advertisement

1 If is an even integer, then could be which of the following? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer B : Choice (B) is correct. The sum of any even number and since options is is even, it follows that so must be odd. Therefore, must be odd. The only odd integer among the could be Explanation for Incorrect Answer A : Choice (A) is not correct. If even. Thus, then would be which is odd, not could not be Explanation for Incorrect Answer C : Choice (C) is not correct. If even. Thus, then would be which is odd, not would be which is odd, not could not be Explanation for Incorrect Answer D : Choice (D) is not correct. If even. Thus, then could not be Explanation for Incorrect Answer E : Choice (E) is not correct. If even. Thus, 2 then would be which is odd, not could not be A certain building has square feet of surface that needs to be painted. If gallon of paint will cover square feet, what is the least whole number of gallons that must be purchased in order to have enough paint to apply one coat to the surface? (Assume that only whole gallons can be purchased.) (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer C : Choice (C) is correct. This problem can be solved by dividing the total number of square feet to be painted by the number of square feet that can be painted with gallon: Since you are asked to find the least whole number of gallons that must be purchased, you would have to buy the next highest whole number of gallons, which is gallons. Explanation for Incorrect Answer A : Choice (A) is not correct. If gallon of paint covers paint cover only to cover square feet. Thus, square feet, then gallons of gallons is not enough paint square feet. Explanation for Incorrect Answer B : Choice (B) is not correct. If gallon of paint covers of paint cover only paint to cover square feet, then square feet. Thus, gallons gallons is not enough square feet. Explanation for Incorrect Answer D : Choice (D) is not correct. It is true that feet, but only gallons are needed. Thus, of gallons that must be purchased. gallons of paint could cover square gallons is not the least whole number Explanation for Incorrect Answer E : Choice (E) is not correct. It is true that feet, but only gallons are needed. Thus, of gallons that must be purchased. gallons of paint could cover square gallons is not the least whole number 3 On a report, the typing begins inch from the left edge of the paper and ends inches from the right edge. If the width of the paper is per line are used for typing? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS inches, how many inches Explanation for Correct Answer E : Choice (E) is correct. The width devoted to typing is the width of the paper, inches, minus the widths of the left and the right margins, or inches. Explanation for Incorrect Answer A : Choice (A) is not correct. inches is the width of the paper minus the width of the left margin only. The number of inches per line used for typing is the width of the paper minus the widths of both the left and the right margins. Explanation for Incorrect Answer B : Choice (B) is not correct. inches is too much by inches. Explanation for Incorrect Answer C : Choice (C) is not correct. inches is the width of the paper minus the width of the right margin only. The number of inches per line used for typing is the width of the paper minus the widths of both the left and the right margins. Explanation for Incorrect Answer D : Choice (D) is not correct. 4 Brenda received pledges from inches is too much by people for a inch. mile bike-a-thon. If Brenda rode miles and each person gave for each mile she rode, which of the following expressions gives the total dollar amount of money Brenda collected? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer A : Choice (A) is correct. Since Brenda rode miles and each person gave each mile she rode, it follows that each person gave were for dollars. Since there people, the total number of dollars that were given was Explanation for Incorrect Answer B : Choice (B) is not correct. The total amount of money that Brenda collected was dollars. The value of is Brenda did not collect dollars; she collected dollars. Explanation for Incorrect Answer C : Choice (C) is not correct. The total amount of money that Brenda collected was dollars. The value of collect dollars; she collected is Brenda did not dollars. Explanation for Incorrect Answer D : Choice (D) is not correct. The total amount of money that Brenda collected was dollars. The value of collect dollars; she collected is Brenda did not dollars. Explanation for Incorrect Answer E : Choice (E) is not correct. The total amount of money that Brenda collected was dollars. The value of collect 5 dollars; she collected is Brenda did not dollars. During a sale at a music store, if a customer buys one tape at full price, the customer is given a percent discount on a second tape of equal or lesser value. If Linda buys two tapes that have full prices of two tapes reduced during the sale? and by what percent is the total cost of the (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer B : Choice (B) is correct. This question involves calculating the percent decrease, which is given by discount is The full price for both tapes before the The tape is discounted, because the discounted tape must be of equal or lesser value. With the reduction of discount, the tape costs This is a Therefore, the percent decrease for the total purchase is Explanation for Incorrect Answer A : Choice (A) is not correct. The reduction in price is equal to decrease. Explanation for Incorrect Answer C : but this is not a Choice (C) is not correct. The reduction in price would be had the same value. only if the two tapes Explanation for Incorrect Answer D : Choice (D) is not correct. This answer corresponds to a discount for the more expensive tape, but only the tape of lesser value will be discounted. Explanation for Incorrect Answer E : Choice (E) is not correct. This answer corresponds to a but only the tape of lesser value will be discounted. 6 If microns equal centimeter and discount for both tapes, angstrom units equal centimeter, how many angstrom units equal micron? