In the xy-plane, line k has a positive x-intercept and a slope of 2.Which of the following equations represents a line that intersects line k at a point above the x-axis?
Answer(s): D
Line k has equation y = 2x - 2a, where a > 0 is its (positive) x-intercept.For a second line to intersect k above the x-axis, their intersection must give y > 0.y = x + 1Set 2x - 2a = x + 1.Then x = 2a + 1 and y = x + 1 = 2a + 2.Because a > 0, we have y = 2a + 2 > 0; hence the lines meet at a point above the x-axis.
FILL BLANKThe average (arithmetic mean) of the numbers in list A is 20, and the average of the numbers in list B is 8. List A has 3 times as many numbers as list B.What is the average of the numbers in lists A and B combined?
Answer(s): A
Let list B contain n numbers.Sum of list B: 8n.List A has three times as many numbers, i.e., 3n numbers.Sum of list A: 20 × 3n = 60n.Combined lists:Total number of values: n + 3n = 4n.Total sum of values: 8n + 60n = 68n.Average of the combined lists: 68n ÷ 4n = 17
Eugene and Penny started a job in sales on the same day. Eugene's sales for the first month were r dollars, and each month after the first his sales for that month were twice his sales for the preceding month. Penny's sales for the first month were 10r dollars, and each month after the first her sales for that month were 10r dollars more than her sales for the preceding month.Which of the following statements are true? (Choose all that apply.)
Answer(s): A,B,C
Eugene's monthly sales: E1=r, En = 2n-1r (doubling each month).Penny's monthly sales: P1 = 10r, Pn = Pn - 1 + 10r = 10nr (increase by the constant 10r each month).P2 = 20r = 10 × 2r = 10E2P4 = 40r = 5 × 8r = 5E4E8 = 128r > 80r = P8
Working alone at its constant rate, machine A produces n envelopes in 10 minutes. Working simultaneously at their respective constant rates, machine A and machine B produce n envelopes in 8 minutes.How many minutes does it take machine B, working alone at its constant rate, to produce n envelopes?
Answer(s): E
Machine A's rateWorking together, A + B produce n envelopes in 8 min, soHence machine B's rate isTime for machine B alone to produce n envelopes:
In a certain university, 60 percent of all sophomores are liberal arts majors, 24 percent are education majors, and the rest are majoring in other areas or have not yet chosen a major. At the university, 55 percent of all sophomores are currently taking a psychology course.If x percent of all sophomores are liberal arts majors who are currently taking a psychology course, what is the least possible value of x?
Of all sophomores,Liberal-arts majors: 60 %Psychology students: 55 %To make the overlap as small as possible, assign psychology students first to the 40 % of sophomores who are not liberal-arts majors.After filling that 40 %, there remain55% - 40% = 15%who must be both liberal-arts majors and psychology students.Thus, the least possible value of x is 15 %.
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