Choose the best answer. If necessary, use the paper you were given.
The average of 5 numbers is 11. Which of the following must be true?
- A. The range of the 5 numbers is 55.
- B. The sum of the 5 numbers is 55.
- C. The median of 5 numbers is 11.
- D. The mode of 5 numbers is 11.
Correct Answer & Rationale
Correct Answer: B
To find the average of 5 numbers, you sum them and divide by 5. If the average is 11, the total sum must be 5 times 11, which equals 55. Thus, option B is true. Option A is incorrect; the range is the difference between the highest and lowest numbers, which can vary regardless of the average. Option C is also not necessarily true; the median, or middle value, can differ from the average depending on the distribution of the numbers. Option D is incorrect; the mode, or most frequently occurring number, does not have to be the same as the average.
To find the average of 5 numbers, you sum them and divide by 5. If the average is 11, the total sum must be 5 times 11, which equals 55. Thus, option B is true. Option A is incorrect; the range is the difference between the highest and lowest numbers, which can vary regardless of the average. Option C is also not necessarily true; the median, or middle value, can differ from the average depending on the distribution of the numbers. Option D is incorrect; the mode, or most frequently occurring number, does not have to be the same as the average.
Other Related Questions
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.
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.
Which equation is a correct way to calculate x?
- A. sin x=5,000 /7,000
- B. sin x= 7,000 /5,000
- C. tan x= 5,000/7,000
- D. tan x=7,000/5,000
Correct Answer & Rationale
Correct Answer: C
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
To solve for \( x \), the correct relationship involves the tangent function, as \( \tan \) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Option C, \( \tan x = \frac{5,000}{7,000} \), accurately represents this ratio. Option A misapplies the sine function, which should represent the ratio of the opposite side to the hypotenuse, not the adjacent side. Similarly, option B incorrectly uses sine but with the sides reversed, leading to an inaccurate representation. Option D misuses tangent, suggesting the opposite and adjacent sides are swapped, which does not align with the definition of tangent. Thus, only option C correctly applies the tangent function to find \( x \).
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.