26 Mathprobs

Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion.
Explain your reasoning.
1. Each time Monica kicks a ball up in the air, it returns to the ground. So, the next
time Monica kicks a ball up in the air, it will return to the ground.
ANSWER
Inductive reasoning, because a pattern is used to reach the conclusion.
2. All reptiles are cold-blooded. Parrots are not cold-blooded. Sue’s pet parrot is not a reptile.
ANSWER
Deductive reasoning, because facts about animals and the laws of logic are used to reach
the conclusion.
3. Since some grapes are purple, and all grapes are fruit, some fruit is purple.
ANSWER
Deductive Reasoning, because facts about fruits and the law of logic is used to reach the
conclusion.
4. All math teachers have a great sense of humor. Patrick is a math teacher. Therefore, Patrick
must have a great sense of humor.
ANSWER
Because the reasoning goes from general to specific, deductive reasoning was used.
Identify each premise and the conclusion in each of the following arguments. Then tell
whether each argument is an example of inductive or deductive reasoning.
5. Our house is made of redwood. Both of my next-door neighbors have redwood houses.
Therefore, all houses in our neighborhood are made of redwood.
ANSWER
The premises are "Our house is made of redwood" and "Both of my next-door neighbors have
redwood houses." The conclusion is "Therefore, all houses in our neighborhood are made of
redwood." Since the reasoning goes from specific examples to a general statement, the argument
is an example of inductive reasoning.
6. All word processors will type the symbol @.I have a word processor. I can type the symbol @.
ANSWER
Here the premises are "All word processors will type the symbol @" and "I have a word
processor." The conclusion is "I can type the symbol @." This reasoning goes from general to
specific, so deductive reasoning was used.
7. Today is Friday. Tomorrow will be Saturday.

ANSWER
There is only one premise here, "Today is Friday." The conclusion is "Tomorrow will be
Saturday." The fact that Saturday follows Friday is being used, even though this fact is not
explicitly stated. Since the conclusion comes from general facts that apply to this special case,
deductive reasoning was used.
Read the following arguments and determine whether they use inductive or deductive
reasoning:
8. A child examines ten tulips, all of which are red, and concludes that all tulips must
be red.
ANSWER
Inductive Reasoning
9. If an isosceles triangle has at least two sides congruent, then an equilateral triangle
is also isosceles.
ANSWER
Deductive Reasoning
10. If 5x = 25, then x =5.
ANSWER
Deductive Reasoning
11. Sandy earned A's on her first six geometry tests so she concludes that she will
always earn A's on geometry tests.
ANSWER
Inductive Reasoning
12. Since it snowed every New Year's Day for the past four years it will snow on New
Year's Day this year.
ANSWER
Inductive Reasoning
Use inductive reasoning to predict the next number in the sequence.
13. 4, 8, 12, 16, ?
ANSWER
Each successive number is 4 larger than the last. Thus we predict that the next number
after 16 is 16 + 4 which is 20.
14. 1, 1, 2, 3, 5, 8,?
ANSWER
The first two numbers in the list is not too helpful, but the succeeding numbers are. Two is
the sum of 1 plus 1, 3 is the sum of 1 plus 2, 5 is the sum of 2 plus 3. The succeeding numbers
after the first two 1's are the sum of two numbers preceeding them. Thus, the next number after 8
is 13, because 8 plus 5 is 13.
15. 2, 9, 16, 23, 30

ANSWER
Each number in the list is obtained by adding 7 to the previous number.
The probable next number is 30 + 7 = 37.
16. 3, 7, 11, 15, 19, 23

ANSWER
Each number in the list is obtained by adding 4 to the previous number. The probable next
number is 23 + 4 = 27.
16. 1, 1,2,3,5, 8, 13, 21

ANSWER
Beginning with the third number in the list, each number is obtained by adding the two previous
numbers in the list. That is, 1 +1= 2, 1 + 2= 3, 2 +3 =5, and so on. The probable next number in
the list is 13 +21 = 34.
17. 1,2,4,8, 16
ANSWER
It appears here that in order to obtain each number after the first, we must double the previous
number. Therefore, the most probable next number is 16 X 2 = 32.
Use deductive reasoning to make conjectures.

18. All integers ending in 0 or 5 are divisible by 5. What conclusion can you make of 1,005?
ANSWER
Our familiarity with integers does not make us doubt the assumptions we have about
integers ending in either 0 or 5. Since 1,005 ends in 5, it is divisible by 5. In addition, our
understanding of being divisible by 5 means there is a natural number n such that 1,005=5n. To
obtain n, we divide 1,005 by 5 which yields 201.
19. All noble gases are stable, and helium is a noble gas. What conjectures can you make of
helium?
ANSWER
Helium is stable. Along with other noble gases in group 18 of the periodic table, its
outer shell holds the maximum number of valence electrons. As a noble gas, it does not easily
react with other elements.
20. To earn a master’s degree in mathematics, a graduate student must earn 42 units. Mr.
Rolando G. Panopio, a mathematics graduate student, has earned 36 units. What conjectures can
you make of Mr. Panopio?
ANSWER
These are the conjectures we can make about Mr. Panopio. He still needs 6 units to earn a
master’s degree, or he is close to finishing his studies to get a master’s degree.
21. Which is an example of a deductive argument?

