1. There are n students in a biology class, and only 6 of them are seniors. If 7 juniors are added to the class, how many students in the class will not be seniors? Answer Choices
; (A) n-3
(B) n-2 ;
; (C) n-1
(D) n+1 ;
; (E) n+2
Choice (D) is correct. At first, there are n students in the class, and 6 of them are seniors. It follows that initially n-6 students are not seniors. If 7 juniors are added to the class, there will be n-6+7, or n+1, students in the class who are not seniors. 2. For how many positive two-digit integers is the ones digit greater than twice the tens digit?
; (A) 16
; (B) 20
; (C) 28
; (D) 32
; (E) 36
Choice (A) is correct. If the tens digit is 1, there are 7 positive two-digit integers with the ones digit greater than twice the tens digit13, 14, 15, 16, 17, 18 and 19. If the tens
digit is 2, there are 5 positive two-digit integers with the ones digit greater than twice
the tens digit 25,26,27,28 and 29. If the tens digit is 3, there are 3 positive two-digit integers with the ones digit greater than twice the tens digit: 37, 38 and 39. If the tens digit is 4, there is 1 positive two-digit integer with the ones digit greater than twice the tens digit: 49. If the tens digit is equal to or greater than 5, then the ones digits, which cannot be greater than 9, is not greater than twice the tens digit. Therefore, there is a total of 7+5+3+1=16 positive two-digit integers for which the ones digit is greater than twice the tens digit.
3. If the graph of the function f is a line with slope 2, which of the following could be the equation of f?
; (A) f(x)=4x-2
; (C)f(x)= -2x-2
; (E)f(x)= -1x/2+1/2
Choice (B) is correct. The graphs of all the given functions are lines, and the equations are all in the form y=mx+b. When a linear equation is in the form y=mx+b, the slope of the graph of the equation is m, the coefficient of x. The graph of the function f is a line with slope 2, so the only choice that could be the equation of f is f(x)=2x+4.
4. A line segment containing the points (0,0) and (12,8) will also contain the point Answer Choices
; (A) (2,3)
; (B) (2,4)
; (C) (3,2)
; (D) (3,4)
; (E) (4,2)
Choice (C) is correct. The slope of the line containing the points (0, 0) and (12, 8) is 8/12, or 2/3. If (x, y) is another point on this line segment, then x and y must satisfy y/x=2/3, or y=2x/3. Of the five choices, (3, 2) is the only point for which y=2x/3. 5. In a community of 416 people, each person owns a dog or a cat or both. If there are 316 dog owners and 280 cat owners, how many of the dog owners own no cat? Answer Choices
; (A) 36
; (B) 100
; (C) 136
; (D) 180
; (E) 316
Choice (C) is correct. To solve this problem, you must realize that some of the 316 dog owners also own cats and some of the 280 cat owners also own dogs. This has to be true because 316+280 is more than 416! The question asks how many of the dog owners do not own cats. To solve the problem, first find the number of people who own both a dog and a cat. To do this, let b equal the number who owns both a dog
and a cat. Then 416=316+280-b. (Note: Adding 316 and 280 counts b twice, so b must be subtracted from the sum to get 416.) Next, the equation 416=596-b can be solved for b, yielding b=180. Finally, the number of dog owners who own no cat is 316-180=136.
The graph of y=f(x) is shown above. If 0 ?t?5, and if (t, v) is on the graph of f, which of the following must be true?
; (A) -10 ? v ? -5
; (B) -5 ? v ? 0
; (C) 0 ? v ? 5
; (D) 5 ? v ? 10
; (E) 10 ? v ? 15
The answer is D. The second coordinate of a point on the graph corresponds to v.
From the graph, you can see that for 0 ? t ? 5, the smallest value of v is 5 and the greatest value is 10, so 5 ? v ? 10.
7. The sum, product, and average (arithmetic mean) of three integers are equal. If two of the integers are 0 and -5, the third integer is
; (A) -5
; (B) 0
; (C) 2
; (D) 5
; (E) 10
The answer is D. You have three integers, 0, -5, and x, whose sum, product, and average are equal. Since 0 is one of the numbers, the product is 0. So the sum and average are also equal to 0. Therefore, -5+x+0=0, and x=5.
