MATH C277: Finite Mathematics

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Free MATH C277: Finite Mathematics Questions

Truncate 348.856 to the nearest tenths place and explain your answer.

348.8; Truncation leaves off digits past the tenths place.
348.9; Truncation rounds and then leaves off digits past the tenths place.
349; Truncation rounds and then leaves off digits starting with the tenths place.
348; Truncation leaves off digits starting with the tenths place.

Explanation

Correct Answer:

348.8; Truncation leaves off digits past the tenths place.

Explanation:

Truncation means cutting off digits without rounding. To truncate 348.856 to the tenths place, we simply keep the tenths digit (which is 8) and remove everything after it (i.e., the hundredths and thousandths, which are 5 and 6). We do not round up based on those digits.

So, 348.856 truncated to the tenths place is 348.8.

Why Other Options Are Wrong:

348.9

This result involves rounding the tenths place (since the hundredths is 5), but truncation does not round — it only chops off digits.

349

This rounds the entire number up and drops all decimal digits, which is incorrect. Truncation at the tenths place does not affect whole numbers.

348

This removes everything from the tenths place onward, which is incorrect for truncation to the tenths place. This would be truncating to the ones place instead.

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

$\frac{1}{1440}$

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 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.

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.

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

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

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

2[18 - 6(3² - 5)/8 + 4]

-113/2
8
20
38

Explanation

Explanation
Start inside the parentheses: 3² = 9, and 9 - 5 = 4. Multiply to get 6 × 4 = 24, then divide by 8 to get 3. Substitute back into the expression: 18 - 3 + 4 = 19. Finally, multiply by 2 to get 2 × 19 = 38.
Correct answer
38

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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