1.5: Problem Solving
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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.
Pólya’s How to Solve It
George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.
In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
Problem Solving Strategy 1 (Guess and Test)
Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.
Example \(\PageIndex{1}\)
Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?
Solution
Step 1: Understanding the problem
We are given in the problem that there are 25 chickens and cows.
All together there are 76 feet.
Chickens have 2 feet and cows have 4 feet.
We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.
Step 2: Devise a plan
Going to use Guess and test along with making a tab
Many times the strategy below is used with guess and test.
Make a table and look for a pattern:
Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.
Step 3: Carry out the plan:
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Chickens |
Cows |
Number of chicken feet |
Number of cow feet |
Total number of feet |
|---|---|---|---|---|
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20 |
5 |
40 |
20 |
60 |
|
21 |
4 |
42 |
16 |
58 |
| Notice we are going in the wrong direction! The total number of feet is decreasing! | ||||
| 19 | 6 | 38 | 24 | 62 |
| Better! The total number of feet are increasing! | ||||
| 15 | 10 | 30 | 40 | 70 |
| 12 | 13 | 24 | 52 | 76 |
Step 4: Looking back:
Check: 12 + 13 = 25 heads
24 + 52 = 76 feet.
We have found a solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.
Check in question 1:
Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31.
Check in question 2:
Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet. How many of each animal does he have? Make sure you use Polya’s 4 problem-solving steps.
Problem Solving Strategy 2 (Using a variable to find the sum of a sequence.)
Gauss's strategy for sequences:
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last term = (fixed number) ( n -1) + first term
The fix number is the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.
Example \(\PageIndex{2}\)
2, 5, 8, ... Find the 200th term.
Solution
Last term = 3(200-1) +2
Last term is 599.
Check in question 3:
Find the 320th term of 7, 10, 13, 16 …
Problem Solving Strategy 3 (Working Backwards)
This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.
Example \(\PageIndex{3}\)
Karen is thinking of a number. If you double it and subtract 7, you obtain 11. What is Karen’s number?
Solution
1. We start with 11 and work backwards.
2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.
3. The opposite of doubling something is dividing by 2. 18/2 = 9
4. This should be our answer. Looking back:
9 x 2 = 18 -7 = 11
5. We have the right answer.
Check in question 4:
Christina is thinking of a number.
If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)
Problem Solving Strategy 4 (Looking for a Pattern)
Definition: Sequence
A sequence is a pattern involving an ordered arrangement of numbers.
We first need to find a pattern.
Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?
Example \(\PageIndex{4}\)
1, 4, 7, 10, 13… Find the next 2 numbers.
Solution
The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.
Example \(\PageIndex{5}\)
1, 4, 9, 16 … Find the next 2 numbers.
Solution
It looks like each number is a perfect square. \(1^2=1\), \(2^2=4\)
\(3^2=9\)
\(4^2=16\)
So the next numbers would be
\(5^2=25\)
\(6^2=36\)
Example \(\PageIndex{6}\)
10, 7, 4, 1, -2… Find the next 2 numbers.
Solution
In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be
-2 -3 = -5
-5 – 3 = -8
Example \(\PageIndex{7}\)
1, 2, 4, 8 …Ffind the next two numbers.
Solution
This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?
1 x 2 = 2
2 x 2 = 4
4 x 2 =8
So each number is being multiplied by 2.
8 x 2 = 16
16 x 2 = 32
Problem Solving Strategy 5 (Make a List)
Example \(\PageIndex{8}\)
Can perfect squares end in a 2 or a 3?
Solution
List all the squares of the numbers 1 to 20.
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.
Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8. But note that this is just an observation. To answer this question, one would need a mathematically rigorous proof.
Example \(\PageIndex{9}\)
How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?
Solution
Quarter’s dimes
0 3 30 cents
1 2 45 cents
2 1 60 cents
3 0 75 cents
Check in question 5:
How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)
Problem Solving Strategy 6 (Process of Elimination)
This strategy can be used when there is only one possible solution.
Example \(\PageIndex{10}\)
I’m thinking of a number.
- The number is odd.
- It is more than 1 but less than 100.
- It is greater than 20.
- It is less than 5 times 7.
- The sum of the digits is 7.
- It is evenly divisible by 5.
Solution
- We know it is an odd number between 1 and 100.
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It is greater than 20 but less than 35
- 21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.