8. Which of the following CANNOT be the lengths of the sides of a triangle? Answer Choices
; (A) 1,1,1
; (B) 1,2,4
; (C) 1,75,75
; (D) 2,3,4
; (E) 5,6,8
The answer is B. The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Four of the answer choices can be eliminated because each set of three numbers can be the lengths of three sides of a triangle. For example, in the third choice (1,75,75), the sum of any two numbers is greater than the third number. However, the second choice (1,2,4) cannot be the lengths of the sides of a
triangle, since 1+2 is not greater than 4. This choice is the correct answer, since the question asked which set of numbers could NOT be lengths of the sides of a triangle. 9. A person bought 12 cards for 30 cents. If the next day the price of the cards was 5 cents each, how much did the person save per card by buying at the earlier price? Answer Choices
; (A) 2 cents
(B) 2.5cents ;
; (C) 3 cents
(D) 3.5cents ;
; (E) 5 cents
The answer is B. You can find the savings per card by subtracting the price per card on the earlier day from the price per card on the later day. Since 12 cards cost 30 cents on the earlier day, the price per card was 30 / 12, or 2.5 cents. The price per card on
2.5 cents = 2.5cents is the amount saved per the later day was 5 cents, so 5 cents –
card by buying at the earlier price.
10. First, 3is subtracted from x and the square root of the difference is taken. Then, 5 is added to the result, giving a final result of 9. What is the value of x? Answer Choices
; (A) 3
; (B) 4
; (C) 5
; (D) 16
; (E) 19
The correct answer is (E). When is subtracted from , the result is . The
square root of this difference is . When is added to , the final
result is . This can be written as . Subtracting from both sides
gives the equation . Squaring both sides of this equation gives
, or .
11. The sum of the positive odd integers less than 50 is subtracted from the sum of the positive even integers less than or equal to 50. What is the resulting difference? Answer Choices
; (A) 0
; (B) 25
; (C) 50
; (D) 100
; (E) 200
The sum of the positive odd integers less than 50 is 1+3+5+…+45+47+49, and the sum of the positive even integers less than or equal to 50 is 2+4+6+…+46+48+50.
When the first sum is subtracted from the second, the terms can be regrouped to form (2-1)+(4-3)+…+(48-47)+(50-49), which is just 1+1+1+…+1, where there are 25 ones.
Therefore, the correct answer is 25.
12. If half the people in a room leave at the end of every five-minute interval and at the end of twenty minutes the next to last person leaves, how many people were in the room to start with? (Assume that no one enters the room once the process begins.) Answer Choices
; (A) 32
; (B) 28
; (C) 16
(D) 12 ;
; (E) 8
At the end of each five-minute interval, half the number of people in the room leave and the other half remain. At the end of twenty minutes, this step of half leaving and half remaining involved only 2 people, 1 leaving and 1 remaining. This means that during the preceding five-minute interval (the interval from minute 15 to minute 20), there were 2 people in the room.
At the end of minute 15, the step of half leaving and half remaining must have involved 4 people, 2 who left and 2 who remained.
By the same reasoning, at the end of minute 10 the step involved 8 people, and at the end of minute 5 it involved 16 people. These 16 people were the ones who were in the room to start with, so the correct answer is 16.
In the figure above, if PQRS is a quadrilateral and TUV is a triangle, what is the sum
of the degree measures of the marked angles?
; (A) 420
; (B) 490
(C) 540 ;
; (D) 560
; (E) 580
The correct answer is (C). The marked angles in the figure are angles P, Q, R, and S, which are the angles of a quadrilateral, and angles T, U, and V, which are the angles of a triangle. The sum of the degree measures of the angles of a quadrilateral is 360, and the sum of the degree measures of the angles of a triangle is 180. Therefore, the sum of the degree measures of the marked angles is 360plus 180, which is 540.
satisfy the inequality above? How many integers
; (A) One
; (B) Two
; (C) Four
; (D) Seven
; (E) Eight
The answer is (D). If , and is an integer, then the possible values of
are the integers greater than or equal to and less than or equal to , so the
possible values for are , , ,, , , and . Therefore, there
are seven integers that satisfy the given inequality.
15. The sum of the digits of a three-digit number is . If the hundreds digit is times the tens digit and the tens digit is the units digit, what is the tens digit of the number?
If x stands for the tens digit, then 3x stands for the hundreds digit and 2x stands for
the units digit. Since the sum of the digits is 12, 3x+x+2x=12. Solving this gives x=2.
16. A train traveling miles per hour for hour covers the same distance as a train traveling miles per hour for how many hours?