Choose the best answer. If necessary, use the paper you were given.
Which of the following is equivalent to 12x +8?
- A. 4(3x+2)
- B. 4(3x+8)
- C. 4(3x+2x)
- D. 20x
Correct Answer & Rationale
Correct Answer: A
To determine the equivalent expression for \(12x + 8\), we can factor out the greatest common factor, which is 4. Option A, \(4(3x + 2)\), simplifies to \(12x + 8\) when distributed, making it equivalent to the original expression. Option B, \(4(3x + 8)\), simplifies to \(12x + 32\), which is not equivalent. Option C, \(4(3x + 2x)\), simplifies to \(4(5x)\) or \(20x\), which is also not equivalent. Option D, \(20x\), does not match the original expression either. Thus, only option A is correct.
To determine the equivalent expression for \(12x + 8\), we can factor out the greatest common factor, which is 4. Option A, \(4(3x + 2)\), simplifies to \(12x + 8\) when distributed, making it equivalent to the original expression. Option B, \(4(3x + 8)\), simplifies to \(12x + 32\), which is not equivalent. Option C, \(4(3x + 2x)\), simplifies to \(4(5x)\) or \(20x\), which is also not equivalent. Option D, \(20x\), does not match the original expression either. Thus, only option A is correct.
Other Related Questions
If a number from set M is selected at random, what is the probability that the number selected will be a factor of 12?
- A. 0.1
- B. 0.2
- C. 0.4
- D. 0.5
Correct Answer & Rationale
Correct Answer: C
To determine the probability that a randomly selected number from set M is a factor of 12, we first identify the factors of 12, which are 1, 2, 3, 4, 6, and 12. If set M consists of 6 numbers (1 through 6), then 4 of these (1, 2, 3, and 4) are factors of 12. Thus, the probability is 4 out of 6, simplifying to 0.4. Option A (0.1) underestimates the number of factors. Option B (0.2) suggests only 2 factors, which is incorrect. Option D (0.5) implies 3 factors, also inaccurate. Therefore, 0.4 accurately represents the proportion of factors of 12 in the set.
To determine the probability that a randomly selected number from set M is a factor of 12, we first identify the factors of 12, which are 1, 2, 3, 4, 6, and 12. If set M consists of 6 numbers (1 through 6), then 4 of these (1, 2, 3, and 4) are factors of 12. Thus, the probability is 4 out of 6, simplifying to 0.4. Option A (0.1) underestimates the number of factors. Option B (0.2) suggests only 2 factors, which is incorrect. Option D (0.5) implies 3 factors, also inaccurate. Therefore, 0.4 accurately represents the proportion of factors of 12 in the set.
Doreen bought a dress priced at $89 and a skirt priced at $36. She paid a total of $135 for the dress and the skirt, including sales tax. What was the sales tax rate?
- A. 6%
- B. 7%
- C. 8%
- D. 9%
Correct Answer & Rationale
Correct Answer: C
To determine the sales tax rate, first calculate the total cost of the dress and skirt without tax: $89 + $36 = $125. Doreen paid $135, which means the sales tax was $135 - $125 = $10. To find the sales tax rate, divide the tax amount by the pre-tax total: $10 / $125 = 0.08, or 8%. Option A (6%) is incorrect as it would result in a lower tax amount. Option B (7%) also yields a tax amount that is too low. Option D (9%) would produce a tax amount exceeding $10, making it incorrect. Thus, the only option that accurately reflects the calculated sales tax rate is 8%.
To determine the sales tax rate, first calculate the total cost of the dress and skirt without tax: $89 + $36 = $125. Doreen paid $135, which means the sales tax was $135 - $125 = $10. To find the sales tax rate, divide the tax amount by the pre-tax total: $10 / $125 = 0.08, or 8%. Option A (6%) is incorrect as it would result in a lower tax amount. Option B (7%) also yields a tax amount that is too low. Option D (9%) would produce a tax amount exceeding $10, making it incorrect. Thus, the only option that accurately reflects the calculated sales tax rate is 8%.
If the values of x and y are negative, which of the following values must be positive?
- A. x²-y²
- B. x/y
- C. x+y
- D. x-y
Correct Answer & Rationale
Correct Answer: B
When both x and y are negative, the quotient \( x/y \) results in a positive value. This is because dividing a negative number by another negative number yields a positive outcome. Option A, \( x^2 - y^2 \), can be either positive or negative depending on the magnitudes of x and y; thus, it is not guaranteed to be positive. Option C, \( x + y \), is the sum of two negative numbers, which will always be negative. Option D, \( x - y \), involves subtracting a negative (y) from another negative (x), which can also yield a negative or zero result, depending on their values. Only \( x/y \) is assuredly positive.
When both x and y are negative, the quotient \( x/y \) results in a positive value. This is because dividing a negative number by another negative number yields a positive outcome. Option A, \( x^2 - y^2 \), can be either positive or negative depending on the magnitudes of x and y; thus, it is not guaranteed to be positive. Option C, \( x + y \), is the sum of two negative numbers, which will always be negative. Option D, \( x - y \), involves subtracting a negative (y) from another negative (x), which can also yield a negative or zero result, depending on their values. Only \( x/y \) is assuredly positive.
If the function g is defined by g (x) = x/(x+1)', which of the following is true?
- A. g (10) <g (20)
- B. g (20) <g (10)
- C. g(0) =1
- D. g(1)=0
Correct Answer & Rationale
Correct Answer: A
To analyze the function \( g(x) = \frac{x}{x+1} \), we first observe its behavior as \( x \) increases. The function \( g(x) \) is a rational function that approaches 1 as \( x \) approaches infinity. For option A, evaluating \( g(10) \) and \( g(20) \): - \( g(10) = \frac{10}{11} \approx 0.909 \) - \( g(20) = \frac{20}{21} \approx 0.952 \) Since \( 0.909 < 0.952 \), option A is true. For option B, it incorrectly suggests \( g(20) < g(10) \), which contradicts the findings. Option C states \( g(0) = 1 \), but \( g(0) = 0 \), making this option false. Option D claims \( g(1) = 0 \), while \( g(1) = \frac{1}{2} \), which is also incorrect. Thus, only option A holds true.
To analyze the function \( g(x) = \frac{x}{x+1} \), we first observe its behavior as \( x \) increases. The function \( g(x) \) is a rational function that approaches 1 as \( x \) approaches infinity. For option A, evaluating \( g(10) \) and \( g(20) \): - \( g(10) = \frac{10}{11} \approx 0.909 \) - \( g(20) = \frac{20}{21} \approx 0.952 \) Since \( 0.909 < 0.952 \), option A is true. For option B, it incorrectly suggests \( g(20) < g(10) \), which contradicts the findings. Option C states \( g(0) = 1 \), but \( g(0) = 0 \), making this option false. Option D claims \( g(1) = 0 \), while \( g(1) = \frac{1}{2} \), which is also incorrect. Thus, only option A holds true.