Directions: Select one or more answer choices according to the specific question directions.
If the question does not specify how many answer choices to select, select all that apply.
- The correct answer may be just one of the choices or as many as all of the choices, depending on the question.
- No credit is given unless you select all of the correct choices and no others.
If the question specifies how many answer choices to select, select exactly that number of choices.
Which two of the following numbers have a product that is between -1 and 0?
Indicate both of the numbers.(A) -20
For this question, you must select a pair of answer choices. The product of the pair must be negative, so the possible products are (–20)(2–4), (–20)(3–2), (–10)(2–4), and (–10)(3–2). The product must also be greater than –1. The first product is , the second product is , and the third product is, , so you can stop there. The correct answer consists of Choices B (–10) and C (2–4).
Which of the following integers are multiples of both 2 and 3?
Indicate all such integers.(A) 8
You can first identify the multiples of 2, which are 8, 12, 18 and 36, and then among the multiples of 2 identify the multiples of 3, which are 12, 18 and 36. Alternatively, if you realize that every number that is a multiple of 2 and 3 is also a multiple of 6, you can identify the choices that are multiples of 6. The correct answer consists of Choices C (12), D (18) and F (36).
Each employee of a certain company is in either Department X or Department Y, and there are more than twice as many employees in Department X as in Department Y. The average (arithmetic mean) salary is $25,000 for the employees in Department X and $35,000 for the employees in Department Y. Which of the following amounts could be the average salary for all of the employees of the company?
Indicate all such amounts.(A) $26,000
One strategy for answering this kind of question is to find the least and/or greatest possible value. Clearly the average salary is between $25,000 and $35,000, and all of the answer choices are in this interval. Since you are told that there are more employees with the lower average salary, the average salary of all employees must be less than the average of $25,000 and $35,000, which is $30,000. If there were exactly twice as many employees in Department X as in Department Y, then the average salary for all employees would be, to the nearest dollar, the following weighted mean,dollars
where the weight for $25,000 is 2 and the weight for $35,000 is 1. Since there are more than twice as many employees in Department X as in Department Y, the actual average salary must be even closer to $25,000 because the weight for $25,000 is greater than 2. This means that $28,333 is the greatest possible average. Among the choices given, the possible values of the average are therefore $26,000 and $28,000. Thus, the correct answer consists of Choices A ($26,000) and B ($28,000).
Intuitively, you might expect that any amount between $25,000 and $28,333 is a possible value of the average salary. To see that $26,000 is possible, in the weighted mean above, use the respective weights 9 and 1 instead of 2 and 1. To see that $28,000 is possible, use the respective weights 7 and 3.
Which of the following could be the units digit of where n is a positive integer?
Indicate all such digits.(A) 0
The units digit of is the same as the units digit of for all positive integers n. To see why this is true for compute by hand and observe how its units digit results from the units digit of Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with and proceeding consecutively, the units digits of 7, and are 7, 9, 3, 1 and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of and therefore of The correct answer consists of Choices B (1), D (3), H (7) and J (9).
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