Choose the best answer. If necessary, use the paper you were given.
A bowl contains 6 green grapes, 10 red grapes, and 8 black grapes.Which of the following is the correct calculation for the probability of choosing a red grape and then without putting the red grape back into the bowl, choosing a green grape?
- A. 10/24+6/24
- B. 10/24+6/23
- C. 10/24*6/24
- D. 10/24*6/23
Correct Answer & Rationale
Correct Answer: D
To determine the probability of selecting a red grape followed by a green grape without replacement, the first step involves calculating the probability of the first event (selecting a red grape). There are 10 red grapes out of a total of 24 grapes, giving a probability of 10/24. After choosing a red grape, there are now 23 grapes left in the bowl, including 6 green grapes. Thus, the probability of then selecting a green grape is 6/23. Option A incorrectly adds the probabilities, which is not appropriate for sequential events. Option B uses the correct second probability but fails to multiply the probabilities of the two events. Option C mistakenly adds both probabilities instead of multiplying them. Only option D correctly multiplies the probabilities of the two dependent events.
To determine the probability of selecting a red grape followed by a green grape without replacement, the first step involves calculating the probability of the first event (selecting a red grape). There are 10 red grapes out of a total of 24 grapes, giving a probability of 10/24. After choosing a red grape, there are now 23 grapes left in the bowl, including 6 green grapes. Thus, the probability of then selecting a green grape is 6/23. Option A incorrectly adds the probabilities, which is not appropriate for sequential events. Option B uses the correct second probability but fails to multiply the probabilities of the two events. Option C mistakenly adds both probabilities instead of multiplying them. Only option D correctly multiplies the probabilities of the two dependent events.
Other Related Questions
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.
The expressions x - 2 and x + 3 represent the length and width of a rectangle, respectively. If the area of the rectangle is 24, what is the perimeter of the rectangle?
- A. 20
- B. 22
- C. 24
- D. 28
Correct Answer & Rationale
Correct Answer: B
To find the perimeter of the rectangle, first calculate its dimensions using the area formula. The area is given by multiplying length and width: \[ (x - 2)(x + 3) = 24 \] Expanding this, we get: \[ x^2 + x - 6 = 24 \implies x^2 + x - 30 = 0 \] Factoring yields: \[ (x - 5)(x + 6) = 0 \implies x = 5 \text{ (valid)} \text{ or } x = -6 \text{ (not valid)} \] Using \(x = 5\), the dimensions are \(3\) (length) and \(8\) (width). The perimeter is: \[ 2(3 + 8) = 22 \] Options A (20), C (24), and D (28) do not match the calculated perimeter of 22, confirming they are incorrect.
To find the perimeter of the rectangle, first calculate its dimensions using the area formula. The area is given by multiplying length and width: \[ (x - 2)(x + 3) = 24 \] Expanding this, we get: \[ x^2 + x - 6 = 24 \implies x^2 + x - 30 = 0 \] Factoring yields: \[ (x - 5)(x + 6) = 0 \implies x = 5 \text{ (valid)} \text{ or } x = -6 \text{ (not valid)} \] Using \(x = 5\), the dimensions are \(3\) (length) and \(8\) (width). The perimeter is: \[ 2(3 + 8) = 22 \] Options A (20), C (24), and D (28) do not match the calculated perimeter of 22, confirming they are incorrect.
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.
Which of the following could be the function graphed above?
- A. f(x)=x+1
- B. f(x)=x-1
- C. f(x)=|x|+1
- D. f(x)=x-1
Correct Answer & Rationale
Correct Answer: C
Option C, \( f(x) = |x| + 1 \), accurately represents a V-shaped graph that opens upwards, with its vertex at (0, 1). This aligns with the characteristics of the graph shown. Option A, \( f(x) = x + 1 \), is a linear function with a slope of 1, resulting in a straight line, which does not match the V-shape. Option B, \( f(x) = x - 1 \), is another linear function with a slope of 1, also producing a straight line that does not fit the graph. Option D, \( f(x) = x - 1 \), is identical to Option B and shares the same linear characteristics, further confirming it cannot represent the V-shaped graph.
Option C, \( f(x) = |x| + 1 \), accurately represents a V-shaped graph that opens upwards, with its vertex at (0, 1). This aligns with the characteristics of the graph shown. Option A, \( f(x) = x + 1 \), is a linear function with a slope of 1, resulting in a straight line, which does not match the V-shape. Option B, \( f(x) = x - 1 \), is another linear function with a slope of 1, also producing a straight line that does not fit the graph. Option D, \( f(x) = x - 1 \), is identical to Option B and shares the same linear characteristics, further confirming it cannot represent the V-shaped graph.