MATH C277: Finite Mathematics

Finite Mathematics – Practice Questions With Answers
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Free MATH C277: Finite Mathematics Questions

Simplify the expression:
$\sqrt{\frac{4}{9} \times \frac{9}{4}}$
Reduce to lowest terms.

1
$\frac{13}{6}$
$\frac{\sqrt{13}}{6}$
$\frac{\sqrt{97}}{6}$

Explanation

Correct Answer:

$\frac{\sqrt{97}}{6}$

Explanation:

Step 1: Add the fractions under the square root

$\frac{4}{9} + \frac{9}{4} = \frac{(4 \times 4) + (9 \times 9)}{36} = \frac{16 + 81}{36} = \frac{97}{36}$

Step 2: Apply the square root

$\sqrt{\frac{97}{36}} = \frac{\sqrt{97}}{\sqrt{36}} = \frac{\sqrt{97}}{6}$

So actually, the simplified result is:

$\frac{\sqrt{97}}{6}$

Why Other Options Are Wrong:

1

This is far too small. The value under the square root is greater than 1, so the square root is also greater than 1.

$\frac{13}{6}$

This implies the square root of a rational number, but $\frac{\sqrt{97}}{6}$ ≠ $\frac{13}{6}$ . That would only be true if the value inside the radical were exactly $\frac{\sqrt{169}}{6}$ , which it isn’t.

$\frac{\sqrt{13}}{6}$

The expression under the radical is $\frac{97}{36}$ , not 13. So this doesn't match and is incorrect.

The property tax on a house with an assessed value of $228,530 is $1,942.51.
What is the assessed value of a house whose property tax is $1,341.64, assuming the same tax rate?
Round your answer to the nearest dollar.

$156,972
$157,840
$114,039
$330,880

Explanation

Correct Answer:

$157,840

Explanation:

First, find the tax rate from the given values:

Tax Rate = $\frac{1, 942.51}{228, 530}$ ≈ 0.0085

Rate ≈ 0.0085

Now use this rate to find the assessed value for the new tax amount:

Assessed Value = $\frac{1, 341.64}{0.0085}$ ≈ 157,840

This matches the option $157,840 when rounded to the nearest dollar.

Why Other Options Are Wrong:

$156,972

This is slightly below the correct computed value. It may result from a rounding or calculation error when estimating the tax rate or assessed value.

$114,039

This is too low and does not align with the tax amount provided. Using the correct tax rate, this amount would result in a much smaller tax.

$330,880

This value is too high. A house assessed this high at the same tax rate would owe over $2,800 in taxes, far above $1,341.64.

$3 \frac{3}{12} + 5 \frac{1}{8}$
Simplify the above expression.

$\frac{11}{8}$
$\frac{101}{12}$
$\frac{29}{12}$
$\frac{67}{8}$

Explanation

Correct Answer:

$\frac{67}{8}$

Explanation:

$3 \frac{3}{12} + 5 \frac{1}{8} = 3 \frac{1}{4} + 5 \frac{1}{8}$

= $3 \frac{2}{8} + 5 \frac{1}{8}$

= $8 \frac{3}{8}$

= $\frac{67}{8}$

Why Other Options Are Wrong:

11/8

This is incorrect because 11/8 is a small fraction (1.375), far less than the sum of two mixed numbers like 3 3/12 and 5 5/8, which should exceed 8. The addition of whole numbers 3 + 5 = 8, plus fractional parts, results in a value much larger than 11/8, indicating this option likely stems from a division error or misinterpretation of the problem as a fraction division rather than addition.

101/12

This is incorrect because 101/12 (approximately 8.4167) is a fraction that does not reflect the sum of 3 3/12 and 5 5/8, which should be around 8 or 9 when considering the whole numbers and fractions. The numerator 101 suggests a miscalculation, possibly from adding numerators incorrectly or confusing the operation, making it an invalid result for this addition problem.

29/12

This is incorrect because 29/12 (approximately 2.4167) is too small to represent the sum of 3 3/12 and 5 5/8, which involves adding numbers greater than 8. This option might arise from an erroneous subtraction or division of the fractions rather than their addition, and it fails to account for the whole number components correctly.

p → q
Select the truth table that represents the statement above.

Explanation

Correct Answer:

The table that shows:

T → T = T

T → F = F

F → T = T

F → F = T

Explanation:

The statement p → q means "if p, then q." In logic, this implication is false only when p is true and q is false. In all other cases, the implication is considered true.

When p is true and q is true, the implication holds, so the result is true.

When p is true and q is false, the implication fails, so the result is false.

When p is false (regardless of q), the implication is considered true because nothing is required of q when p is false.

Therefore, both cases where p is false (F → T and F → F) yield true.

