Problem Solving: A Primer
Incorporating problem solving in throughout the mathematics curriculum makes the mathematics
more concrete for students and helps them connect new concepts to what they already know.
Students cannot become good problem solvers through memorization. However, good problem
solvers do follow a heuristic and do answer the same questions with every problem that is
amenable to solution using mathematics. The heuristic we will use was developed by George
Polya and codified in his book How to Solve It.
The first step is understand the problem. If a problem can be solved using algebraic
techniques, this means you must determine the constants and related variables needed to create a
model of the problem situation. It is often helpful in understanding a problem, if you replace the
variables by a reasonable constant in order to understand the relationship between the constants
and variables.
The second step is devise a plan. For problems that can be solved using algebra, this means
finding a mathematical representation of the problem. This may be an equation, a system of
equations, or a function. With this step you have moved from the real world to the world of
mathematics.
The third step is carry out the plan. This is where you used mathematical tools and your
mathematical model to find solutions that make sense mathematically.
The fourth and final step is look back. In this step you take the mathematical solution(s) and
interpret them with respect to the real world problem. You ask yourself questions such as “What
are the units of the solution?”, “Does the solution make sense in terms of the original situation?”,
and “Have I solved the problem?” You should also check your answer at this step. Finally, you
should reflect on what you learned in solving the problem and whether you could use any of the
strategies you employed in other problem situations.
Example 1: The A-1 Furniture store first adds $10 to their wholesale cost, then doubles the sum
to get the retail price. A certain sofa retails at this store for $880. What was the wholesale cost
of this sofa?
Solution:
1) Understand the problem.
The constants in this problem are $10, 2, and $880. The 2 comes from the word
“doubles” and is an example of a hidden constant. The 1 in A-1 is not one of our
constants, since it is not related to the wholesale cost of the furniture. The only variable in
this problem is the wholesale cost, which we will represent by the letter w. To better
understand the problem we replace the wholesale cost w by $200. The A-1 Furniture
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store owner adds $10 to the $200 to get $210. Then this amount is doubled: 2(210) = 420,
so the retail price would be $420.
2) Devise a plan.
This is an example of a problem that requires us to convert a verbal description to a
symbolic representation. The phrase “adds $10 to their wholesale cost” translates into
w  10 . The phrase “doubles the sum” translates into 2w  10 . Note that we had to use
parentheses to change the order of operations. Without parentheses we would multiply by
2 first, then add 10. Our model is the equation 2w  10  880 .
3) Carry out the plan.
We solve this equation.
2w  10  880

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2
w  10  440
w  10  10  440  10
w  430
4) Look back.
The interpretation is that the wholesale cost of the sofa was $430. This is reasonable,
since a wholesale cost of $200 resulted in a retail price of only $420. If we add $10 and
double the sum we get 2430  10  2440  880 , the desired retail price, so the
answer checks.
For each problem 1) indicate the constants and variables in the problem, 2) write an equation that
models the situation, 3) solve the equation, and 4) interpret the solution in the context of the
problem and check that it does solve the problem.
1. Jill's average score of 82 was 7 more than her score on the last test. What was her score
on the last test?
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2. A local furniture store adds 10% of the wholesale cost of sofas to the wholesale cost, then
doubles the sum to get the retail price. If a sofa retails for $880, what was the wholesale
price?
Some problems require you to know and use a formula to solve the problem. Use a formula to
solve problems 3 and 4. For each problem 1) indicate the constants and variables in the problem,
2) write an equation that models the situation, 3) solve the equation, and 4) interpret the solution
in the context of the problem and check that it does solve the problem.
3. Sarah has 100 feet of edging to go around the circumference of a circular garden she is
building. What radius (to the nearest foot) should she use for the garden, if she wants to
use all of the edging to surround the garden?
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 F  32  is used to convert F degrees Fahrenheit to C degrees
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Celsius. Use this formula to convert 100 C (the boiling point of water at sea level) to
degrees Fahrenheit.
4. The formula C 
Many problems involve rates. You can recognize a rate the use of the word “per” in the units for
the rate. Some examples of rates are $1.99 per pound, 60 miles per hour, and 12 inches per foot.
A rate is always multiplied by an amount with the same units as the unit that comes after the
work “per”. For example, $1.99 per pound is multiplied by the amount of pounds bought to get
the total cost of the item, while 60 miles per hour is multiplied by the amount of hours spent
traveling at that rate to get the total miles traveled. Use this hint in solving the following
problems. Again, for each problem 1) indicate the constants and variables in the problem, 2)
write an equation that models the situation, 3) solve the equation, and 4) interpret the solution in
the context of the problem and check that it does solve the problem.
