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
A playground at a mall is in the shape of a rectangle, and there is a 144 foot long fence around it. If the rectangle is 6 feet longer than it is wide, what is the width, in feet, of the rectangle?
- A. 33
- B. 39
- C. 69
- D. 75
Correct Answer & Rationale
Correct Answer: A
To find the width of the rectangle, let the width be represented as \( w \). The length, being 6 feet longer, can be expressed as \( w + 6 \). The perimeter of a rectangle is given by the formula \( P = 2(l + w) \). Here, the perimeter is 144 feet, leading to the equation \( 2(w + (w + 6)) = 144 \). Simplifying this gives \( 2(2w + 6) = 144 \), which reduces to \( 4w + 12 = 144 \), and further simplifies to \( 4w = 132 \), resulting in \( w = 33 \). Option B (39) is incorrect as it gives a perimeter of 156 feet. Option C (69) would lead to an impossible perimeter of 150 feet. Option D (75) results in a perimeter of 162 feet, which exceeds the given value. Thus, only option A satisfies all conditions, confirming the width as 33 feet.
To find the width of the rectangle, let the width be represented as \( w \). The length, being 6 feet longer, can be expressed as \( w + 6 \). The perimeter of a rectangle is given by the formula \( P = 2(l + w) \). Here, the perimeter is 144 feet, leading to the equation \( 2(w + (w + 6)) = 144 \). Simplifying this gives \( 2(2w + 6) = 144 \), which reduces to \( 4w + 12 = 144 \), and further simplifies to \( 4w = 132 \), resulting in \( w = 33 \). Option B (39) is incorrect as it gives a perimeter of 156 feet. Option C (69) would lead to an impossible perimeter of 150 feet. Option D (75) results in a perimeter of 162 feet, which exceeds the given value. Thus, only option A satisfies all conditions, confirming the width as 33 feet.
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.
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.
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 \( x \).