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer C : Choice (C) is correct. Since centimeter equals both angstrom units, it follows that angstrom units. Therefore, microns and microns are equal to micron is equal to angstrom units. Explanation for Incorrect Answer A : Choice (A) is not correct. A micron is larger than an angstrom unit, so the answer must be greater than Explanation for Incorrect Answer B : Choice (B) is not correct. This is the number of microns that are equal to unit. angstrom Explanation for Incorrect Answer D : Choice (D) is not correct. is too large by a factor of Explanation for Incorrect Answer E : Choice (E) is not correct. correct answer is is but the 7 In a hotel, all rooms are occupied by either occupied by more than person are occupied by person and or people. If of the rooms are of the rooms that are occupied by more than people, how many rooms are occupied by people? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer D : Choice (D) is correct. Since follows that rooms are occupied by more than rooms are occupied by be occupied by of the rooms are occupied by more than people, it follows that person, it person. Since of these of these rooms, or must people. Explanation for Incorrect Answer A : Choice (A) is not correct. See the explanation for the correct response (D). Explanation for Incorrect Answer B : Choice (B) is not correct. See the explanation for the correct response (D). Explanation for Incorrect Answer C : Choice (C) is not correct. See the explanation for the correct response (D). Explanation for Incorrect Answer E : Choice (E) is not correct. This is the number of rooms that are occupied by more than person. According to the problem, more than person could mean either people. The problem asks for the number occupied by 8 or people. If the product of three consecutive integers written in increasing order equals the middle integer, what is the least of the three integers? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer D : Choice (D) is correct. The product of the three consecutive integers starting with Since the product equals the middle integer, Another way to solve the problem is to set up an equation. Let middle number; then divide both sides by to If represent the is not equal to The resulting equation is Because there are no integers is this is correct. you can which simplifies that satisfy this equation, is the only integer solution to Explanation for Incorrect Answer A : Choice (A) is not correct. The product of the three consecutive integers starting with is which does not equal the middle integer, the correct choice. Therefore, is not Explanation for Incorrect Answer B : Choice (B) is not correct. The product of the three consecutive integers starting with is which does not equal the middle integer, correct choice. Therefore, is not the Explanation for Incorrect Answer C : Choice (C) is not correct. The product of the three consecutive integers starting with is correct choice. which does not equal the middle integer, Therefore, is not the Explanation for Incorrect Answer E : Choice (E) is not correct. The product of the three consecutive integers starting with is which does not equal the middle integer, not the correct choice. 9 If is divided by the remainder is What is the remainder if Therefore, is is divided by (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer C : Choice (C) is correct. Since the remainder is quotient written as such that Thus, when is divided by there is some which can be Therefore, the remainder when is divided by is Explanation for Incorrect Answer A : Choice (A) is not correct. If value of is is divided by the remainder is The remainder when so one possible is divided by is not Explanation for Incorrect Answer B : Choice (B) is not correct. If value of is is divided by the remainder is The remainder when so one possible is divided by is not Explanation for Incorrect Answer D : Choice (D) is not correct. If value of is is divided by the remainder is The remainder when so one possible is divided by is not Explanation for Incorrect Answer E : Choice (E) is not correct. If value of is is divided by the remainder is The remainder when so one possible is divided by is not 10 Which of the following has the same effect as dividing by and then multiplying by (A) Multiplying by (B) Multiplying by (C) Dividing by (D) Dividing by (E) Dividing by ANSWERS AND EXPLANATIONS Explanation for Correct Answer A : Choice (A) is correct. If a number is divided by this number is then multiplied by the result is Therefore, dividing by the result is and then multiplying the result by If has the same effect as multiplying by Explanation for Incorrect Answer B : Choice (B) is not correct. Multiplying by has the same effect as multiplying by and then dividing the result by Explanation for Incorrect Answer C : Choice (C) is not correct. Dividing by has the same effect as multiplying by and then dividing the result by Explanation for Incorrect Answer D : Choice (D) is not correct. See the explanation for the correct response (A). Explanation for Incorrect Answer E : Choice (E) is not correct. See the explanation for the correct response (A). 11 A certain computer chip processes million instructions per second. Which of the following is another way of writing the number of instructions per second that the computer chip can process? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer C : Choice (C) is correct. One million is Therefore, million is or, in scientific notation, or Explanation for Incorrect Answer A : Choice (A) is not correct. or million. Explanation for Incorrect Answer B : Choice (B) is not correct. Explanation for Incorrect Answer D : or million. Choice (D) is not correct. or billion. Explanation for Incorrect Answer E : Choice (E) is not correct. or billion. 