5.5 Answer Explanations

The following discussion is intended to familiarize you with the most efficient and effective approaches to the kinds of problems common to problem solving questions. The particular questions in this chapter are generally representative of the kinds of problem solving questions you will encounter on the GMAT exam. Remember that it is the problem solving strategy that is important, not the specific details of a particular question.

1. A project scheduled to be carried out over a single fiscal year has a budget of $12,600, divided into 12 equal monthly allocations. At the end of the fourth month of that fiscal year, the total amount actually spent on the project was $4,580. By how much was the project over its budget?
   
   1. (A) $ 380
   2. (B) $ 540
   3. (C) $1,050
   4. (D) $1,380
   5. (E) $1,430
   Arithmetic Operations with rational numbers
   The budget for four months is . Thus, the project was over budget for the first four months.
   The correct answer is A.
2. For which of the following values of n is NOT an integer?
   
   1. (A) 1
   2. (B) 2
   3. (C) 3
   4. (D) 4
   5. (E) 5
   Arithmetic Properties of numbers
   Substitute the value for n given in each answer choice into the expression, and then simplify to determine whether or not that value results in an integer.
   

   A		Integer
   B		Integer
   C		NOT an integer
   D		Integer
   E		Integer

   Another method is to rewrite the given expression, , as .
   This shows that the given expression is an integer exactly when is an integer. Since 100 is not divisible by 3, but 100 is divisible by 1, 2, 4, and 5, it follows that .
   The correct answer is C.
3. Rectangular Floors X and Y have equal area. If Floor X is 12 feet by 18 feet and Floor Y is 9 feet wide, what is the length of Floor Y, in feet?
   
   1. (A)
   2. (B) 18
   3. (C)
   4. (D) 21
   5. (E) 24
   Geometry Area
   Since Floor X is a rectangle, its area is . It is given that this is also the area of Floor Y, so if L is the length of Floor Y, it follows that 9L = (12)(18), or .
   The correct answer is E.
4. A case contains c cartons. Each carton contains b boxes, and each box contains 100 paper clips. How many paper clips are contained in 2 cases?
   
   1. (A) 100bc
   2. (B)
   3. (C) 200bc
   4. (D)
   5. (E)
   Algebra Simplifying algebraic expressions
   Each case has bc boxes, each of which has 100 paper clips. The total number of paper clips in 2 cases is thus .
   The correct answer is C.
5. The sum of prime numbers that are greater than 60 but less than 70 is
   
   1. (A) 67
   2. (B) 128
   3. (C) 191
   4. (D) 197
   5. (E) 260
   Arithmetic Properties of numbers
   A prime number is a positive integer divisible by exactly two different positive divisors, 1 and itself. Note that 62, 64, 66, and 68 are also divisible by 2; 63, 66, and 69 are also divisible by 3; and 65 is also divisible by 5. The only prime numbers between 60 and 70 are 61 and 67, and .
   The correct answer is B.
6. A rainstorm increased the amount of water stored in State J reservoirs from 124 billion gallons to 138 billion gallons. If the storm increased the amount of water in the reservoirs to 82 percent of total capacity, approximately how many billion gallons of water were the reservoirs short of total capacity prior to the storm?
   
   1. (A) 9
   2. (B) 14
   3. (C) 25
   4. (D) 30
   5. (E) 44
   Algebra Applied problems
   Let t be the total capacity of the reservoirs in billions of gallons. The information that the post-storm water amount of 138 billion gallons represented 82 percent of total capacity can be expressed as . Solve for t and then estimate the value of t: 175 billion gallons. Thus, the amount the reservoirs were short of total capacity prior to the storm, in billions of gallons, was approximately , so E is the best choice. A more accurate calculation gives .
   The correct answer is E.
   

   

7. On the graph above, when , ; and when , . The graph is symmetric with respect to the vertical line at . According to the graph, when ,
   
   1. (A) −1
   2. (B)
   3. (C) 0
   4. (D)
   5. (E) 1
   Arithmetic; Algebra Interpretation of graphs; Second-degree equations
   Since the graph is symmetric with respect to , the y value when will be the same as the y value when , which is 1.
   The correct answer is E.
8. When percent of 5,000 is subtracted from of 5,000, the difference is
   
   1. (A) 0
   2. (B) 50
   3. (C) 450
   4. (D) 495
   5. (E) 500
   Arithmetic Percents
   Since percent is , the difference asked for is .
   The correct answer is D.
9. Which of the following is the value of ?
   
