Dr. Antonio Quesada – Director, Project AMP
Project AMP
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Problem Solving
Activity: Using Trial & Error to Solve Problems
Team members’ names:___________________________________________________
Goal: In this activity you will learn how to use trial and error and a table to organize
your work to solve word problems.
Let us start with an example.
Example. Joe has $10 more than his friend Paul. Together they have $40. How much
money does each one have?
First Approach to solve the problem. The trial & error method consists of guessing
what the answer might be using an initial educated guess, and subsequently refining your
next guess by taking into consideration the results obtained.
First, you need to organize your work. For this it is recommended that you use a table.
The headings for the table consist typically of the names of the variables involved. What
will you choose for the headings in this problem? _____________________
The headings of the columns in the table that follows is appropriate for this problem.
Joe’s money Paul’s money Total money
Next, you need to make an educated guess. For instance, would it be reasonable to guess
that Joe has $50? ________ Why? _______________________________________
What is the largest amount of money that Joe can have? _______
What is the smallest amount of money that Joe can have? _______
So, an educated guess is one that does not contradict the information given in the
statement of the problem.
Once you make an educated guess, you then proceed to fill all the entries of the table
keeping in mind the relationships about them established in the problem. For instance, if
you guess that Joe has $20, how much money will Paul have? ________ (Did you keep
in mind the difference in the amount of money that Joe and Paul have? If not, guess
again the amount that Paul will have.)
How much will they have together in this case? ________
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A Quesada Director Project AMP
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Dr. Antonio Quesada – Director, Project AMP
2
At this point, your table will look like the following:
Joe’s money Paul’s money Total money
$20
$10
$30
Analyzing your last answer, did you fall short or did you guess too much for the amount
that Joe has? _________
Make a new guess for the amount that Joe has and continue to fill in the second row of
your table. Using the new total obtained, increase or decrease the amount that Joe has for
your entries in the new row. Continue this process until you obtain the correct answers.
What relation given in the statement of the problem have you used to decide if your
answer is correct? ____________________________________________________
Let us try a second approach to solve the problem.
You could have started by writing $40 in the column with heading “Total money.” Then,
after initially guessing that Joe has $20, you could have calculated Paul’s money taken
into consideration the “Total money” that both have together. Following this reasoning,
how much money does Paul have? ______.
At this point, your table will look like the following:
Joe’s money Paul’s money Total money
$20
$20
$40
What is wrong with your answer in this case? ___________________________________
Did you fall short or did you guess too much for the amount that Joe has? ____________
Why? __________________________________________________________________
Rectify your initial guess as needed and calculate Paul’s money based on your new guess.
Then, observing if the difference between John and Paul’s monies is $10, increase or
decrease the amount that Joe has for your entries in the new row. Continue this process
until you obtain the correct answers.
When will you know that you have the correct answers?
What relation given in the statement of the problem have you used this time to decide if
your answer is correct? ____________________________________________________
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A Quesada Director Project AMP
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Dr. Antonio Quesada – Director, Project AMP
3
How do these approaches differ? _____________________________________________
Remarks.
a) Making an educated guess helps you to identify the existing relationships among
the different variables.
b) You may use any of the relations given in the problem to make your calculation
and another relation to test each guess.
c) Your goal is to solve the problem; the number of guesses that you have to make is
irrelevant.
d) The more you use this method the more proficient you will become and you will
find that you won’t need as many guesses or rows in the table.
Additional practice problems. Draw the appropriate table and solve by trial & error.
1. Debbie has $5 less than Sue. Together they have $61. How much money
does each person have?
2. Ed has twice as much money as Frank. Together they have $55.50. How
much money does each person have?
3. Rebecca has one-third as much money as Sarah. Together they have
$77.60. How much money does each person have?
4. There are 560 third- and fourth-grade students in a certain elementary
school. If there are 80 more third-graders than fourth-graders, how
many third-graders are in the school?
5. When two pieces of rope are placed end to end, their combined length is
150 feet. When the two pieces are placed side by side, one is 26 feet
longer than the other. What are the lengths of the two pieces?
Extension: There are three numbers. The first number is twice the
second number. The third number is twice the first number. Their sum
is 112. What are the numbers?
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A Quesada Director Project AMP