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
In triangle ABC above, AC ||DE. If AD = 2x - 1 and AC = 3x - 1 , what is the value of x ?
- A. 3
- B. 4
- C. 5
- D. 6
Correct Answer & Rationale
Correct Answer: A
In triangle ABC, since AC is parallel to DE, the segments AD and AC are proportional. This relationship can be expressed as AD = AC. Substituting the expressions gives us the equation: 2x - 1 = 3x - 1. Solving for x, we simplify to 2x - 3x = -1 + 1, leading to -x = 0, or x = 3. Option B (4), C (5), and D (6) do not satisfy the equation derived from the parallel lines, making them incorrect. Only x = 3 maintains the equality, confirming the proportional relationship in the triangle.
In triangle ABC, since AC is parallel to DE, the segments AD and AC are proportional. This relationship can be expressed as AD = AC. Substituting the expressions gives us the equation: 2x - 1 = 3x - 1. Solving for x, we simplify to 2x - 3x = -1 + 1, leading to -x = 0, or x = 3. Option B (4), C (5), and D (6) do not satisfy the equation derived from the parallel lines, making them incorrect. Only x = 3 maintains the equality, confirming the proportional relationship in the triangle.
The sum of n and the product 3 times n is 12. What is the value of n?
- A. 2
- B. 3
- C. 4
- D. 4 ½
Correct Answer & Rationale
Correct Answer: B
To solve the equation formed by the problem statement, we express it as \( n + 3n = 12 \), which simplifies to \( 4n = 12 \). Dividing both sides by 4 gives \( n = 3 \). Option A (2) does not satisfy the equation, as substituting it results in \( 2 + 6 = 8 \), which is incorrect. Option C (4) leads to \( 4 + 12 = 16 \), also incorrect. Option D (4 ½) results in \( 4.5 + 13.5 = 18 \), which is too high. Thus, only \( n = 3 \) fulfills the original equation, confirming its validity.
To solve the equation formed by the problem statement, we express it as \( n + 3n = 12 \), which simplifies to \( 4n = 12 \). Dividing both sides by 4 gives \( n = 3 \). Option A (2) does not satisfy the equation, as substituting it results in \( 2 + 6 = 8 \), which is incorrect. Option C (4) leads to \( 4 + 12 = 16 \), also incorrect. Option D (4 ½) results in \( 4.5 + 13.5 = 18 \), which is too high. Thus, only \( n = 3 \) fulfills the original equation, confirming its validity.
During a sale, the regular price of a pair of running shoes is reduced by 20 percent. $64.00, what is the regular price of the running shoes?
- A. $48.00
- B. $51.20
- C. $76.80
- D. $80.00
Correct Answer & Rationale
Correct Answer: D
To find the regular price of the running shoes, we need to determine what amount, when reduced by 20%, equals $64.00. This can be calculated using the formula: Sale Price = Regular Price × (1 - Discount Rate). Here, the discount rate is 20%, or 0.20. Therefore, the equation becomes $64.00 = Regular Price × 0.80. Solving for Regular Price gives us $64.00 / 0.80 = $80.00. Option A ($48.00) is incorrect because it suggests a much larger discount than 20%. Option B ($51.20) miscalculates the reduction, indicating a 36% discount. Option C ($76.80) inaccurately reflects a smaller discount, resulting in an incorrect sale price. Thus, only option D correctly represents the regular price before the 20% reduction.
To find the regular price of the running shoes, we need to determine what amount, when reduced by 20%, equals $64.00. This can be calculated using the formula: Sale Price = Regular Price × (1 - Discount Rate). Here, the discount rate is 20%, or 0.20. Therefore, the equation becomes $64.00 = Regular Price × 0.80. Solving for Regular Price gives us $64.00 / 0.80 = $80.00. Option A ($48.00) is incorrect because it suggests a much larger discount than 20%. Option B ($51.20) miscalculates the reduction, indicating a 36% discount. Option C ($76.80) inaccurately reflects a smaller discount, resulting in an incorrect sale price. Thus, only option D correctly represents the regular price before the 20% reduction.
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.