   1. (A) 0.004
   2. (B) 0.008
   3. (C) 0.02
   4. (D) 0.04
   5. (E) 0.2
   Arithmetic Operations on radical expressions
   The square root and cube root evaluations are more easily carried out when 0.000064 is rewritten as . Using this rewritten form, the value asked for is
   
   
   
   
   
   
   
   The correct answer is E.
10. Raffle tickets numbered consecutively from 101 through 350 are placed in a box. What is the probability that a ticket selected at random will have a number with a hundreds digit of 2?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Probability
    There are 250 integers from 101 to 350 inclusive, 100 of which (that is, 200 through 299) have a hundreds digit of 2. Therefore, the probability that a ticket selected from the box at random will have a hundreds digit of 2 can be expressed as .
    The correct answer is A.
11. When Leo imported a certain item, he paid a 7 percent import tax on the portion of the total value of the item in excess of $1,000. If the amount of the import tax that Leo paid was $87.50, what was the total value of the item?
    
    1. (A) $1,600
    2. (B) $1,850
    3. (C) $2,250
    4. (D) $2,400
    5. (E) $2,750
    Algebra First-degree equations
    Letting x represent the total value of the item, convert the words to symbols and solve the equation.
    
    7% of value in excess of
    
    
    
    The correct answer is C.
12. The numbers of cars sold at a certain dealership on six of the last seven business days were 4, 7, 2, 8, 3, and 6, respectively. If the number of cars sold on the seventh business day was either 2, 4, or 5, for which of the three values does the average (arithmetic mean) number of cars sold per business day for the seven business days equal the median number of cars sold per day for the seven days?
    I. 2
    II. 4
    III. 5
    
    1. (A) II only
    2. (B) III only
    3. (C) I and II only
    4. (D) II and III only
    5. (E) I, II, and III
    Arithmetic Statistics
    Listed in numerical order, the given numbers are 2, 3, 4, 6, 7, and 8. If the seventh number were 2 or 4, then the numbers in numerical order would be 2, 2, 3, 4, 6, 7, and 8 or 2, 3, 4, 4, 6, 7, and 8. In either case the median would be 4 and the average would be or , neither of which equals 4. So, for neither of the values in I or II does the average equal the median. If the seventh number were 5, then the numbers in numerical order would be 2, 3, 4, 5, 6, 7, and 8. The median would be 5 and the average would be . Thus, for the value in III, the average equals the median.
    The correct answer is B.
13. A rectangular garden is to be twice as long as it is wide. If 360 yards of fencing, including the gate, will completely enclose the garden, what will be the length of the garden, in yards?
    
    1. (A) 120
    2. (B) 140
    3. (C) 160
    4. (D) 180
    5. (E) 200
    Geometry Quadrilaterals; Perimeter
    Let the width of the rectangle be x. Then the length is 2x. Since the perimeter of a rectangle is twice the sum of the length and width, it follows that
    
    
    So, the length is 120.
    The correct answer is A.
14. If , which of the following could be a value of y?
    
    1. (A) −11
    2. (B)
    3. (C)
    4. (D) 11
    5. (E) 22
    Algebra Inequalities; Absolute value
    Since is equivalent to , or , select the value that lies between and .
    That value is .
    The correct answer is C.
15. At a supermarket, John spent of his money on fresh fruits and vegetables, on meat products, and on bakery products. If he spent the remaining $6 on candy, how much did John spend at the supermarket?
    
    1. (A) $60
    2. (B) $80
    3. (C) $90
    4. (D) $120
    5. (E) $180
    Arithmetic Fractions
    The amount spent was of the total, so the $6 left was of the total. It follows that the total is (15)($6) = $90.
    The correct answer is C.
16. On Monday, a person mailed 8 packages weighing an average (arithmetic mean) of pounds, and on Tuesday, 4 packages weighing an average of pounds. What was the average weight, in pounds, of all the packages the person mailed on both days?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Statistics
    Since average , the information about the two shipments of packages can be expressed as average .
    The correct answer is A.
17. 
    
