If the length of a rectangle is increased by 30% and the width of the same rectangle is decreased by 30%, what is the effect on the area of the rectangle?
- A. It is increased by 60%.
- B. It is unchanged.
- C. It is decreased by 15%.
- D. It is decreased by 9%.
Correct Answer & Rationale
Correct Answer: D
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
Increasing the length of a rectangle by 30% results in a new length of 1.3L, while decreasing the width by 30% gives a new width of 0.7W. The new area can be calculated as A' = (1.3L)(0.7W) = 0.91LW, indicating a decrease in area. Option A is incorrect because a 60% increase does not occur; the area actually decreases. Option B is wrong as the area changes due to the modifications in dimensions. Option C suggests a decrease of 15%, which miscalculates the area change. The area decreases by 9%, confirming the effect of the opposing percentage changes in length and width.
Other Related Questions
A salesperson's commission is k percent of the selling price of a car. Which of the following represents the commission, in dollars, on 2 cars that sold for $14,000 each?
- A. 280k
- B. 28,000k
- C. 14,000/(100+2k)
- D. (28,000+k)/100
Correct Answer & Rationale
Correct Answer: A
To determine the commission on 2 cars sold for $14,000 each, first calculate the total selling price: 2 × $14,000 = $28,000. The commission, being k percent of this total, is expressed as (k/100) × $28,000, which simplifies to $280k. Option B, 28,000k, incorrectly suggests the commission is k percent of the total without dividing by 100. Option C, 14,000/(100+2k), misrepresents the calculation entirely by altering the formula. Option D, (28,000+k)/100, incorrectly adds k to the total selling price before calculating the percentage, which is not aligned with commission calculation principles.
To determine the commission on 2 cars sold for $14,000 each, first calculate the total selling price: 2 × $14,000 = $28,000. The commission, being k percent of this total, is expressed as (k/100) × $28,000, which simplifies to $280k. Option B, 28,000k, incorrectly suggests the commission is k percent of the total without dividing by 100. Option C, 14,000/(100+2k), misrepresents the calculation entirely by altering the formula. Option D, (28,000+k)/100, incorrectly adds k to the total selling price before calculating the percentage, which is not aligned with commission calculation principles.
Malia collected information about whether the members of the 36 households on her block subscribed to cable television and home phone services. Her results are shown in the table below.\nIf a household on Malia's block is selected at random and does subscribe to cable television, what is the probability the members of the household also subscribe to home phone service?
- A. 14/18
- B. 14/26
- C. 18/36
- D. 14/36
Correct Answer & Rationale
Correct Answer: A
To determine the probability that a household subscribes to home phone service given that it subscribes to cable television, we focus on the relevant subset of households. Malia found 18 households that subscribe to cable, out of which 14 also subscribe to home phone service. Thus, the probability is calculated as the number of households with both services (14) divided by the total number of households with cable (18), resulting in 14/18. Option B (14/26) incorrectly uses the total number of households with home phone service instead of just those with cable. Option C (18/36) misinterprets the probability as a ratio of all households rather than those who subscribe to cable. Option D (14/36) inaccurately represents the total number of households instead of focusing on the cable subscribers.
To determine the probability that a household subscribes to home phone service given that it subscribes to cable television, we focus on the relevant subset of households. Malia found 18 households that subscribe to cable, out of which 14 also subscribe to home phone service. Thus, the probability is calculated as the number of households with both services (14) divided by the total number of households with cable (18), resulting in 14/18. Option B (14/26) incorrectly uses the total number of households with home phone service instead of just those with cable. Option C (18/36) misinterprets the probability as a ratio of all households rather than those who subscribe to cable. Option D (14/36) inaccurately represents the total number of households instead of focusing on the cable subscribers.
Square S has area 2√2 square units. What is the length of a side of square S?
- A. ∜128
- B. ∜32
- C. ∜8
- D. ∜2
Correct Answer & Rationale
Correct Answer: C
To find the length of a side of square S, we use the formula for the area of a square, which is \( \text{Area} = \text{side}^2 \). Given that the area is \( 2\sqrt{2} \), we set up the equation \( \text{side}^2 = 2\sqrt{2} \). Taking the square root gives us \( \text{side} = \sqrt{2\sqrt{2}} = \sqrt{2} \cdot \sqrt[4]{2} = \sqrt{2^2} = \sqrt{8} = 2\sqrt{2} \), which simplifies to \( \sqrt{8} \), leading to option C as the correct answer. Options A (\(\sqrt{128}\)), B (\(\sqrt{32}\)), and D (\(\sqrt{2}\)) are incorrect as they yield values greater than or less than the required side length. Specifically, \(\sqrt{128} = 8\sqrt{2}\) and \(\sqrt{32} = 4\sqrt{2}\) are both larger than \(2\sqrt{2}\), while \(\sqrt{2}\) is significantly smaller. Thus, option C accurately represents the side length of square S.
To find the length of a side of square S, we use the formula for the area of a square, which is \( \text{Area} = \text{side}^2 \). Given that the area is \( 2\sqrt{2} \), we set up the equation \( \text{side}^2 = 2\sqrt{2} \). Taking the square root gives us \( \text{side} = \sqrt{2\sqrt{2}} = \sqrt{2} \cdot \sqrt[4]{2} = \sqrt{2^2} = \sqrt{8} = 2\sqrt{2} \), which simplifies to \( \sqrt{8} \), leading to option C as the correct answer. Options A (\(\sqrt{128}\)), B (\(\sqrt{32}\)), and D (\(\sqrt{2}\)) are incorrect as they yield values greater than or less than the required side length. Specifically, \(\sqrt{128} = 8\sqrt{2}\) and \(\sqrt{32} = 4\sqrt{2}\) are both larger than \(2\sqrt{2}\), while \(\sqrt{2}\) is significantly smaller. Thus, option C accurately represents the side length of square S.
The x-and y- coordinates of point P are each to be chosen at random from the set of integers 1 through 10. What is the probability that P will be in quadrant II?
- B. 01-Oct
- C. 01-Apr
- D. 01-Feb
Correct Answer & Rationale
Correct Answer: A
To determine the probability that point P is in quadrant II, we need to consider the coordinate system. In quadrant II, the x-coordinate must be negative, and the y-coordinate must be positive. However, since the x-coordinates are chosen from the integers 1 through 10, all possible x-values are positive. This means point P cannot be in quadrant II, making the probability 0. Option A correctly reflects this conclusion with a probability of 0. Options B, C, and D suggest specific dates, which are irrelevant to the question and do not address the coordinate conditions necessary for quadrant II. Thus, they are incorrect.
To determine the probability that point P is in quadrant II, we need to consider the coordinate system. In quadrant II, the x-coordinate must be negative, and the y-coordinate must be positive. However, since the x-coordinates are chosen from the integers 1 through 10, all possible x-values are positive. This means point P cannot be in quadrant II, making the probability 0. Option A correctly reflects this conclusion with a probability of 0. Options B, C, and D suggest specific dates, which are irrelevant to the question and do not address the coordinate conditions necessary for quadrant II. Thus, they are incorrect.