A sales representative's monthly earnings consist of a base salary of $3,500 plus a commission equal to 5 percent of monthly sales in excess of $50,000.If monthly sales total s dollars and s > $50,000, what are the sales representative's monthly earnings in terms of s, in dollars?
Answer(s): A
3500 + 0.05(s - 50,000) = 3500 + 0.05s - 2500 = 0.05s + 1000
All of the 80 science students at a certain school are enrolled in at least one of three science courses: biology, chemistry, and physics. There are 60 students enrolled in biology, 50 students enrolled in chemistry, and 35 students enrolled in physics. None of the students are enrolled in all three courses.Which of the following could be the number of students enrolled in both chemistry and physics? (Choose all that apply.)
Answer(s): B,C,D,E
Let B, C, and P be the sets of students in biology, chemistry, and physics.We know:|B| = 60, |C| = 50, |P| = 35Total students = 80No student is in all three courses.By inclusionexclusion (with no triple intersection):Let CP be the number of students in both chemistry and physics. We must have nonnegative counts in each category and not exceed course totals. This leads to the constraints:CP 5BP 15BP + CP 3515 BP + CP 35Checking each answer choice for CP:0: violates CP 5 not possible.5: choose BP = 15 BP + CP = 20 (valid), yields consistent nonnegative counts possible.10: choose BP = 15 BP + CP = 25 (valid) possible.15: choose BP = 15 BP + CP = 30 (valid) possible.20: choose BP = 15 BP + CP = 35 (valid) possible.25, 30, 35: no BP 15 satisfies BP + CP 35 not possible.Thus, the values that could be the number of students enrolled in both chemistry and physics are 5, 10, 15, and 20.
The average (arithmetic mean) of the hourly salaries of Ted and Bill is $8.50. Mary and Margaret each have an hourly salary of $8.50.Quantity AThe average of the hourly salaries of Ted, Bill, Mary, and MargaretQuantity BThe median of the hourly salaries of Ted, Bill, Mary, and Margaret
Answer(s): D
Ted and Bill's average is $8.50, soMary and Margaret each earn $8.50.The four salaries are T, B, 8.5, 8.5.Quantity A (the mean):The mean is always $8.50, no matter what Ted and Bill individually earn.Quantity B (the median):The median depends on whether T and B are above or below 8.5:If both T and B are greater than 8.5 median > 8.5 B > AIf both are less than 8.5 median < 8.5 A > BIf one is above and one below 8.5 median = 8.5 A = BAll scenarios are possible because only T+B=17T + B = 17T+B=17 is fixed.
Quantity AThe length of ADQuantity BThe length of BC
Answer(s): B
AD and BC rise from the same base DC to the parallel top AB, so their vertical height is the same (call it h).AD forms a 65° angle with DC AD = h / sin 65°.BC forms a 62° angle with DC BC = h / sin 62°.Since sin 65° > sin 62°, dividing by the larger sine gives the shorter length, so AD < BC.Thus, Quantity B (BC) is greater.
a > b > 0 and k > 0Quantity AQuantity B
a(b + k) = ab + ak > ab + bk = b(a+ k) because a > b and k > 0; therefore
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