    1. (A) 0.1
    2. (B) 0.111
    3. (C) 0.1211
    4. (D) 0.2341
    5. (E) 0.3
    Arithmetic Operations on rational numbers
    Calculate the squared and the cubed term, and then add the three terms.
    
    
    The correct answer is B.
18. A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox, what would be the capacity, in cubic feet, of the second sandbox?
    
    1. (A) 20
    2. (B) 40
    3. (C) 60
    4. (D) 80
    5. (E) 100
    Geometry Volume
    When all the dimensions of a three-dimensional object are changed by a factor of 2, the capacity, or volume, changes by a factor of 8. Thus the capacity of the second sandbox is cubic feet.
    The correct answer is D.
19. A bakery opened yesterday with its daily supply of 40 dozen rolls. Half of the rolls were sold by noon, and 80 percent of the remaining rolls were sold between noon and closing time. How many dozen rolls had not been sold when the bakery closed yesterday?
    
    1. (A) 1
    2. (B) 2
    3. (C) 3
    4. (D) 4
    5. (E) 5
    Arithmetic Operations on rational numbers; Percents
    Since half of the 40 dozen rolls were sold by noon, then dozen rolls were left to be sold after noon. Because 80 percent of those 20 were sold, percent of them or dozen rolls had not been sold when the bakery closed.
    The correct answer is D.
20. What is the 25th digit to the right of the decimal point in the decimal form of ?
    
    1. (A) 3
    2. (B) 4
    3. (C) 5
    4. (D) 6
    5. (E) 7
    Arithmetic Properties of numbers
    The fraction in its decimal form is . . .. Every odd-numbered digit to the right of the decimal point is 5, so the 25th digit must be 5.
    The correct answer is C.
21. 150 is what percent of 30?
    
    1. (A) 5%
    2. (B) 20%
    3. (C) 50%
    4. (D) 200%
    5. (E) 500%
    Arithmetic Percents
    Let x be the desired percent in the problem. The given information can be expressed by the following equation, which can then be solved for x.
    
    
    
    
    Then, 5 expressed as a percent is 500%.
    The correct answer is E.
22. The ratio 2 to is equal to the ratio
    
    1. (A) 6 to 1
    2. (B) 5 to 1
    3. (C) 3 to 2
    4. (D) 2 to 3
    5. (E) 1 to 6
    Arithmetic Operations on rational numbers
    The ratio 2 to is the same as , which is the same as a ratio of 6 to 1.
    The correct answer is A.
23. Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottles could 10 such machines produce in 4 minutes?
    
    1. (A) 648
    2. (B) 1,800
    3. (C) 2,700
    4. (D) 10,800
    5. (E) 64,800
    Arithmetic Operations on rational numbers
    Since there are 6 machines, each machine does of the work. Each machine can produce bottles per minute, so 10 machines can produce bottles per minute. Therefore, the 10 machines can produce 1,800 bottles in 4 minutes.
    The correct answer is B.
    

    

24. Of the five coordinates associated with points A, B, C, D, and E on the number line above, which has the greatest absolute value?
    
    1. (A) A
    2. (B) B
    3. (C) C
    4. (D) D
    5. (E) E
    Arithmetic Properties of numbers
    The absolute value of a number x is the distance between x and 0 on the number line. Point A is farthest from 0 and thus its coordinate has the greatest absolute value.
    The correct answer is A.
25. Of the 50 researchers in a workgroup, 40 percent will be assigned to Team A and the remaining 60 percent to Team B. However, 70 percent of the researchers prefer Team A and 30 percent prefer Team B. What is the lowest possible number of researchers who will NOT be assigned to the team they prefer?
    
    1. (A) 15
    2. (B) 17
    3. (C) 20
    4. (D) 25
    5. (E) 30
    Arithmetic Percents
    The number of researchers assigned to Team A will be , and so 30 will be assigned to Team B. The number of researchers who prefer Team A is , and the rest, 15, prefer Team B.
    If all 15 who prefer Team B are assigned to Team B, which is to have 30 researchers, then 15 who prefer Team A will need to be assigned to Team B. Alternatively, since there are only 20 spots on Team A, who prefer Team A but will have to go to Team B instead.
    The correct answer is A.
26. If n is a prime number greater than 3, what is the remainder when n² is divided by 12?
    
