x is an even positive integer.Quantity AThe number of odd positive integers less than xQuantity BThe number of even positive integers less than x + 1
Answer(s): C
Since x is an even positive integer, we can assume x = 2k, where k is a positive integer.Quantity A: The number of odd positive integers less than x is the number of odd integers from 1 to x - 1.Since x is even, the odd integers are 1, 3, 5, ..., x - 1. There are k such odd integers, as every second integer is odd.Quantity B: The number of even positive integers less than x + 1 is the number of even integers from 2 to x.Since x is even, the even integers are 2, 4, 6, ..., x. There are k such even integers, as every second integer is even.
(2x - 1)(y - 4) = 0Quantity AxyQuantity B2
The equation (2x - 1)(y - 4) = 0 implies that either 2x - 1 = 0 or y - 4 = 0.· If 2x - 1 = 0, then x = 1/2.· If y - 4 = 0, then y = 4.The possible values for x and y are x = 1/2 and y = 4. We now calculate the value of xy:Quantity A is xy = 2, and Quantity B is also 2.
Lines l and m are parallel.90 < c < 180Quantity AcQuantity Ba + b
Answer(s): B
In the given image, lines l and m are parallel, and the angles a°, b°, and c° are formed by intersecting lines.Given that 90° < c < 180°, it implies that c is an obtuse angle. The angles a and b are supplementary to c, meaning a + c = 180° and b + c = 180°, because they form linear pairs with the angle c.Since c is between 90 and 180 degrees, a + b = 180°.Quantity A is c, and Quantity B is a + b, which equals 180.
The semicircle has center O and radius r.CF = OFBD || AEQuantity ABDQuantity B
Answer(s): D
· The semicircle has center O and radius r.· CF = OF suggests that F is the midpoint of CO.· BD || AE, meaning BD and AE are horizontal lines.Since CF = OF, point F is the midpoint of CO. This means BD lies symmetrically within the semicircle. The exact length of BD is not explicitly defined, but it depends on the radius r of the semicircle and how high above O the segment BD is placed. The proportion of BD relative to AE could change based on the given radius and specific distances. Since the exact measurement of BD is unknown and no direct numerical comparison is provided, we cannot definitively determine the relationship between the two quantities.
In a survey of n families, x families had at least 1 child, and y families had at least 2 children.Quantity AThe number of families in the survey that had exactly 1 childQuantity Bx - y
Quantity A refers to the number of families that had exactly 1 child. This can be calculated as the difference between the number of families with at least 1 child (x) and the number of families with at least 2 children (y).So, the number of families with exactly 1 child is x - y, which matches Quantity B.
r > sQuantity Ar + sQuantity B
The expression in Quantity B can be simplified:Since r > s, we can cancel out r - s in the numerator and denominator, leaving us with:r + s
The two circles have centers at B and C, respectively, and are mutually tangent. Each circle has radius r.Quantity AThe perimeter of quadrilateral ABCDQuantity B8r
Answer(s): A
The two circles are mutually tangent, meaning they touch at exactly one point.Each circle has a radius r.The quadrilateral ABCD consists of two segments (AB and CD) that are straight lines connecting the points where the circles touch the perimeter.The total length of the perimeter of quadrilateral ABCD can be broken down as:1. The sum of the lengths of the straight segments AB and CD (each equal to 2r because they are the sum of the radii of the two circles).2. The length of the arc from the circumference of each circle: since each circle has a radius r, the circumference of each circle is 2r. Half of the circumference will be r, which is the arc length from B to C and from A to D.Thus, the perimeter of quadrilateral ABCD is:2r + 2r + r + r = 4r + 2rNow, compare the two quantities: 4r + 2r (Quantity A) and 8r (Quantity B) Since 2 6.28, we know that 4r + 2r 4r + 6.28r = 10.28r, which is greater than 8r.
The product of the positive numbers x and y is less than the sum of x and y.Quantity AThe sum of the reciprocal of x and the reciprocal of yQuantity B1
We are given the inequality:xy < x + yWe need to compare:Quantity A: The sum of the reciprocals of x and y, which is:Quantity B: 1From the given inequality xy < x + y, dividing both sides by xy (which is positive since x, y > 0) gives:
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