Quantity AThe average (arithmetic mean) of the length and the width of a rectangle whose perimeter is 18Quantity B
Answer(s): B
The perimeter P of a rectangle is given by the formula:P = 2 × (length + width)We are told that the perimeter is 18, so:18 = 2 × (length + width)length + width = 9Quantity A asks for the average (arithmetic mean) of the length and the width, which is:4.5 (Quantity A) < 5 (Quantity B)
x-1y-1 > 0Quantity AQuantity B
Answer(s): D
This inequality implies that the product of the reciprocals of x and y is positive. This means both x-1 and y-1 must either both be positive or both be negative. In other words, x and y must have the same sign, meaning both x > 0 and y > 0, or both x < 0 and y < 0.Quantity AQuantity BSince x and y have the same sign, we know that both quantities are positive, but we cannot definitively say which one is greater without knowing the exact values of x and y.
According to a certain model, the number of residents in City W, in thousands of people, n years from the present will be0.1n2 + 20n + 1,000Quantity AThe number of residents in City W 10 years from the present, according to the modelQuantity B1,200,000
Answer(s): A
Substitute n = 10 into the formula:Number of residents = 0.1(10)2 + 20(10) + 1,000.Number of residents = 10 + 200 + 1,000 = 1,210.This means the number of residents 10 years from the present is 1,210 thousand, or 1,210,000 people.1,210,000 > 1,200,000
Line l in the xy-plane contains the points (a, 0) and (0, 6), where b < a < 0.Quantity AThe slope of line lQuantity B-1
The formula for the slope of a line passing through two points (x1,y1) and (x2,y2) is:For the points (a,0) and (0,6), the slope is:We are given that b < a < 0, which means a is a negative number.Since a is negative, the slope will be positive because dividing by a negative number changes the sign.Now, we compare this slope with Quantity B, which is -1. The slope is positive and the magnitude of a is less than 6, implying a value greater than 1 for the slope.
Polygon P has n sides (n > 4).Quantity A180 more than the sum of the degree measures of the interior angles of PQuantity BThe sum of the degree measures of the interior angles of a polygon with n + 1 sides
Answer(s): C
The sum of the interior angles of a polygon with n sides is given by the formula:Sum of interior angles = 180(n - 2).Quantity A asks for 180 more than the sum of the interior angles of polygon P, which has n sides. Using the formula for the sum of the interior angles, we get:Quantity A = 180(n - 2) + 180 = 180n - 360 + 180 = 180n - 180.Quantity B is the sum of the interior angles of a polygon with n + 1 sides. Using the formula for the sum of the interior angles for a polygon with n + 1 sides:Quantity B = 180((n + 1) - 2) = 180(n - 1) = 180n - 180.
Of the 200 integers in a set, the 65th percentile is 20.Quantity AThe arithmetic mean of the integers in the setQuantity B20
The 65th percentile of a set means that 65% of the numbers in the set are less than or equal to the given value, which in this case is 20. This means that 130 out of the 200 integers in the set are less than or equal to 20, while the remaining 70 integers are greater than or equal to 20.Understanding Quantity AThe arithmetic mean of a set is the sum of all numbers divided by the total number of numbers. The 65th percentile tells us how many numbers are above and below 20 but does not provide information about the actual values of those numbers. The mean depends on the distribution of all values in the set, and we do not have enough information to determine whether the mean is greater than, less than, or equal to 20.Compare with Quantity BSince we do not have the full distribution of numbers in the set, we cannot determine if the mean is greater than, less than, or equal to 20. For example, if the numbers above 20 are much larger, the mean could be greater than 20. If the numbers are more evenly spread, the mean could be equal to or less than 20.Because the relationship between the mean and 20 depends on the distribution of the numbers, we do not have enough information to determine that relationship.
Quantity AQuantity B2-54
227 - 224 = 224(23 - 1) = 224(8 - 1) = 224 × 7827 = (23)27 = 281826 = (23)26 = 278281 - 278 = 278(23 - 1) = 278(8 - 1) = 278 × 7224-78 = 2-54
The sum of -5 and k equals m2.Quantity AkQuantity B4
-5 + k = m2k = m2 + 5m2 + 5 vs 4m2 + 5 > 4 m2 > -1Since m2 is always non-negative (because it is a square of a real number), we always have:m2 0k = m2 + 5 5Since k 5, it' is possible that k > 4, but we cannot conclude if k is always greater that 4 without knowing specific values for m.
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