5. Sal's Pizza charges $8 for a large pizza. Yesterday, $728 was made from sales of large pizzas.
How many large pizzas did they sell?
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6. Paul's Plumbing charges $35 for a service call plus $40 per hour. The total charge to
install new pipe for Sally was $135. How many hours did it take to install the pipe?
7. Carl leased a computer system for his business for $100 per month with $1000 paid at
signing. He calculates that he could have bought the computer system for $5000 and had
maintenance charges of $50 per month. How long should the system last for the cost of
the lease to equal the cost of buying and maintaining the system.
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If there are two variables in a problem, you will need to find a system of two equations in two
variables to model the problem situation. You can then solve one of the equations for a variable
and use the substitution property to change the other equation to an equation in one variable that
you can solve. This answer can then be used to find the value of the other variable.
Example 2: Macadamia nuts costing $9 per pound will be mixed with peanuts costing $3 per
pound to obtain 12 pounds of a mixture costing $4 per pound. How many pounds of each type of
nut should be used?
Solution:
1) Understand the problem.
Constants
$9/lb, $3/lb, 12 lbs, $4/lb
Variables
amount of macadamia nuts = x lbs
amount of peanuts = y lbs
2) Devise a plan.
Because there are multiply rates in this problem, we organize the constants and variables
in a RAP (Rates – Amounts – Products) Chart. The first row is for the macadamia nuts,
the second row for peanuts, and the last row is always the totals. We put in the rates, the
constants or variables with units that include the word “per”, in the first column. The
amounts in the second column have to be in pounds, since the rates were all “per pound”.
We get the product column by multiplying each rate by the associated amount.
Rates
Amounts
Products
9 $/lb
x lbs
$ 9x
3 $/lb
y lbs
$ 3y
4 $/lb
12 lbs
$ 48
The rates cannot be added, but we know that the total amount of pounds is the sum of the
first two amounts and the total value is the sum of the first two products. We obtain a
model consisting of two equations in two variables from the chart.
A: x  y  12
B: 9 x  3 y  48
3) Carry out the plan.
We solve equation A (the simpler equation) for y getting y  12  x . We then substitute
the expression 12  x for y in equation B: 9x  312  x  48 . We can now solve
equation B for x.
9x  312  x  48
9 x  36  3x  48
Use the distributive property to remove parentheses.
6 x  36  48
Use the commutative property and combine.
6 x  12
x2
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We now replace x by 2 in equation A. 2  y  12 . Solving for y, we get y = 10.
4) Look back.
We interpret x  2 and y = 10 as meaning we should mix 2 pounds of macadamia nuts
with 10 pounds of peanuts to get 12 pounds worth $4 per pound. These are reasonable
numbers. If we check our solution, we see the total is 12 pounds of nuts and the value is
92  310  48 dollars.
For the following problems find a system of equations and use the substitution property to solve
each system. For each problem 1) indicate the constants and variables in the problem, 2) write a
system of equations that models the situation, 3) solve the system of equations, and 4) interpret
the solution in the context of the problem and check that it does solve the problem.
8. For part of a 4 hour trip Jan averaged 60 mph, while for the rest of the trip she averaged
40 mph. She traveled a total of 220 miles. How long was she traveling at 60 mph? at 40
mph?
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9. The Jones family invested $10,000 in two investments. The more risky investment paid
8% interest the first year, while the safer investment paid 5% interest. They earned $680
interest for the year. How much did they invest at each rate?
10. Stu bought some box seats and bleacher seats for a minor league baseball game for 13
people. Box seats cost $20 while bleacher seats were $10. He spent $210. How many of
each type of seat did he purchase?
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11. A field is seven feet less than twice as long as it is wide. A drainage pipe is placed along
the perimeter of the field and 346 feet of pipe are needed. What are the dimensions of the
field?
12. The fish market sells haddock for $6 per pound and tilapia for $4.50 per pound. Their
total revenue from selling these two fish in one day is $330. The wholesale cost per
pound for haddock is $5 per pound, while the wholesale cost per pound for tilapia is
$3.50 per pound. The total wholesale cost for the fish sold on this day is $265. How
many pounds of each type of fish did they sell?
End of Project
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