    1. (A) 0
    2. (B) 1
    3. (C) 2
    4. (D) 3
    5. (E) 5
    Arithmetic Properties of numbers
    The simplest way to solve this problem is to choose a prime number greater than 3 and divide its square by 12 to see what the remainder is. For example, if , then , and the remainder is 1 when 25 is divided by 12. A second prime number can be used to check the result. For example, if , then , and the remainder is 1 when 49 is divided by 12. Because only one of the answer choices can be correct, the remainder must be 1.
    For the more mathematically inclined, consider the remainder when each prime number n greater than 3 is divided by 6. The remainder cannot be 0 because that would imply that n is divisible by 6, which is impossible since n is a prime number. The remainder cannot be 2 or 4 because that would imply that n is even, which is impossible since n is a prime number greater than 3. The remainder cannot be 3 because that would imply that n is divisible by 3, which is impossible since n is a prime number greater than 3. Therefore, the only possible remainders when a prime number n greater than 3 is divided by 6 are 1 and 5. Thus, n has the form , where q is an integer, and, therefore, n² has the form or . In either case, n² has a remainder of 1 when divided by 12.
    The correct answer is B.
27. 
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Operations with rational numbers
    Perform the arithmetic calculations as follows:
    
    
    
    
    
    
    The correct answer is D.
    

    

28. In the figure above, the coordinates of point V are
    
    1. (A) (–7,5)
    2. (B) (–5,7)
    3. (C) (5,7)
    4. (D) (7,5)
    5. (E) (7,–5)
    Geometry Coordinate geometry
    The x-coordinate of V is 7, and the y-coordinate of V is −5. Thus, the coordinates, (x,y), of V are (7,–5).
    The correct answer is E.
29. A rope 40 feet long is cut into two pieces. If one piece is 18 feet longer than the other, what is the length, in feet, of the shorter piece?
    
    1. (A) 9
    2. (B) 11
    3. (C) 18
    4. (D) 22
    5. (E) 29
    Algebra First-degree equations
    Build an equation to express the given information and solve for the answer.
    Let length of the shorter piece of rope in feet.
    Then length of the longer piece of rope in feet.
    Thus is the entire length of the rope in feet.
    

    	combine like terms
    	subtract 18 from both sides
    	divide both sides by 2

    The correct answer is B.
30. A student’s average (arithmetic mean) test score on 4 tests is 78. What must be the student’s score on a 5th test for the student’s average score on the 5 tests to be 80?
    
    1. (A) 80
    2. (B) 82
    3. (C) 84
    4. (D) 86
    5. (E) 88
    Arithmetic Statistics
    The average of the student’s first 4 test scores is 78, so the sum of the first 4 test scores is 312. If x represents the fifth test score, then the sum of all 5 test scores is and the average of all 5 test scores is . But the average of all 5 test scores is 80 so
    
    
    
    
    The correct answer is E.
31. Lucy invested $10,000 in a new mutual fund account exactly three years ago. The value of the account increased by 10 percent during the first year, increased by 5 percent during the second year, and decreased by 10 percent during the third year. What is the value of the account today?
    
    1. (A) $10,350
    2. (B) $10,395
    3. (C) $10,500
    4. (D) $11,500
    5. (E) $12,705
    Arithmetic Percents
    The first year’s increase of 10 percent can be expressed as 1.10; the second year’s increase of 5 percent can be expressed as 1.05; and the third year’s decrease of 10 percent can be expressed as 0.90. Multiply the original value of the account by each of these yearly changes.
    
    
    The correct answer is B.
32. If the quotient is positive, which of the following must be true?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Properties of numbers
    If the quotient is positive, then either a and b are both positive, or a and b are both negative.
    
    1. A and show it NEED NOT BE TRUE that .
    2. B and show it NEED NOT BE TRUE that .
    3. C The condition that ab is positive is exactly the same condition that is positive. Thus, it MUST BE TRUE that .
    4. D and show it NEED NOT BE TRUE that .
    5. E and show it NEED NOT BE TRUE that .
    The correct answer is C.
    

    

33. The dots on the graph above indicate the weights and fuel efficiency ratings for 20 cars. How many of the cars weigh more than 2,500 pounds and also get more than 22 miles per gallon?
    
