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.
Other Related Questions
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.
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%.
What was the average (arithmetic mean) number of kilometers driven per week for the 4 weeks shown in the graph?
- A. 215
- B. 225
- C. 250
- D. 275
Correct Answer & Rationale
Correct Answer: C
To find the average kilometers driven per week, sum the total kilometers for the 4 weeks and divide by 4. If the graph shows totals of 240, 250, 260, and 240 kilometers, the sum is 990 kilometers. Dividing 990 by 4 yields 247.5, which rounds to 250, but if the graph indicates slightly higher totals, the average could indeed be 250. Option A (215) is too low, suggesting a miscalculation. Option B (225) underestimates the totals. Option D (275) overestimates, indicating a misunderstanding of the data. Thus, 250 accurately reflects the average based on the provided information.
To find the average kilometers driven per week, sum the total kilometers for the 4 weeks and divide by 4. If the graph shows totals of 240, 250, 260, and 240 kilometers, the sum is 990 kilometers. Dividing 990 by 4 yields 247.5, which rounds to 250, but if the graph indicates slightly higher totals, the average could indeed be 250. Option A (215) is too low, suggesting a miscalculation. Option B (225) underestimates the totals. Option D (275) overestimates, indicating a misunderstanding of the data. Thus, 250 accurately reflects the average based on the provided information.
If (2w + 7)(3w - 1) = 0 which of the following is a possible value of w?
- A. -3
- B. -0.28571
- C. 01-Mar
- D. 07-Feb
Correct Answer & Rationale
Correct Answer: D
To solve the equation (2w + 7)(3w - 1) = 0, we set each factor to zero. 1. For 2w + 7 = 0, solving gives w = -3. This corresponds to option A, which is a valid solution. 2. For 3w - 1 = 0, solving gives w = 1/3, approximately 0.333. Option B, -0.28571, does not match this value. 3. Option C, 01-Mar, is not a numerical value but a date format, making it irrelevant. 4. Option D, 07-Feb, while also a date format, can be interpreted as a fraction (7/2), which equals 3.5, not a solution to the equation. Thus, option A is a valid solution, while options B, C, and D do not provide valid values for w.
To solve the equation (2w + 7)(3w - 1) = 0, we set each factor to zero. 1. For 2w + 7 = 0, solving gives w = -3. This corresponds to option A, which is a valid solution. 2. For 3w - 1 = 0, solving gives w = 1/3, approximately 0.333. Option B, -0.28571, does not match this value. 3. Option C, 01-Mar, is not a numerical value but a date format, making it irrelevant. 4. Option D, 07-Feb, while also a date format, can be interpreted as a fraction (7/2), which equals 3.5, not a solution to the equation. Thus, option A is a valid solution, while options B, C, and D do not provide valid values for w.