Why Other Options Are Wrong:

Option one: This option is incorrect because it fails to mark the statement as true when both p and q are true. In the case of p → q, if both are true, the implication is valid and the result must be true. The presence of any incorrect truth value makes the entire truth table invalid.

Option two: This option is incorrect because it shows the implication as false when both p and q are false. However, in logic, p → q is true when p is false, regardless of q’s value. Therefore, marking this case as false is logically incorrect.

Wildlife biologists tagged 68 fish and released them into a lake. Later, they caught 119 fish and found that 17 of them were tagged.
Approximately how many fish are in the lake?

7,956
476
1,156
204
85

Explanation

Correct Answer:

476

Explanation:

First calculating the ratio of the tagged fish that were caught to the total fish tagged.

119 ÷ 17 = 4

Hence there will be 4 times more fish in the lake compared to the one caught.

119 * 4 = 476.

Why Other Options Are Wrong:

7,956

This is far too high and would suggest a tagging rate that doesn’t match the 17 tagged fish found in the sample.

1,156

Also too high; it results from miscalculating the proportion or misusing the cross-multiplication step.

204

Too low; this would imply an unrealistically high tag rate in the sample, inconsistent with 17 out of 119 being tagged.

85

This is almost the number of tagged fish, not the total population estimate. It significantly underrepresents the lake population.

Simplify the expression:
$\frac{\frac{\frac{2.4 \times 10^{- 7}}{8.0 \times 10^{- 7}}}{3.0 \times 10^{9}}}{6.0 \times 10^{- 3}}$

6.0 × 10³
6.0 × 10⁻¹³
1.5 × 10⁻¹³
1.5 × 10³

Explanation

Correct Answer:

6.0 × 10⁻¹³

Explanation:

Begin by simplifying the first part of the expression (the numerator):

$\frac{2.4 \times 10^{- 7}}{8.0 \times 10^{- 7}}$

$(2.4 \div 8.0) \times 10^{- 7 - (- 7)} = 0.3$

Next, simplify the second part (the denominator):

$\frac{3.0 \times 10^{9}}{6.0 \times 10^{- 3}}$

$(3.0 \div 6.0) \times 10^{9 - (- 3)} = 0.5 \times 10^{12}$

Now divide the simplified parts:

$\frac{0.3}{0.5 \times 10^{12}} = (\frac{0.3}{0.5}) \times 10^{12}$

$0.6 \times 10^{- 12} = 6 \times 10^{- 13}$

$6 \times 10^{- 13}$

Thus, the simplified expression is 6.0 × 10⁻¹³.

Why Other Options Are Wrong:

6.0 × 10³

This represents a very large number, which contradicts the result of dividing very small numbers by very large numbers. The actual operation yields a small value due to negative exponents, not a large one.

1.5 × 10⁻¹³

This value suggests a miscalculation in the base coefficient. The correct simplified result comes from 0.3 ÷ 0.5 = 0.6, not 1.5. The exponent may be close, but the base number is incorrect.

1.5 × 10³

This is incorrect both in magnitude and exponent direction. The correct simplification results in a very small number (negative exponent), not a large positive one. It also shows an incorrect coefficient of 1.5 instead of the accurate 0.6.

Which list of numbers is in order from least to greatest?

2.23565, 2.2356, 2.2359, √5
5, 2.23565, 2.2359, 2.2356
√5, 2.2359, 2.2356, 2.23565
2.2356, 2.23565, 2.2359, √5

Explanation

Correct Answer:

2.2356, 2.23565, 2.2359, √5

Explanation:

To determine the correct order from least to greatest, compare the decimal numbers and the square root of 5 (approximately 2.23607). Start with the decimals: 2.2356 is the smallest as its digits are 2.2356, followed by 2.23565 (which adds a 5 in the fifth decimal place), then 2.2359 (which has a 9 in the fourth decimal place, making it larger than the others). Next, compare these to √5 (2.23607), which is greater than 2.2359 because 2.23607 exceeds 2.2359 at the fourth decimal place. Thus, the order 2.2356, 2.23565, 2.2359, √5 is correct as it progresses from the smallest to the largest value.

Why Other Options Are Wrong:

2.23565, 2.2356, 2.2359, √5

This is incorrect because 2.2356 is listed before 2.23565, but 2.2356 is less than 2.23565 due to the additional 5 in the fifth decimal place of 2.23565. The sequence should start with the smallest number, 2.2356, making this order invalid as it reverses the correct progression.

5, 2.23565, 2.2359, 2.2356

This is incorrect because 5 is much larger than the other numbers, which are all around 2.23 to 2.24, including √5 (approximately 2.23607). Placing 5 first violates the least-to-greatest order, as it should be the last number if included, rendering this sequence incorrect.