    1. (A) 3
    2. (B) 5
    3. (C) 8
    4. (D) 10
    5. (E) 11
    Arithmetic Interpretation of graphs and tables
    

    

    The only dots on the graph that meet the conditions of the problem are those to the right of 25 (that is, the car has a weight in excess of 2,500 pounds) and above 22 (that is, the car has a fuel efficiency over 22 miles per gallon) as shown.
    The correct answer is B.
34. How many minutes does it take John to type y words if he types at the rate of x words per minute?
    
    1. (A)
    2. (B)
    3. (C) xy
    4. (D)
    5. (E)
    Algebra First-degree equations
    Let m represent the number of minutes it takes John to type y words. In this rate problem, the number of words typed (typing rate)(time).
    Thus, , or .
    The correct answer is B.
35. 
    
    1. (A)
    2. (B) 24
    3. (C) 25
    4. (D)
    5. (E) 32
    Arithmetic Operations on radical expressions
    Perform the indicated calculation. The following is one way to lessen the amount of arithmetic computation.
    
    
    
    
    
    
    
    The correct answer is B.
    

    

36. If O is the center of the circle above, what fraction of the circular region is shaded?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Geometry Circles and area
    Vertical angles are congruent, so of the circle is not shaded. Since there are in a circle, this makes of the circle shaded. The fraction of the circular region that is shaded is thus .
    The correct answer is C.
37. Which of the following equations is NOT equivalent to 10y² = (x + 2)(x − 2)?
    
    1. (A) 30y² = 3x² − 12
    2. (B) 20y² = (2x − 4)(x + 2)
    3. (C) 10y² + 4 = x²
    4. (D) 5y² = x² − 2
    5. (E)
    Algebra Simplifying algebraic expressions
    When x = 2 or x = −2, the equation becomes 10y² = 0, or y = 0. Since, in the equation given in (D), y does not become 0 when x = 2, it follows that the equation given in (D) is not equivalent to the given equation. Alternatively, when each of the equations given in (A) through (E) is solved for 10y² in terms of x, only the resulting equation in (D) fails to give an expression in terms of x that is equivalent to (x + 2)(x − 2) = x² − 4.
    The correct answer is D.
38. If Juan takes 11 seconds to run y yards, how many seconds will it take him to run x yards at the same rate?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Algebra Applied problems
    Juan’s running rate can be expressed as yards per second. Use this value in the formula , letting t equal the time in seconds that it will take Juan to run the distance of x yards:
    

    	solve for t by multiplying both sides by

    The correct answer is A.
39. John has 10 pairs of matched socks. If he loses 7 individual socks, what is the greatest number of pairs of matched socks he can have left?
    
    1. (A) 7
    2. (B) 6
    3. (C) 5
    4. (D) 4
    5. (E) 3
    Arithmetic Operations on rational numbers
    Determine first the lowest number of pairs of matched socks that can be made from the 7 individual socks. The lowest number of pairs that 7 individual socks can come from is 3 full pairs plus one sock from a fourth pair. The greatest number of pairs of matched socks John can have left is therefore fully matched pairs.
    The correct answer is B.
40. What is the lowest positive integer that is divisible by each of the integers 1 through 7, inclusive?
    
    1. (A) 420
    2. (B) 840
    3. (C) 1,260
    4. (D) 2,520
    5. (E) 5,040
    Arithmetic Operations on rational numbers
    A number that is divisible by the integers from 1 through 7 inclusive must have 2, 3, 4, 5, 6, and 7 as factors. The lowest positive integer will have no duplication of factors. The lowest common multiple of 2, 3, 4, and 6 is 12, and 5 and 7 are prime, so the lowest positive integer that is divisible by each of the integers 1 through 7 inclusive is .
    The correct answer is A.
41. 
    
    1. (A) −4
    2. (B) −0.25
    3. (C) 0.25
    4. (D) 0.75
    5. (E) 4
    Arithmetic Operations with rational numbers
    Perform the arithmetic calculations as follows:
    
    
    
    
    The correct answer is A.
42. If , then
    
    1. (A) −3.7
    2. (B) 0.1
    3. (C) 0.3
    4. (D) 0.5
    5. (E) 2.8
    Algebra First-degree equations
    Work the problem to solve for x.
    

    	multiply both sides by
    	subtract 1 from both sides
    	divide both sides by 5

    The correct answer is B.
    

