MATH C277: Finite Mathematics

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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 following numbers are listed according to a pattern: $\frac{1}{2}, \frac{1}{4}, \frac{1}{12}, \frac{1}{48}, \frac{1}{240}$ … Which of the following could be the next number in the pattern?

$\frac{1}{480}$
$\frac{1}{960}$
$\frac{1}{1200}$
$\frac{1}{1440}$

Explanation

Explanation

Look at how the denominators change: 2 → 4 → 12 → 48 → 240. Each step multiplies by the next integer: 2 × 2 = 4, 4 × 3 = 12, 12 × 4 = 48, 48 × 5=240. So the next denominator should be 240 × 6 = 1440. Therefore, the next term is $\frac{1}{1440}$ .

Correct answer

$\frac{1}{1440}$

Let p and q represent the following simple statements:
p: They are eating lunch.
q: It is noon.
Match each English statement with its symbolic equivalent from the options below.
Each symbolic option corresponds to a logical expression involving p and q.
A. p→q
B. q→p
C. ∼p→∼q
D. ∼q→∼p
E. q↔p
F. p↔q
G. ∼p↔∼q
Match:
They eat lunch if and only if it is noon
If it is noon, then they are eating lunch
They are not eating lunch if it is not noon
If they are eating lunch, then it is noon

F, B, D, A
E, A, D, B
G, C, B, D
F, C, E, A

Explanation

Correct Answer:

F, B, D, A

Explanation:

1. They eat lunch if and only if it is noon

This is a biconditional statement, meaning both p and q must be either true or false for the statement to be true.

Symbolically: p ↔ q or q ↔ p — both are logically equivalent.

The matching option is F: p ↔ q

2. If it is noon, then they are eating lunch

This is a conditional statement where q → p.

The matching option is B: q → p

3. They are not eating lunch if it is not noon

This is another conditional statement written contrapositively as:

If ~q, then ~p → ~q → ~p

The matching option is D: ~q → ~p

4. If they are eating lunch, then it is noon

This is a standard conditional: p → q

The matching option is A: p → q

Why Other Options Are Wrong:

E, A, D, B:

Incorrect because option 1 ("They eat lunch if and only if it is noon") is matched with E, which uses q ↔ p. While this is logically equivalent to p ↔ q, the remaining matches do not align correctly: "If they are eating lunch, then it is noon" corresponds to A: p → q, but in this answer it's placed last, after a mismatch.

G, C, B, D:

Incorrect because option 1 is matched with G, which uses ~p ↔ ~q—this is the inverse, not logically equivalent to the biconditional p ↔ q. Also, the third match is B: q → p, which misrepresents "They are not eating lunch if it is not noon."

F, C, E, A:

Incorrect because C: ~p → ~q is not the correct contrapositive form of p → q. Additionally, E: q ↔ p in the third match is incorrectly matched to “They are not eating lunch if it is not noon,” which requires a conditional, not a biconditional.

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.

What is the least common multiple of 504 and 540?

272,160
7,560
50,400
4,540

Explanation

Correct Answer:

7,560

Explanation:

To find the least common multiple (LCM) of 504 and 540, use the formula:

$L . C . M (a, b) = \frac{a \times b}{GCF (a, b)}$

First, find the greatest common factor (GCF).

504 = 2³× 3² × 7

540 = 2²× 3³ ×5

GCF = 2²×3²= 36

Now calculate the LCM:

LCM = $\frac{504 \times 540}{36}$ =7,560

Why Other Options Are Wrong:

272,160

This is the product of 504 × 540 without dividing by the GCF. It is the maximum common multiple, not the least. It represents the numerator before simplification.

50,400

This is a multiple of both numbers but not the smallest one. It results from incorrect simplification and exceeds the correct LCM.

4,540

This number is neither a multiple of 504 nor 540. It is unrelated to the LCM calculation and likely included to mislead based on its similar digits.

Place the following real numbers in increasing order:
$\frac{16}{\sqrt{32}}$ , $\frac{12}{\sqrt{8}}$ , $\frac{2}{\sqrt{2}}$ , $\frac{16}{\sqrt{8}}$

$\frac{2}{\sqrt{2}}, \frac{12}{\sqrt{8}}, \frac{16}{\sqrt{32}}, \frac{16}{\sqrt{8}}$
$\frac{16}{\sqrt{32}}, \frac{2}{\sqrt{2}}, \frac{12}{\sqrt{8}}, \frac{16}{\sqrt{8}}$
$\frac{2}{\sqrt{2}}, \frac{16}{\sqrt{32}}, \frac{12}{\sqrt{8}}, \frac{16}{\sqrt{8}}$
$\frac{2}{\sqrt{2}}, \frac{16}{\sqrt{8}}, \frac{12}{\sqrt{8}}, \frac{16}{\sqrt{32}}$

Explanation

Simplify $\frac{5 \times 10^{2}}{2 \times 10^{5}}$

2.5 × 10³
2.5 × 10^-3
2.5 × 10⁷
2.5 × 10^-7

Explanation

Explanation
First divide the coefficients: 5/2 = 2.5. Next apply the laws of exponents by subtracting the exponents when dividing powers of ten: 10² ÷ 10⁵ = 10^2-5 = 10^-3. Combine the results to get 2.5 × 10^-3, which matches one of the answer choices.
Correct answer
2.5 × 10^-3

Which property in the real number system is illustrated in this problem?
7⊕8=8⊕7
5 = 5

Associative Property
Closure
Commutative Property
Distributive Property

Explanation

Correct Answer:

Commutative Property

Explanation:

The commutative property states that changing the order of numbers in an operation does not affect the result. This applies to both addition and multiplication. In the expression shown, 7⊕8=8⊕7, the numbers are reversed in order, yet the equality remains true, demonstrating that the operation is commutative.

Why Other Options Are Wrong:

Associative Property

This property refers to changing grouping (parentheses), not order. It involves expressions like (a + b) + c = a + (b + c). Since no grouping is shown, it doesn’t apply here.

Closure

Closure means that performing an operation on two real numbers results in another real number. While true in general, it doesn’t explain the equality shown in this problem.

Distributive Property

The distributive property involves expressions like a (b + c) = ab + ac. This has nothing to do with reversing operands, so it is not applicable here.

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.

10.

$3 \frac{3}{12} + 5 \frac{1}{8}$
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.

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