√5, 2.2359, 2.2356, 2.23565

This is incorrect because √5 (approximately 2.23607) is greater than 2.2359, 2.2356, and 2.23565, so it should be last in the sequence. Starting with √5 places the largest value first, which contradicts the requirement for order from least to greatest.

1, 1, 2, 4, 7, 13, 24, . . .
Find the next number in the sequence.

Explanation

Correct Answer:

44

Explanation:

From the sequence:

1 + 1 = 2

1 + 2 = 3 → Add 1 more to make 4 (next in sequence)

2 + 4 = 6 → Add 1 more to make 7 (next in sequence)

4 + 7 = 11 → Add 2 more to make 13

7 + 13 = 20 → Add 4 more to make 24

The extra numbers being added (1, 1, 2, 4) are themselves part of the original sequence: 1, 1, 2, 4. To continue this pattern, the next number to add is the next in the sequence after 4, which is 7. So:

13 + 24 = 37 → Add 7 more = 44

Why Other Options Are Wrong:

24

This number already appears in the sequence and does not follow the forward-moving additive pattern. Repeating it would break the logic that introduces a new number based on cumulative growth.

37

This is only the result of 13 + 24. The established pattern shows that we must also add a number from earlier in the sequence—specifically 7 in this case—making 44 the actual next number.

47

Although this seems like a plausible progression, it does not follow the underlying pattern. The logic requires adding a previous value from the sequence, not simply continuing numerical growth. 47 skips the required additive structure.

Simplify the expression using the order of operations:
$10^{2} + 100 \div 5^{2} \times (- 2)$

Explanation

Correct Answer:

92

Explanation:

Using BODMAS

100 ÷ 5² = 100÷25 = 4

Next multiplication:

4 × (-2) = - 8

Finally:

102 + (-8) = 100 -8 = 92

Why Other Options Are Wrong:

0

This would result only if all terms cancel out, which they don’t.

100

This would be the result if the negative term were ignored.

108

This would occur if the multiplication was treated as positive (i.e., 4×2 instead of −2).

10.

Which of the following are equivalent form of $\frac{22}{7}$ .

3.14
π
$3 \frac{1}{7}$
$2 \frac{11}{7}$
$1 \frac{15}{7}$
$\frac{- 1}{\frac{- 7}{22}}$

Explanation

Correct Answer:

$3 \frac{1}{7}$

$1 \frac{15}{7}$

$\frac{- 1}{\frac{- 7}{22}}$

Explanation:

To determine equivalent forms of $\frac{22}{7}$ , we evaluate each option:

$3 \frac{1}{7}$ is a mixed number equivalent to $\frac{22}{7}$ since 3 × 7 + 1= 22, and $\frac{22}{7}$ = $\frac{22}{7}$ .

$1 \frac{15}{7}$ equals $\frac{7 + 15}{7} = \frac{22}{7}$ , so it is also equivalent.

$\frac{- 1}{\frac{- 7}{22}} = \frac{22}{7}$ after simplifying double negatives and dividing by a fraction.

Why Other Options Are Wrong:

3.14

This is a rounded decimal approximation of π ≈ 3.1416, while $\frac{22}{7}$ ≠3.142 but ≈ 3.142 Meaning approximately. Although close, it isnot exactly equal to $\frac{22}{7}$ .

π

$\frac{22}{7}$ is a common approximation of π, but π is an irrational number with a non-repeating, infinite decimal expansion. Therefore, they are not exactly equal.

$2 \frac{11}{7}$

This mixed number equals $\frac{25}{7}$ , which is greater than $\frac{22}{7}$ and therefore not an equivalent form.

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MATH 1709 C277: Finite Mathematics – Comprehensive Study Notes

1. Reasoning and Conjectures

Conjecture
A conjecture is a conclusion formed using inductive reasoning. Since it is based on observation rather than proof, it may or may not be true.

Inductive Reasoning
This reasoning method involves drawing general conclusions from specific examples or patterns. For instance, predicting the next number in a sequence based on observed trends.

Counterexample
A single instance that disproves a conjecture. If even one counterexample exists, the statement is considered false.

Deductive Reasoning
Deductive reasoning reaches a conclusion using established principles or general rules. It’s the opposite of inductive reasoning and is commonly used in solving logic puzzles and mathematical proofs.

2. Sequences and Pattern Recognition

Sequence
An ordered list of numbers (e.g., 5, 14, 27, 44...) where each number follows a particular rule or pattern.

Terms of a Sequence
The individual numbers in the sequence. For example, in the sequence above, 5 is the first term, 14 is the second, and so on.

nth Term Notation
Written as aₙ, this represents the nth term of a sequence. For example, a₁ is the first term, a₂ the second, etc.