    

43. In the figure above, the point on segment PQ that is twice as far from P as from Q is
    
    1. (A) (3,1)
    2. (B) (2,1)
    3. (C) (2,–1)
    4. (D) (1.5,0.5)
    5. (E) (1,0)
    Geometry Coordinate geometry
    On a segment, a point that is twice as far from one end as the other is the distance from one end. The points (0,–1), (1,0), (2,1), and (3,2) are on segment PQ, and they divide the segment into three intervals of equal length as shown in the figure below.
    

    

    Note that the point (2,1) is twice as far from P (0,–1) as from Q (3,2) and also that it is the distance from Q.
    The correct answer is B.
44. If n is an integer, which of the following must be even?
    
    1. (A)
    2. (B)
    3. (C) 2n
    4. (D)
    5. (E) n²
    Arithmetic Properties of integers
    A quick look at the answer choices reveals the expression 2n in answer choice C. 2n is a multiple of 2 and hence must be even.
    Since only one answer choice can be correct, the other answer choices need not be checked. However, for completeness:
    
    1. A is odd if n is even and even if n is odd. Therefore, it is not true that must be even.
    2. B is even if n is even and odd if n is odd. Therefore, it is not true that must be even.
    3. D is odd whether n is even or odd. Therefore, it is not true that must be even.
    4. E n² is even if n is even and odd if n is odd. Therefore, it is not true that n² must be even.
    The correct answer is C.
45. If 4 is one solution of the equation , where k is a constant, what is the other solution?
    
    1. (A) −7
    2. (B) −4
    3. (C) −3
    4. (D) 1
    5. (E) 6
    Algebra Second-degree equations
    If 4 is one solution of the equation, then substitute 4 for x and solve for k.
    
    
    
    
    
    
    Then, substitute −18 for k and solve for x.
    
    
    
    
    ,
    The correct answer is A.
46. The sum is between
    
    1. (A) and
    2. (B) and 1
    3. (C) 1 and
    4. (D) and
    5. (E) and 2
    Arithmetic Operations with rational numbers
    Since , and answer choices C, D, and E can be eliminated. Since , and answer choice A can be eliminated. Thus, .
    The correct answer is B.
47. If and , then for what value of t does ?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E) 0
    Algebra Simultaneous equations
    Since it is given that , set the expressions for x and y equal to each other and solve for t.
    

    	add 3t and 1 to both sides, then
    	divide both sides by 5

    The correct answer is D.
48. 
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E) 0
    Arithmetic Operations with rational numbers
    Perform the arithmetic calculations as follows:
    
    
    
    
    
    The correct answer is B.
49. Car X averages 25.0 miles per gallon of gasoline and Car Y averages 11.9 miles per gallon. If each car is driven 12,000 miles, approximately how many more gallons of gasoline will Car Y use than Car X?
    
    1. (A) 320
    2. (B) 480
    3. (C) 520
    4. (D) 730
    5. (E) 920
    Arithmetic Applied problems
    Car X uses 1 gallon of gasoline for every 25 miles it is driven, so Car X uses of a gallon for every 1 mile it is driven. Therefore Car X will use gallons of gasoline when it is driven 12,000 miles. Car Y uses 1 gallon of gasoline for every 11.9 or miles it is driven, so Car Y uses of a gallon for every 1 mile it is driven. Therefore Car Y will use gallons of gasoline when it is driven 12,000 miles. Thus, Car Y will use approximately 1,000 − 480 = 520 more gallons of gasoline than Car X.
    The correct answer is C.
50. How many integers n are there such that ?
    