12 In the sequence above, the first term is and each term after the first is previous term. Which of the following expresses the is a positive integer? times the term of this sequence, where (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer B : Choice (B) is correct. This sequence is an example of a geometric sequence. The ratio of adjacent terms is constant. The as where term of any geometric sequence can be written is the constant ratio of adjacent terms, and the sequence. Since we know the first term of the sequence (it’s constant ratio (it’s ), the is the first term of ) and we know the term can be written as Explanation for Incorrect Answer A : Choice (A) is not correct. If the first term would be term of the sequence were then the but the first term is Explanation for Incorrect Answer C : Choice (C) is not correct. If the term would be term of the sequence were but the second term is then the second Explanation for Incorrect Answer D : Choice (D) is not correct. If the term would be term of the sequence were then the first but the first term is Explanation for Incorrect Answer E : Choice (E) is not correct. If the first term would be term of the sequence were then the but the first term is 13 In the rectangular box above, how many different paths are there of total length from to along the edges? (A) Three (B) Four (C) Five (D) Six (E) Eight ANSWERS AND EXPLANATIONS Explanation for Correct Answer D : Choice (D) is correct. The box has edges of lengths and Since the box is rectangular, the edges parallel to the edges with lengths marked same lengths. Also, each path of length length one of length (it is true that to from and one of length have the must contain one edge of to have total length because but there are clearly no paths from consisting of three edges of length the first edge of the path beginning at to and ). There are possible edges to choose for Then, from any of the three vertices reached, there are possible edges for the second edge of the path (namely, the edges that have a different length from that of the first edge used). Finally, from each of the vertices reached, there is only edge possible to complete the path to Therefore, the total number of different paths of length from to is Explanation for Incorrect Answer A : Choice (A) is not correct. See the explanation for the correct response (D). Explanation for Incorrect Answer B : Choice (B) is not correct. See the explanation for the correct response (D). Explanation for Incorrect Answer C : Choice (C) is not correct. See the explanation for the correct response (D). Explanation for Incorrect Answer E : Choice (E) is not correct. See the explanation for the correct response (D). 14 If and are integers, , and which of the following must be true? (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer E : Choice (E) is correct. Since and it follows that and signs: one positive, the other negative. Therefore, the product of have opposite and is negative: Explanation for Incorrect Answer A : Choice (A) is not correct. The statement could be false. For example, if and is not true. then and but Explanation for Incorrect Answer B : Choice (B) is not correct. The statement and then and but could be false. For instance, if is not true. Explanation for Incorrect Answer C : Choice (C) is not correct. The statement implies that Therefore, is equal to, not less than, Explanation for Incorrect Answer D : Choice (D) is not correct. The statement and , then and could be false. For instance, if but is not true, because 15 Set is the set of positive even integers less than squares of the members of set Set is the set of the How many integers are members of both set and set (A) (B) (C) (D) (E) ANSWERS AND EXPLANATIONS Explanation for Correct Answer A : Choice (A) is correct. Set is . Set is which is The question asks for the number of integers that are members of both set that is, the number of members in the intersection of set have to be less than these criteria: set and set perfect squares, and even. There are and Thus, there are and set These integers integers that meet members in the intersection of and set Explanation for Incorrect Answer B : Choice (B) is not correct. See the explanation for the correct response (A). Explanation for Incorrect Answer C : Choice (C) is not correct. See the explanation for the correct response (A). Explanation for Incorrect Answer D : Choice (D) is not correct. See the explanation for the correct response (A). Explanation for Incorrect Answer E : Choice (E) is not correct. See the explanation for the correct response (A). 16 In the sequence of integers above, each term after the first term, integer that is greater than value of Your Response: is the least of the immediately preceding term. What is one possible Correct Response(s): 114 or 115 or 116 or 117 Explanation: The possible correct answers are values of than or By the rule for generating the sequence, it follows that is the least integer greater Therefore, From these values for so you can find the possible values for could be If is so could be or If Therefore, so could be or Therefore, On a certain map, a distance of miles are represented by or then Therefore, could be any of the four integers 17 . First, find the possible is then and miles is represented by centimeter. How many centimeters on this map? Your Response: Correct Response(s): 82.5 Explanation: 18 The correct answer is . Each centimeter represents represents miles. miles, so centimeters How many different assignments of campers, Alice, Barbara, and Connie, to different tents are possible if each tent must contain at least one of these campers? Your Response: Correct Response(s): 6 Explanation: The correct answer is . Since each tent must contain at least one camper, one camper is assigned to one tent and two campers are assigned to the other tent. Let and stand for Alice, Barbara, and Connie, respectively. If there is only one camper in the first tent, then assignments are possible. With two campers in the first tent, there are likewise Therefore, a total of 19 different assignments are possible. In an election, a total of Candidate received and candidate assignments possible. votes were cast for the three candidates, votes. If candidate received more votes than candidate received more votes than candidate of votes that candidate and what is the least number could have received? Your Response: Correct Response(s): 2051 Explanation: The correct answer is together candidate votes than . Since candidate and candidate it follows that If votes. Since the least number of votes candidate received is 20 received of the votes, received more must have received more than half of those votes. Since than received (In this case, received could have votes, which is more received.) percent of is equal to percent of , then is equal to what number? Your Response: Correct Response(s): 4 Explanation: The correct answer is . To find the value of equation. Since percent means "per hundred," and translates into the equation percent of translate the given sentence into an percent of is and solving for is Therefore, the sentence gives