Quantitative Reasoning Name: ___________________________
Practice Exam 1 - Solutions Guide Date:_____________________________
Show all work for full credit, correct answers without supporting work are worth 1 point.
1.) Identify a counterexample for the given general statements. (2 pts./ea.)
a. The sum of two even numbers is odd.
Note: Specific counter examples may vary.
A counter example for this statement would be a case in which we add two even numbers
together (satisfying the premises) but the sum is not an odd number (conclusion is not satisfied).
Sample: 12 + 26 = 38 The even numbers 12 and 26 are added together and resulted in a value
that is not odd. Since the premises are satisfied and the conclusion is not, this is a solid counter
example.
b. All artists paint.
Note: Specific counter examples may vary.
A counter example for this statement would need to identify either a specific artist or a specific
art form that is not painting.
Samples: Sculptors are artists that do not paint, they shape forms through other mediums.
Musicians are artists that may not also paint. If we use a specific person for the counter
example; an artist that does not paint, then we must be careful not to over presume. For
instance, the comedian/actor Jim Carrey may be most known for his performance art, but he also
paints. While acting is an artform other than painting, the specific case of Jim Carrey would not
be a valid counter example to the general claim that all artists paint.
2.) Identify patterns for the given sequences and use it to identify the next term in the sequence. (3 pts./ea.)
a. 3 2 9 4 27 _____ Two potential answers: 8 or 6
There are a number of patterns we could identify for this particular sequence, the challenge is that it
is not a singular pattern for each step. An appropriate term for this type of sequence is an alternating
geometric sequence. However, there are other approaches we could also use: see below.
Patterns for reaching 8:
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, Option 1: We can separate the terms by odd or even index. Hence: 3, 9, 27 form one portion of
the sequence and 2, 4, ____ for the other. We may note that 3 –> 9 -> 27 can be accomplished
by multiplying the previous odd indexed term by 3. 3 x 3 = 9 and 9 x 3 = 27. Similar, we see
that our other portion of the sequence: 2, 4, _____ could be accomplished by multiplying the
previous term even indexed term by 2. 2 x 2 = 4 so 4 x 2 = 8. We conclude the next term is 8.
Option 2: Another way we could analyze the sequence is by exponents. Lets review the
sequence again and compare each term to powers of 3 and 2, respectively.
1 1 2 2 3
3=3 ; 2=2 ; 9=3 ; 4=2 ; 27=3
This pattern illustrates that the terms, while still alternating between powers of 3 and powers of
2, are following a fixed pattern. We would conclude that the next term is found by calculating
3
2 , or simply 8.
Pattern for reaching 6: Using the alternating split found in Option 1 above, we could analyze the even
terms using addition instead of multiplication. 2, 4, ____ could be thought of as following the pattern:
2, 2+2 = 4, 4 + 2 = 6. We would conclude the next term is 6.
b. 6 36 216 1296 7776 _46656__
Pattern: Each term is found by multiplying the previous term by 6.
3.) For each of the following, circle whether the conclusion is being made through inductive or deductive
reasoning (2 pts./ea.)
a. A given MATH 1001 student scores an 83% on the first exam of the semester, that student will
get a B in the class at the end of the semester.
*Inductive* / Deductive
Notice: The given sample of reasoning is using a specific instance (the grade on an exam) to
form a general conclusion (the class grade for entire semester), this is inductive reasoning.
b. All fish breath through gills and have skeletons composed of cartilage. The hammerhead shark
is a fish. Therefore, the hammerhead shark must breath through gills and have a skeleton
composed of cartilage.
Inductive / *Deductive*
Notice: The given sample of reasoning is using a general rule or pattern (that all fish breath
through gills and all fish have bones composed of cartilage) and using it to make draw a
conclusion for a specific type of fish (the hammer head shark), this is deductive reasoning.
This study source was downloaded by 100000838401522 from CourseHero.com on 04-28-2022 05:02:07 GMT -05:00
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Practice Exam 1 - Solutions Guide Date:_____________________________
Show all work for full credit, correct answers without supporting work are worth 1 point.