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The sum of the digits is 7
- 21 (2+1=3) No
- 23 (2+3 = 5) No
- 25 (2 + 5= 7) Yes
Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.
Reference
- References (2)
Exercises
Fallacies in Common Language
For each of the following statements, name the type of logical fallacy being used.
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If you don’t want to drive from Boston to New York, then you will have to take the train.
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Every time I go to Dodger Stadium, the Dodgers win. I should go there for every game.
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New England Patriots quarterback Tom Brady likes his footballs slightly underinflated. The “Cheatriots” have a history of bending or breaking the rules, so Brady must have told the equipment manager to make sure that the footballs were underinflated.
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What you are doing is clearly illegal because it’s against the law.
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The county supervisor voted against the new education tax. He must not believe in education.
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“Apples a day keeps doctors away.” No one has said apples are bad, so this old saying must be true.
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Wine has to be good for your health because… I mean, can you imagine a life without wine?
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Studies show that slightly overweight senior citizens live longer than underweight ones. The extra weight must make people live longer.
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Whenever our smoke detector beeps, my kids eat cereal for dinner. The loud beeping sound must make them want to eat cereal for some reason.
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There is a scientist who works at a really good university, and he says there is no strong evidence for climate change, especially global warming. Some politicians also question climate change. So I don’t really believe it.
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My neighbor cheats on his tax returns. I don’t believe anything he says.
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A: “Don’t fight over small things. Just let them go.”
B: “What exactly are ‘small things’? How do I know what is small and what isn’t?”
A: “Well, small things are things you really don’t want to fight over.”
Propositions and Logic
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List the set of integers that satisfy the following statement: A positive multiple of 5 and not a multiple of 2
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List the set of integers that satisfy the following statement: Greater than 12 and less than or equal to 18
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List the set of integers that satisfy the following statement: Even number less than 10 or odd number between 12 and 10
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You qualify for a special discount if you are either
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a full-time student in the state of California or
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at least 18 and your income is less than $20,000 a year.
For each person below, determine if the person qualifies for this discount. If more information is needed, indicate that.
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A 17-year-old full-time student at a California community college with no job
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A 28-year-old man earning $50,000 a year
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A 60-year-old grandmother who does not work and does not go to school
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A boy in first grade
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A mother making $18,000 a year and not enrolled in any college
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A 22-year-old full-time student at Arizona State University with no job
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A 18-year-old earning exactly $20,000 a year while attending UCLA part-time
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Write the negation: Everyone failed the quiz today.
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Write the negation: Someone in the car needs to use the restroom.
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How can you prove this statement wrong?
“Everyone who ate at that restaurant got sick.”
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How can you prove this statement wrong?
“There was someone who ate at that restaurant and got sick.”
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How can you prove this statement wrong?
“There is no baseball player who can excel at both pitching and hitting. Everyone must choose one or the other.”
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How can you prove this statement wrong?
“Every student must have an ID number before registering for classes.”
Truth Tables
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Translate each statement from symbolic notation into English sentences. Let A represent “Elvis is alive” and let G represent “Elvis gained weight.”
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A ⋁ G
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~( A ⋀ G )
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G → ~ A
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A ↔ ~ G
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A ⋀ ~ G
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~( A ⋁ G )
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Create a truth table for each statement below.
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A ⋀ ~ B
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~(~ A ⋁ B )
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( A ⋀ B ) → C
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( A ⋁ B ) → ~ C
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Complete the truth table for ( A ⋁ B ) ⋀ ~( A ⋀ B ).
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A |
B |
A ⋁ B |
A ⋀ B |
~(A ⋀ B) |
(A ⋁ B) ⋀ ~(A ⋀ B) |
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T |
T |
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T |
F |
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F |
T |
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F |
F |
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We have been studying the inclusive or, which allows both A and B to be true. The exclusive or does not allow both to be true; it translates to “either A or B , but not both.” For each situation, decide whether the “or” is most likely exclusive or inclusive.
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An entrée at a restaurant includes soup or a salad.
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You should bring an umbrella or a raincoat with you.
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We can keep driving on I-5 or get on I-405 at the next exit.
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Use Gate 1 if you are at least 35 years old, or Gate 2 if you are younger.
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You should save this document on your computer or a flash drive.
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I am not sure if my pregnant wife is going to have a boy or a girl.
