MATH C277: Finite Mathematics

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Free MATH C277: Finite Mathematics Questions

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.

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.

Round 35,325.738 to the nearest hundredth and explain your answer.

35,325.74; The 8 to the right of the 3 rounds the 3 up to a 4, and digits to the right of this decimal position are then left off.
35,300; To round, change the digits after the hundreds place to zeros.
35,300; The 2 to the right of 3 is less than 5, so 3 does not change.
35,330; The 5 to the right of the 2 rounds the 2 up to a 3, and digits to the right of this decimal position are then changed to zeros.

Explanation

Explanation
The hundredth place in 35,325.738 is the second digit to the right of the decimal, which is 3. The digit immediately to its right (the thousandth place) is 8. Since 8 is greater than or equal to 5, the hundredth digit increases from 3 to 4. All digits to the right are dropped.
Correct answer
35,325.74

Which property is illustrated?
( a(bc) = (ab)c )

Associative Property
Closure Property
Commutative Property
Distributive Property

Explanation

Explanation
The equation shows that when multiplying three numbers, the way the numbers are grouped does not change the result. In ( a(bc) ) and ( (ab)c ), the factors stay in the same order, but the grouping changes. This demonstrates that multiplication is associative, meaning parentheses can be moved without affecting the product.
Correct answer
Associative Property

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

$\frac{1}{1440}$

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.

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 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.

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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