a. There are 25 CDs on the top shelf of my bookcase and 14 on the lower shelf. There are no
other CDs in my bookcase. Therefore, there are 39 CDs in my bookcase.
b. Topeka is either in Kansas or Honduras. If Topeka is in Kansas, then Topeka is in North
America. If Topeka is in Honduras, then Topeka is in Central America. Therefore, Topeka is in
Kansas.
c. No one got an A on yesterday's test. Jimmy wasn't in school yesterday. Jimmy will make up
the test today and get an A.
d. All human beings are in favor of world peace. Terrorists don't care about world peace.
Terrorists bring about destruction.
ANSWER
The answer is a, because it has two premises and a conclusion that follows logically from them.
Choice b has three premises and the conclusion does not follow from them. Choices c and d have
conclusions that do not follow the premises.
22. Finding a counter-example to each answer choice may be the fastest way to solve the
problem. Remember, one counter-example to a statement is enough to disprove it.
Which of the following numbers is a counter-example to the following claim?
If p is an odd prime, then p + 2 is also a prime.
(A) 3
(B) 5
(C) 7
(D) 9
(E) 11
ANSWER
The correct answer is C. Because we are looking for a counter-example, we must find an odd
prime number p, such that p+2 is NOT a prime. We analyze each of the choices.
(A) 3 is a prime, and 3+2=5 is a prime. This is not a counter-example.

(B) 5 is a prime, and 5+2=7 is a prime. This is not a counter-example.
(C) 7 is a prime, and 7+2=9 is NOT a prime. This is a counter-example.
(D) 9 is a not a prime. This is not a counter-example.
(E) 11 is a prime, and 11+2=13 is a prime. This is not a counter-example.
Hence, the answer is (C).

23. Which of the following numbers is a counter-example to the following claim?
If n is an integer, then n2 + 1 is a prime.
(A) 1
(B) 2
(C) 3
(D) 4
(E)
ANSWER
The correct answer is C. Because we are looking for a counter-example, we must find an integer
n such that n2 + 1 is NOT a prime. We analyze each of the choices.
(A) 1 is an integer, and 12 + 1 = 2, which is a prime. This is not a counter-example.

(B) 2 is an integer, and 22 + 1= 5, which is a prime. This is not a counter-example.
(C) 3 is an integer, and 32 + 1= 10 = 2×5, which is NOT a prime. This is a counter-example.
(D) 4 is an integer, and 42+1 = 17, which is a prime. This is not a counter-example.
(E) is not an integer. This is not a counter-example. Thus, the answer is (C).
24. Find a counterexample to disprove the conjecture.

Conjecture The sum of two numbers is always more than the greater number.
ANSWER
To find a counterexample, you need to fi nd a sum that is less than the greater number.
−2 + (−3) = −5
−5 ≯ −2
Because a counterexample exists, the conjecture is false.
25. What conclusion can you make about the product of an even integer and any other integer?
ANSWER
Step 1 Look for a pattern in several examples. Use inductive reasoning to make a
conjecture.
(−2)(2) = −4 (−1)(2) = −2 2(2) = 4 3(2) = 6
(−2)(−4) = 8 (−1)(−4) = 4 2(−4) = −8 3(−4) = −12
Conjecture Even integer • Any integer = Even integer
Step 2 Let n and m each be any integer. Use deductive reasoning to show that the
conjecture is true.
2n is an even integer because any integer multiplied by 2 is even.
2nm represents the product of an even integer 2n and any integer m.
2nm is the product of 2 and an integer nm. So, 2nm is an even integer.
The product of an even integer and any integer is an even integer.
26. There are three boxes that contain books. one is labelled math only; another is labeled
science only. The last box is labeled math science. However, each box is labeled incorrectly. If
you are to pick only one box, how will you determine the correct label for each?
ANSWER
Method: Deductive Reasoning
Open the box that is labeled math science. Since it is incorrectly labeled, then it must contain
either a math or a science book. Without loss of generality, assume that this contains math book;
therefore, it must be labeled math only. Then, the box labeled math only contains a science book,
and the box labeled science only contains both math and science books.
References:
http://www.crossroadsacademy.org/crossroads/wp-content/uploads/2016//05/Logical-
Reasoning-in-Mathematics.pdf
https://nj01001706.schoolwires.net/cms/lib/NJ01001706/Centricity/Domain/288/Geometry
%25202.2.pdf
http://www1.lpssonline.com/uploads/13cDeductiveReasoningPracticeExercise.pdf
http://www1.lpssonline.com/uploads/13cDeductiveReasoningPracticeExercise.pdf
https://brilliant.org/wiki/sat-counter-examples/
https://www.google.com/amp/s/www.thatquiz.org
Mathematics in the Modern World, 2018,Librarcy Services & Publishing Inc.,Juan Apolinario
C. Reyes
Mathematics in the Modern World, 2018, Mutya Publishing House, Inc., Aurora Rosalie P.
Tolentino, Miraflor C. Gutierez, Floryfe G. Hernandez, Romell A. Ramos, Jason T. Hortelano,
Teresita O. Pante
Mathematics
in the
Modern World
Chapter 3: Problem Solving
Submitted by:
Buen, Patricia Ann G.
BSA 1104
Submitted to:
Mr. Romer C. Castillo

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Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion.

Explain your reasoning.

7. Today is Friday. Tomorrow will be Saturday.

10. If 5x = 25, then x =5.

15. 2, 9, 16, 23, 30

16. 3, 7, 11, 15, 19, 23

16. 1, 1,2,3,5, 8, 13, 21

Use deductive reasoning to make conjectures.

21. Which is an example of a deductive argument?

(A) 3 is a prime, and 3+2=5 is a prime. This is not a counter-example.

Hence, the answer is (C).

(A) 1 is an integer, and 12 + 1 = 2, which is a prime. This is not a counter-example.

24. Find a counterexample to disprove the conjecture.

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