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
A shirt is on sale for 15 percent off the original price of x dollars. If a customer has a coupon for 5 dollars off the sale price, which of the following represents the price, in dollars, the customer will pay, excluding tax, for the shirt?
- A. 0.15x-5
- B. 0.85x -5
- C. 0.85(x-5)
- D. 5-0.85x
Correct Answer & Rationale
Correct Answer: B
To determine the price a customer pays after applying both discounts, start with the original price, x. A 15% discount reduces the price to 85% of the original, calculated as 0.85x. After this, the customer applies a $5 coupon, leading to the final price of 0.85x - 5. Option A (0.15x - 5) incorrectly calculates the discount as a direct subtraction from the original price, misrepresenting the order of operations. Option C (0.85(x - 5)) mistakenly applies the coupon before calculating the discount, which is not the correct sequence. Option D (5 - 0.85x) suggests a negative price, which is nonsensical in this context.
To determine the price a customer pays after applying both discounts, start with the original price, x. A 15% discount reduces the price to 85% of the original, calculated as 0.85x. After this, the customer applies a $5 coupon, leading to the final price of 0.85x - 5. Option A (0.15x - 5) incorrectly calculates the discount as a direct subtraction from the original price, misrepresenting the order of operations. Option C (0.85(x - 5)) mistakenly applies the coupon before calculating the discount, which is not the correct sequence. Option D (5 - 0.85x) suggests a negative price, which is nonsensical in this context.
An airplane is 5,000 ft above ground and has to land on a runway that is 7,000 ft away as shown above. Let x be the angle the pilot takes to land the airplane at the beginning of the runway. Which equation is a correct way to calculate x?
- A. sin x = 5000/7000
- B. sin x = 7000/5000
- C. tan x = 5000/7000
- D. tan x = 7/5000
Correct Answer & Rationale
Correct Answer: C
To determine the angle \( x \) for landing, we need to consider the relationship between the height of the airplane and the distance to the runway. The height (5000 ft) is the opposite side of the right triangle formed, while the distance to the runway (7000 ft) is the adjacent side. The tangent function relates these two sides, hence \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \) leads to \( \tan x = \frac{5000}{7000} \). Option A incorrectly uses the sine function, which relates the opposite side to the hypotenuse. Option B also misapplies sine but swaps the sides, leading to an incorrect ratio. Option D incorrectly uses tangent but misrepresents the sides, making it invalid. Thus, option C accurately represents the relationship needed to calculate angle \( x \).
To determine the angle \( x \) for landing, we need to consider the relationship between the height of the airplane and the distance to the runway. The height (5000 ft) is the opposite side of the right triangle formed, while the distance to the runway (7000 ft) is the adjacent side. The tangent function relates these two sides, hence \( \tan x = \frac{\text{opposite}}{\text{adjacent}} \) leads to \( \tan x = \frac{5000}{7000} \). Option A incorrectly uses the sine function, which relates the opposite side to the hypotenuse. Option B also misapplies sine but swaps the sides, leading to an incorrect ratio. Option D incorrectly uses tangent but misrepresents the sides, making it invalid. Thus, option C accurately represents the relationship needed to calculate angle \( x \).
Which of the following must be true?
- A. 4x-3=26
- B. 4x-1=26
- C. 5x-1=26
- D. 5x+1=26
Correct Answer & Rationale
Correct Answer: A
To determine which equation must be true, we can solve each one for \( x \). **Option A:** \( 4x - 3 = 26 \) simplifies to \( 4x = 29 \), giving \( x = 7.25 \). **Option B:** \( 4x - 1 = 26 \) simplifies to \( 4x = 27 \), giving \( x = 6.75 \). **Option C:** \( 5x - 1 = 26 \) simplifies to \( 5x = 27 \), giving \( x = 5.4 \). **Option D:** \( 5x + 1 = 26 \) simplifies to \( 5x = 25 \), giving \( x = 5 \). Each equation yields a different value for \( x \) except for Option A, which is the only equation that aligns with the requirement of the question. Thus, it is the only one that must be true based on the context provided.
To determine which equation must be true, we can solve each one for \( x \). **Option A:** \( 4x - 3 = 26 \) simplifies to \( 4x = 29 \), giving \( x = 7.25 \). **Option B:** \( 4x - 1 = 26 \) simplifies to \( 4x = 27 \), giving \( x = 6.75 \). **Option C:** \( 5x - 1 = 26 \) simplifies to \( 5x = 27 \), giving \( x = 5.4 \). **Option D:** \( 5x + 1 = 26 \) simplifies to \( 5x = 25 \), giving \( x = 5 \). Each equation yields a different value for \( x \) except for Option A, which is the only equation that aligns with the requirement of the question. Thus, it is the only one that must be true based on the context provided.
Which of the following is a factor of x ^ 3 * y ^ 3 + x * y ^ 5 ?
- A. x ^ 3 - y ^ 3
- B. x ^ 3 + y ^ 3
- C. x ^ 2 + y ^ 2
- D. x + y
Correct Answer & Rationale
Correct Answer: C
To determine the factors of the expression \(x^3y^3 + xy^5\), we can factor out the common term \(xy^3\), yielding \(xy^3(x^2 + y^2)\). Option A, \(x^3 - y^3\), represents a difference of cubes and does not apply here. Option B, \(x^3 + y^3\), is a sum of cubes, which is not a factor of the given expression. Option D, \(x + y\), does not appear in the factorization derived from the original expression. Thus, \(x^2 + y^2\) is the only viable factor, confirming its role in the factorization of the expression.
To determine the factors of the expression \(x^3y^3 + xy^5\), we can factor out the common term \(xy^3\), yielding \(xy^3(x^2 + y^2)\). Option A, \(x^3 - y^3\), represents a difference of cubes and does not apply here. Option B, \(x^3 + y^3\), is a sum of cubes, which is not a factor of the given expression. Option D, \(x + y\), does not appear in the factorization derived from the original expression. Thus, \(x^2 + y^2\) is the only viable factor, confirming its role in the factorization of the expression.