Difference Table
A method to analyze sequences by calculating the differences between successive terms, which helps in identifying linear, quadratic, or other polynomial patterns.

First, Second, Third Differences
These are found in successive rows of a difference table. First differences show how the terms change, second differences show changes of the first differences, and so on.

nth-Term Formula
Used to calculate or predict any term in a sequence, particularly once a pattern is identified.

Recursive Definition
Defines each term based on previous terms. Example: the Fibonacci sequence uses recursion.

Fibonacci Sequence
A special sequence where each term is the sum of the two preceding terms. Defined as:
F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3.

3. Problem-Solving Strategies

Polya’s Four-Step Strategy

Understand the problem
Devise a plan
Carry out the plan
Review the solution

This strategy provides a structured approach for solving word problems and logical reasoning tasks.

4. Sets and Set Theory

Set
A collection of distinct objects, called elements or members.

Ways to Represent Sets

Word Description: Describing the elements using words
Roster Method: Listing elements in braces: {1, 2, 3}
Set Builder Notation: Describes elements using a rule: {x | x is a natural number < 10}

Types of Numbers (Real Number System)

Natural Numbers (N): {1, 2, 3,...}
Whole Numbers (W): {0, 1, 2,...}
Integers (I): {..., -3, -2, -1, 0, 1, 2,...}
Rational Numbers (Q): Fractions or repeating/terminating decimals
Irrational Numbers (I): Non-repeating, non-terminating decimals (e.g., √2, π)

Special Sets

Empty/Null Set (∅): A set with no elements
Well-defined Set: Clear membership criteria
Finite Set: Has a countable number of elements
Cardinal Number: Number of elements in a finite set, denoted n(A)

5. Set Operations and Properties

Equal Sets: A = B if both have identical elements
Equivalent Sets: A ~ B if both have the same number of elements
Complement (A′): All elements in the universal set U not in A
Subset (A⊆B): Every element in A is also in B
Proper Subset (A⊂B): A⊆B but A ≠ B
Number of Subsets: A set with n elements has 2ⁿ subsets

Venn Diagrams
Used to visualize sets and their relationships.

Rectangle = Universal set
Circles = Subsets

Set Operations

Union (A∪B): All elements in A or B or both
Intersection (A∩B): Elements common to both A and B
Difference (A − B): Elements in A but not in B

De Morgan’s Laws

(A ∪ B)′ = A′ ∩ B′
(A ∩ B)′ = A′ ∪ B′

Properties of Sets

Commutative: A ∪ B = B ∪ A; A ∩ B = B ∩ A
Associative: (A ∪ B) ∪ C = A ∪ (B ∪ C)
Distributive: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

Inclusion-Exclusion Principle
For finite sets A and B:
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

6. Infinite Sets and Cardinality

One-to-One Correspondence
A pairing of elements from one set to another such that each element is matched with exactly one element from the other set.

Infinite Set
A set is infinite if it can be placed in one-to-one correspondence with a proper subset of itself.

Transfinite Numbers
ℵ₀ (Aleph-null) represents the cardinality of countably infinite sets (like natural numbers).

Countable Set
A set that is either finite or equivalent to the set of natural numbers.

Cantor’s Theorem
The set of all subsets of any set S has a larger cardinality than S itself.

7. Logic and Statements

Statement
A declarative sentence that is either true or false.

Compound Statement
Formed by combining statements using logical connectors like and (∧), or (∨), not (¬), and if…then (→).

Truth Tables
Show the truth values for compound statements based on all possible combinations of component truths.

Conditional Statements
"If p, then q" is written symbolically as p → q.

Antecedent = p
Consequent = q
Truth Value: False only when p is true and q is false.

Negation of Conditional
~(p → q) ≡ p ∧ ~q

Biconditional (p ↔ q)
True when both statements are either true or both false. Equivalent to (p → q) ∧ (q → p)

Related Conditional Forms

Converse: q → p
Inverse: ~p → ~q
Contrapositive: ~q → ~p

8. Arguments and Validity

Argument Structure
Consists of premises and a conclusion. Validity means the conclusion is logically supported if the premises are all true.

Truth Table for Validity

Translate the argument into symbols.
Build a truth table.
Check that whenever all premises are true, the conclusion is also true.

MATH C277: Finite Mathematics

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Free MATH C277: Finite Mathematics Questions

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1. Reasoning and Conjectures

2. Sequences and Pattern Recognition

3. Problem-Solving Strategies

4. Sets and Set Theory

5. Set Operations and Properties

6. Infinite Sets and Cardinality

7. Logic and Statements

8. Arguments and Validity

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