the problem. Label the information
given.
• Make a list of the given facts that are pertinent to the
problem.
• Determine if you have sufficient information to solve the
problem.
1.3-4

5. 2. Devise a Plan to Solve the Problem.
• Can you relate this problem to a previous problem that
you’ve worked before?
• Can you express the problem in terms of an algebraic
equation?
• Look for patterns or relationships.
• Can you express the problem more simply?
• Can you substitute smaller or simpler numbers to make the
problem more understandable?
• Will listing the information in a table help in solving the
problem?
• Can you make an educated guess at the solution? You can
work backward to determine the correct procedure.
1.3-5

6. 3. Carrying Out the Plan.
• Use the plan you devised in step 2 to solve the problem.
1.3-6

7. 4. Check the Results.
• Ask yourself, “Does the answer make sense?” and “Is it reasonable?”
 If the answer is not reasonable, recheck your method for
solving the problem and your calculations.
• Can you check the solution using the original statement?
• Is there an alternative method to arrive at the same conclusion?
• Can the results of this problem be used to solve other problems?
1.3-7

8. Example 2: Shuttle Revenue
Airport Express shuttle service provides service
from San Antonio International Airport to
downtown hotels, approximately 10 miles away.
One particular shuttle makes 16 round trips per
day, carrying 5 passengers per trip. The fare each
way is $18. What are the receipts from one day’s
operation for this particular shuttle?
1.3-8

9. Example 2: Shuttle Revenue
Solution:
List all the information:
• Distance from airport to downtown hotels ≈ 10 miles
• * Number of round trips per day = 16
• * Number of passengers per trip = 5
• * Fare each way = $18
We need only the information with a *.
1.3-9

10. Example 2: Shuttle Revenue
Solution:
Round trip costs 2 x $18 = $36
The total receipts for one day is the product of
the number of round trips, the number of
passengers, and the cost per round trip
= 16 × 5 × $36 = $2880
1.3-
10

11. Example 3: Retirement
It is never too early to start planning for
retirement. U.S. Census Bureau data indicate
that at age 65 the average woman will live
another 20 years and the average man will live
another 17.2 years. The data also indicate that
about 33% of the average person’s retirement
income will come from Social Security.
1.3-
11

12. Example 3: Retirement
When discussing retirement planning, many
investment firms and financial planners use the
graph in the figure on the next slide which shows
how long a typical retiree’s assets (or “nest egg”)
will last based on the percentage of the assets
withdrawn each year.
1.3-
12

13. Example 3: Retirement
1.3-
13

14. Example 3: Retirement
a) If a typical retiree has retirement assets of
$500,000, how much can he or she withdraw
annually if he or she wishes the assets to last
21 years?
1.3-
14

15. Example 3: Retirement
Solution
a) From the graph, we can see that for assets to last
21 years, about 6% of the assets can be withdrawn
annually. The amount that can be withdrawn is:
1.3-
15
Amount = 6% of assets
Amount = 0.06(500,000) = $30,000
Thus, about $30,000 can be withdrawn
annually.

16. Example 3: Retirement
b) How much should a retiree have in assets if he
or she wishes to withdraw $25,000 annually
and wishes his or her assets to last 18 years?
1.3-
16

17. Example 3: Retirement
Solution
b) From the graph, we can see that for assets to last 18 years,
about 7% of the assets can be withdrawn annually. That
amount has to equal $25,000:
1.3-
17
7% of assets = $25,000
0.07  assets = $25,000
assets = 25,000 ÷ 0.07
= 357,142.86
The retiree needs $357,142.86

18. Example 4: Determining a Tip
The cost of Freddie Rojo’s meal before tax is
$28.00.
a) If a 6 ½ % sales tax is added to his bill,
determine the total cost of the meal including
tax.
1.3-
18

19. Example 4: Determining a Tip
Solution:
Change 6 ½ to a decimal: 0.065
Sales tax = 6 ½ % of meal
Sales tax = 0.065(28.00) = 1.82
The total bill = cost of meal + sales tax
Total bill = 28.00 + 1.82 = $29.82
The bill including sales tax is $29.82
1.3-
19