1.) Identify a counterexample for the given general statements. (2 pts./ea.)
a. The sum of two even numbers is odd.
Note: Specific counter examples may vary.
A counter example for this statement would be a case in which we add two even numbers
together (satisfying the premises) but the sum is not an odd number (conclusion is not satisfied).
Sample: 12 + 26 = 38 The even numbers 12 and 26 are added together and resulted in a value
that is not odd. Since the premises are satisfied and the conclusion is not, this is a solid counter
example.
b. All artists paint.
Note: Specific counter examples may vary.
A counter example for this statement would need to identify either a specific artist or a specific
art form that is not painting.
Samples: Sculptors are artists that do not paint, they shape forms through other mediums.
Musicians are artists that may not also paint. If we use a specific person for the counter
example; an artist that does not paint, then we must be careful not to over presume. For
instance, the comedian/actor Jim Carrey may be most known for his performance art, but he also
paints. While acting is an artform other than painting, the specific case of Jim Carrey would not
be a valid counter example to the general claim that all artists paint.
2.) Identify patterns for the given sequences and use it to identify the next term in the sequence. (3 pts./ea.)
a. 3 2 9 4 27 _____ Two potential answers: 8 or 6
There are a number of patterns we could identify for this particular sequence, the challenge is that it
is not a singular pattern for each step. An appropriate term for this type of sequence is an alternating
geometric sequence. However, there are other approaches we could also use: see below.
Patterns for reaching 8:
This study source was downloaded by 100000838401522 from CourseHero.com on 04-28-2022 05:02:07 GMT -05:00
https://www.coursehero.com/file/70949514/MATH-1001-Practice-Exam-1-Solutions-Guidedocx/
, Option 1: We can separate the terms by odd or even index. Hence: 3, 9, 27 form one portion of
the sequence and 2, 4, ____ for the other. We may note that 3 –> 9 -> 27 can be accomplished
by multiplying the previous odd indexed term by 3. 3 x 3 = 9 and 9 x 3 = 27. Similar, we see
that our other portion of the sequence: 2, 4, _____ could be accomplished by multiplying the
previous term even indexed term by 2. 2 x 2 = 4 so 4 x 2 = 8. We conclude the next term is 8.
Option 2: Another way we could analyze the sequence is by exponents. Lets review the
sequence again and compare each term to powers of 3 and 2, respectively.
1 1 2 2 3
3=3 ; 2=2 ; 9=3 ; 4=2 ; 27=3
This pattern illustrates that the terms, while still alternating between powers of 3 and powers of
2, are following a fixed pattern. We would conclude that the next term is found by calculating
3
2 , or simply 8.
Pattern for reaching 6: Using the alternating split found in Option 1 above, we could analyze the even
terms using addition instead of multiplication. 2, 4, ____ could be thought of as following the pattern:
2, 2+2 = 4, 4 + 2 = 6. We would conclude the next term is 6.
b. 6 36 216 1296 7776 _46656__
Pattern: Each term is found by multiplying the previous term by 6.
3.) For each of the following, circle whether the conclusion is being made through inductive or deductive
reasoning (2 pts./ea.)
a. A given MATH 1001 student scores an 83% on the first exam of the semester, that student will
get a B in the class at the end of the semester.
*Inductive* / Deductive
Notice: The given sample of reasoning is using a specific instance (the grade on an exam) to
form a general conclusion (the class grade for entire semester), this is inductive reasoning.
b. All fish breath through gills and have skeletons composed of cartilage. The hammerhead shark
is a fish. Therefore, the hammerhead shark must breath through gills and have a skeleton
composed of cartilage.
Inductive / *Deductive*
Notice: The given sample of reasoning is using a general rule or pattern (that all fish breath
through gills and all fish have bones composed of cartilage) and using it to make draw a
conclusion for a specific type of fish (the hammer head shark), this is deductive reasoning.
This study source was downloaded by 100000838401522 from CourseHero.com on 04-28-2022 05:02:07 GMT -05:00
https://www.coursehero.com/file/70949514/MATH-1001-Practice-Exam-1-Solutions-Guidedocx/
