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.
Other Related Questions
If an item regularly costs d dollars and is discounted 12 percent, which of the following represents the discounted price in dollars?
- A. 0.12d
- B. 0.88d
- C. 1.12d
- D. d-0.12
Correct Answer & Rationale
Correct Answer: B
To find the discounted price after a 12 percent discount on an item that costs d dollars, we first calculate the amount of the discount, which is 12% of d, or 0.12d. To determine the final price, we subtract this discount from the original price: d - 0.12d = 0.88d. Option A (0.12d) represents only the discount amount, not the final price. Option C (1.12d) incorrectly suggests an increase in price. Option D (d - 0.12) does not account for the percentage; it inaccurately represents the discount as a flat dollar amount rather than a percentage of the original price. Thus, 0.88d correctly reflects the discounted price.
To find the discounted price after a 12 percent discount on an item that costs d dollars, we first calculate the amount of the discount, which is 12% of d, or 0.12d. To determine the final price, we subtract this discount from the original price: d - 0.12d = 0.88d. Option A (0.12d) represents only the discount amount, not the final price. Option C (1.12d) incorrectly suggests an increase in price. Option D (d - 0.12) does not account for the percentage; it inaccurately represents the discount as a flat dollar amount rather than a percentage of the original price. Thus, 0.88d correctly reflects the discounted price.
In the figure above, what is the average (arithmetic mean) of w, x, y, and z?
- A. 90
- B. 100
- C. 120
- D. It cannot be determined from the information given.
Correct Answer & Rationale
Correct Answer: D
To find the average of w, x, y, and z, all values must be known. Option D is valid since the problem does not provide specific values or relationships between these variables, making it impossible to calculate their average. Option A (90), Option B (100), and Option C (120) suggest definitive averages, but without concrete data on w, x, y, and z, these answers cannot be substantiated. Each of these options assumes values that may not exist or be accurate, highlighting the necessity of complete information for such calculations.
To find the average of w, x, y, and z, all values must be known. Option D is valid since the problem does not provide specific values or relationships between these variables, making it impossible to calculate their average. Option A (90), Option B (100), and Option C (120) suggest definitive averages, but without concrete data on w, x, y, and z, these answers cannot be substantiated. Each of these options assumes values that may not exist or be accurate, highlighting the necessity of complete information for such calculations.
How many cups of peanut butter must be used in order to make exactly enough peanut butter balls for the children at the party?
- A. 10
- B. 12
- C. 18
- D. 24
Correct Answer & Rationale
Correct Answer: C
To determine the number of cups of peanut butter needed for the peanut butter balls, one must consider the recipe's requirements and the number of children attending the party. Option C (18 cups) aligns with the recipe's proportion to yield the exact quantity necessary for all children. Option A (10 cups) is insufficient, likely resulting in fewer peanut butter balls than required. Option B (12 cups) may also fall short, leading to a shortage. Option D (24 cups) exceeds the needed amount, creating waste. Thus, C is the optimal choice, ensuring each child receives a peanut butter ball without excess or deficit.
To determine the number of cups of peanut butter needed for the peanut butter balls, one must consider the recipe's requirements and the number of children attending the party. Option C (18 cups) aligns with the recipe's proportion to yield the exact quantity necessary for all children. Option A (10 cups) is insufficient, likely resulting in fewer peanut butter balls than required. Option B (12 cups) may also fall short, leading to a shortage. Option D (24 cups) exceeds the needed amount, creating waste. Thus, C is the optimal choice, ensuring each child receives a peanut butter ball without excess or deficit.
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 \( x \).