20. Example 4: Determining a Tip
The cost of Freddie Rojo’s meal before tax is
$28.00.
b) If Freddie wants to leave a 10% tip on the
pretax cost of the meal, how much should he
leave?
1.3-
20

21. Example 4: Determining a Tip
Solution:
To find 10% of any number, we multiply the number by 0.10
10% of pretax cost = 0.10(28.00) = 2.80
A simple way to find 10% of any number is to move the decimal
point in the number one place to the left:
28.00  2.80
1.3-
21

22. Example 4: Determining a Tip
The cost of Freddie Rojo’s meal before tax is
$28.00.
c) If he wants to leave a 15% tip on the pretax cost
of the meal, how much should he leave?
1.3-
22

23. Example 4: Determining a Tip
Solution:
To find 15% of $28.00, multiply:
15% of $28.00 – 0.15(28.00) = 4.20
Or to find 15%, find 10% and add it to ½ that amount:
$2.80 + = $2.80 + $1.40 = $4.20
1.3-
23
$2.80
2

24. Example 6: Spraying Weed Killer
The instructions on the Ortho Weed-B-Gon lawn
weed killer indicate that to cover 1000 square
feet (ft2) of lawn, 20 teaspoons (tsp) of the weed
killer should be mixed in 5 gallons (gal) of water.
Ron Haines wishes to spray his lawn with the weed
killer using his pressurized sprayer.
a) How much weed killer should be mixed with
8 gal of water to get a solution of the proper
strength? 1.3-
24

25. Example 6: Spraying Weed Killer
Solution
Use: 20 teaspoons of weed killer is to be mixed
with 5 gal water
1.3-
25
Note: teaspoons and gallons are placed in the same
relative positions.
Replace the ? with an x.
20 tsp
5 gal water

? tsp
8 gal

26. Example 6: Spraying Weed Killer
Solution:
1.3-
26
Ron must mix 32 tsp of weed killer with 8 gal of
water.
(32 tsp = 10 2/3 tablespoons = 2/3 cup)
20
5

x
8
20 8
  5x
160  5x
x 
160
5
 32

27. Example 6: Spraying Weed Killer
The instructions on the Ortho Weed-B-Gon lawn
weed killer indicate that to cover 1000 square feet
(ft2) of lawn, 20 teaspoons (tsp) of the weed killer
should be mixed in 5 gallons (gal) of water. Ron
Haines wishes to spray his lawn with the weed
killer using his pressurized sprayer.
b) How much weed killer is needed to cover an
area of 2820 ft2 of lawn?
1.3-
27

28. Example 6: Spraying Weed Killer
Solution:
Use: 1000 ft2 requires 20 teaspoons of weed killer
1.3-
28
Areas may be on top or bottom as long as they
are in the same relative position.
Replace the ? with an x.
1000 sq ft
20 tsp

2820 sq ft
? tsp

29. Example 6: Spraying Weed Killer
Solution:
1.3-
29
Thus, 56.4 tsp are needed.
1000
20

2820
x
1000 x
  20 2820
 
1000x  56,400
x 
56,400
1000
 56.4

30. Exercises

31. A paper fan and a visor together
cost P100.00. The visor costs P60.00
more than the paper fan. What are
the individual costs of the visor and
the fan?
Problem no. 1

32. The product of ages, in years of
three teenagers is 4590. None of the
teens are of the same age. What are the
ages of the teenagers?
Problem no. 2

33. The number of ducks and pigs in a field
totals 35. The total number of legs among them is
98. Assuming each duck has exactly two legs and
each pig has exactly four legs, determine how
many ducks and how many pigs are in the field.
Problem no. 3

34. Gardo picks a number. He doubles the number,
squares the result, divides the square by three,
subtracts 30 from the quotient, and gets 18. What are
the possible numbers that Gardo could have picked?
What operation does Gardo perform which prevents us
from knowing the exact number he picked?
Problem no. 4

35. Katrina sells eggs by piece. On the first day, she
sold a half more than half the number of eggs for sale. On
the second day, she sold a half more than half the
remaining number of eggs on the first day. On the third
day, she again sold a half more than half the remaining
number of eggs on the second day. Only a dozen eggs
were left for the fourth day. How many eggs did she have
originally?
Problem no. 5