MATH C277: Finite Mathematics

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Free MATH C277: Finite Mathematics Questions

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.

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

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.

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.

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).

Multiply 25.14 by 62.13.
Round your answer to the nearest hundredth and explain the answer.

1,561.948; because the 8 does not change because 2 is less than 5
1561.94; because digits to the right of 4 are dropped
1,561.95; because the 4 rounds up to 5
1,600; because the 5 rounds up to 6

Explanation

Correct Answer:

1,561.95; because the 4 rounds up to 5

Explanation:

Step 1: Multiply

25.14 × 62.13 = 1,561.9482

Step 2: Round to the nearest hundredth

The hundredth digit is 4 and the digit after it (the thousandths place) is 8, which is greater than 5.

So we round the 4 up to 5, giving:

1,561.95

Why Other Options Are Wrong:

1,561.948

This is the unrounded product. The question specifically asks for rounding to the nearest hundredth.

1561.94

This would be correct if the digit after the hundredths place were less than 5. But here it is 8, so we must round up.

1,600

This is a rough estimate, not a precise rounding to the hundredth. The correct rounding process requires attention to decimal places, not nearest hundred.

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.

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.

10.

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.

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MATH C277: Finite Mathematics

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