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Consider the statement “If you are under age 17, then you cannot attend this movie.”
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Write the converse.
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Write the inverse.
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Write the contrapositive.
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Consider the statement “If you have a house in Beverly Hills, you are rich.”
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Write the converse.
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Write the inverse.
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Write the contrapositive.
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Assume that the statement “If you swear, then you will get your mouth washed out with soap” is true. Which of the following statements must also be true?
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If you don’t swear, then you won’t get your mouth washed out with soap.
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If you don’t get your mouth washed out with soap, then you didn’t swear.
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If you get your mouth washed out with soap, then you swore.
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Write the negation: If Luke faces Vader, then Obi-Wan cannot interfere.
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Write the negation: If you look both ways before crossing the street, then you will not get hit by a car.
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Write the negation: If you weren’t talking, then you wouldn’t have missed the instructions.
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Write the negation: If you score a goal now, we will win.
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Assume that the biconditional statement “You will play in the game if and only if you attend all practices this week” is true. Which of the following situations could NOT happen?
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You attended all practices this week and didn’t play in the game.
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You didn’t attend all practices this week and played in the game.
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You didn’t attend all practices this week and didn’t play in the game.
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Use De Morgan’s Laws to rewrite the disjunction as a conjunction: It is not true that Tina likes Sprite or 7-Up.
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Use De Morgan’s Laws to rewrite the disjunction as a conjunction: It is not true that the father or the mother of that child will be required to testify.
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Use De Morgan’s Laws to rewrite the conjunction as a disjunction: It is not the case that both the House and the Senate passed the bill.
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Use De Morgan’s Laws to rewrite the conjunction as a disjunction: It is not the case that you need a dated receipt and your credit card to return this item.
Analyzing Arguments
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Determine whether each of the following is an inductive or deductive argument:
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The new medicine works. We tried it on 100 patients, and all of them were cured.
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Every student has an ID number. Sandra is a student, so she has an ID number.
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Every angle of a rectangle is 90 degrees. A soccer field (pitch) is rectangular, so every corner is 90 degrees.
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Everything that goes up comes down. I throw a ball up. It must come down.
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Every time it rains, my grass grows fast. Rain speeds up the growth of grass.
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Sports makes a person strong. My daughter plays basketball. She will be strong.
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Analyze the validity of the argument.
Everyone who gets a degree in science will get a good job. I got a science degree. Therefore, I will get a good job.
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Analyze the validity of the argument.
If someone turns off the switch, the lights will not be on. The lights are off. Therefore, someone must have turned off the switch.
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Suppose the statement: “If today is Dec. 25, then the store is closed.”
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“Today is Dec. 25. Thus, the store is closed.” Which property was used? Is this valid?
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“The store is not closed. So today is not Dec. 25.” Which property was used? Is this valid?
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“The store is closed. Therefore, today must be Dec. 25.” Which property was used? Is this valid?
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“Today is not Dec. 25. Thus, the store is open.” Which property was used? Is this valid?
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For the following questions, use a Venn diagram or a truth table to determine the validity.
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If a person is on this reality show, they must be self-absorbed. Laura is not self-absorbed. Therefore, Laura cannot be on this reality show.
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If you are a triathlete, then you have outstanding endurance. LeBron James is not a triathlete. Therefore, LeBron does not have outstanding endurance.
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Jamie must scrub the toilets or hose down the garbage cans. Jamie refuses to scrub the toilets. Therefore, Jamie will hose down the garbage cans.
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Some of these kids are rude. Jimmy is one of these kids. Therefore, Jimmy is rude!
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Every student brought a pencil or a pen. Marcie brought a pencil. Therefore, Marcie did not bring a pen.
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If a creature is a chimpanzee, then it is a primate. If a creature is a primate, then it is a mammal. Bobo is a mammal. Therefore, Bobo is a chimpanzee.
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Every cripsee is a domwow. Mekep is not a domwow. Therefore, Mekep is not a cripsee. (This sentence has a lot of made-up words, but it is still possible to check for the validity of the argument. This is a good practice for abstract thinking.)
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Whoever dephels a kipoc will be bopied. I did not dephel any kipoc. Therefore, I will not be bopied. (Again, you do not need to know the meaning of each word to do this exercise.)
Problem Solving
For the following exercises, apply any problem-solving strategies and your critical-thinking skills to solve various types of problems. There is single formula or procedure to follow. Be flexible and consider all possibilities.
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There are 13 postage stamps on the table. Some are 20-cent stamps while others are 45-cent stamps. The total postage value of these stamps is $4.10.
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If they were all 20-cent stamps, would 13 of them add up to $4.10?
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If there were five 20-cent stamps, would these stamps add up to $4.10?
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How about ten 20-cent stamps? OK, you probably got some idea now.
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How many 20-cent stamps are there?
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Can you think of another way to solve the problem?
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What would you say to a friend of yours who tries to help you out by writing a system of two equations to solve this problem?
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Find the next two terms of each of the following sequences and explain why.
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9, 7, 5, 3, …
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0, 1, 4, 9, 16, 25, …
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3, 6, 12, 24, …
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0, 1, 3, 6, 10, 15, …
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5, 7, 5, 5, 7, 5, 5, 7, 5, 5, …
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1, 1, 2, 3, 5, 8, 13, 21, … (Fibonacci Sequence) We will study it later.
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A man bought an old car for $2000. He fixed it up and sold it for $2,500. But he missed it, so he brought it back for $3,200. Later, he sold it for $4,000. How much did he make in these transactions?
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Is it $4,000 - $2,000 = $2,000?
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He made $500 on the first sale and then $800 on the second. But he lost $700 in between when he re-purchased it. So is it $800 - $700 = $100?
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Is it $500 + $800 = $1,300?
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See the pattern below. Can you make a conjecture? Is it true? Can you prove it?
1=1 (=12).
1+3=4 (=22).
1+3+5=9 (=32).
1+3+5+7=16 (=42).
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A heart surgeon is about to perform a medical procedure on a boy. The surgeon told the nurses, “I want you to know that this is my son. I am operating on my own child today.” Everyone there knew that the boy was the surgeon’s son. But the surgeon was not the boy’s father. How can this be true?
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What can “H.D.” in this little story be?
“H.D. sat on a wall. H.D. had a great fall. All the president’s horses and all the president’s men couldn’t put H.D. together again.”
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You have decided to work out at the gym at least twice a week, but never on two consecutive days. If you are to keep the same schedule every week, list all possible days of the week) you can exercise at the gym.
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A man must be married to have a mother-in-law but can be single and have a brother-in-law. Explain why.
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Your sprinklers are set to water your plants at 6 am every morning during summer, when the daylight saving time is in effect. Once the time goes back to regular time (in November, say from PDT to PST), what time do your sprinklers start? (Hint: it’s either 5 am or 7 am.) What is the best way to explain this to your friend?
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You are to visit a friend who lives 300 miles away. You drive to his house early in the morning, averaging 60 mph, but you return in the afternoon, when the freeway is jammed, at an average speed of 40 mph. Is the overall average speed 50 mph? Why or why not?
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You have 10 identical pairs of black socks and 9 identical parts of white socks in your drawer, except each pair is not “paired up,” i.e., each sock is in the drawer, separated from all others. The room is completely dark, and you cannot see which socks you are taking out. How many socks do you have to pull out if you want to be sure that you get
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A pair of white socks?
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A pair of black socks?
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Any matching pair?
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A divorced 49-year-old man with a 25-year-old son marries a young woman whose mother is a widow. The 25-year-old son marries the widow and have a baby girl.
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How is that baby related to the 49-year-old man? (The baby is his son’s daughter.)
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How else is that baby related to the 49-year-old man? (The baby is his wife’s mother’s daughter.)
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Does that baby have a step-brother? If so, under what conditions?
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Who is that baby’s step-sister?
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Combine your answers to describe how the 49-year-old man is related to the baby.
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A person can be ½ Chinese and ½ Italian. Are the following cases possible?
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½ Japanese, ¼ Russian, ¼ Irish
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3/8 Scottish, 3/8 Vietnamese, ¼ Spanish
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3/8 Mexican, ¼ Turkish, and ½ Norwegian
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1/6 French, ½ Canadian, 1/3 Brazilian
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½ Armenian, ¼ German, ½ Swedish
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John says, “I don’t have any brothers, sisters, step-brothers, or step-sisters. See that tall woman? Her father is my mother’s child.” Who is the tall woman?
Contributors and Attributions
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Saburo Matsumoto
CC-BY-4.0