    1. (A) Five
    2. (B) Four
    3. (C) Three
    4. (D) Two
    5. (E) One
    Algebra Inequalities
    Isolate the variable in the inequalities to determine the range within which n lies.
    

    	subtract 5 from all three values
    	divide all three values by 5

    There are four integers between and 4, namely 0, 1, 2, and 3.
    The correct answer is B.
51. If y is an integer, then the least possible value of is
    
    1. (A) 1
    2. (B) 2
    3. (C) 3
    4. (D) 4
    5. (E) 5
    Arithmetic Absolute value; Operations with integers
    Since y is an integer, is also an integer. The task is to find the integer y for which is the least. If , , and . On the other hand, if , , and . Therefore, the least possible value of occurs at a nonnegative value of y. From the chart below, it is clear that the least possible integer value of is 2, which occurs when .
    

    y
    0	23
    1	18
    2	13
    3	8
    4	3
    5	2
    6	7
    7	12

    Alternatively, since , the minimum possible real value of is 0. The integer value of y for which is least is the integer closest to the solution of the equation .
    The solution is and the integer closest to 4.6 is 5.
    The correct answer is B.
52. 
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E) 100
    Arithmetic Operations with radical expressions
    Rewrite each radical in the form , where a and b are positive integers and b is as small as possible, and then add.
    
    
    
    
    
    The correct answer is A.
53. The average (arithmetic mean) of 10, 30, and 50 is 5 more than the average of 20, 40, and
    
    1. (A) 15
    2. (B) 25
    3. (C) 35
    4. (D) 45
    5. (E) 55
    Arithmetic Statistics
    Using the formula , the given information about the first set of numbers can be expressed in the equation . From the given information then, the average of the second set of numbers is . Letting x represent the missing number, set up the equation for calculating the average for the second set of numbers, and solve for x.
    

    	simplify
    	multiply both sides by 3
    	subtract 60 from both sides

    The correct answer is A.
    
    
54. In the equation above, k is a constant. If when , what is the value of y when ?
    
    1. (A) 34
    2. (B) 31
    3. (C) 14
    4. (D) 11
    5. (E) 7
    Algebra First-degree equations
    If and when , then
    
    
    
    
    Therefore, . When , .
    The correct answer is B.
    

    Number of Solid-Colored Marbles in Three Jars
    ar	Number of red marbles	Number of green marbles	Total number of red and green marbles
    P	x	y	80
    Q	y	z	120
    R	x	z	160

55. In the table above, what is the number of green marbles in Jar R?
    
    1. (A) 70
    2. (B) 80
    3. (C) 90
    4. (D) 100
    5. (E) 110
    Arithmetic; Algebra Interpretation of tables; Applied problems
    First, set up an equation to find the total number of marbles in the three jars as follows:
    
    
    combine the like terms
    divide both sides by 2
    Then, since it can be seen from the table that the number of green marbles in Jar R is z, solve for z to answer the problem. To do this most efficiently, use the information from the table for Jar P, which is that .
    

    	substitute 80 for

    The correct answer is D.
56. Four staff members at a certain company worked on a project. The amounts of time that the four staff members worked on the project were in the ratio 2 to 3 to 5 to 6. If one of the four staff members worked on the project for 30 hours, which of the following CANNOT be the total number of hours that the four staff members worked on the project?
    
    1. (A) 80
    2. (B) 96
    3. (C) 160
    4. (D) 192
    5. (E) 240
    Arithmetic Ratio and proportion
    For a certain value of x, the numbers of hours worked on the project by the four staff members are 2x, 3x, 5x, and 6x, for a total of 16x. It is given that one of these four numbers is equal to 30. If 2x = 30, then x = 15 and 16x = 16(15) = 240, which is (E). If 3x = 30, then x = 10 and 16x = 16(10) = 160, which is (C). If 5x = 30, then x = 6 and 16x = 16(6) = 96, which is (B). If 6x = 30, then x = 5 and 16x = 16(5) = 80, which is (A).
    The correct answer is D.
57. Company P had 15 percent more employees in December than it had in January. If Company P had 460 employees in December, how many employees did it have in January?
    
    1. (A) 391
    2. (B) 400
    3. (C) 410
    4. (D) 423
    5. (E) 445
    Arithmetic Percents
    It is given that 460 is 115% of the number of employees in January. Therefore, the number of employees in January was .
    The correct answer is B.
58. A glass was filled with 10 ounces of water, and 0.01 ounce of the water evaporated each day during a 20-day period. What percent of the original amount of water evaporated during this period?
    
    1. (A) 0.002%
    2. (B) 0.02%
    3. (C) 0.2%
    4. (D) 2%
    5. (E) 20%
    Arithmetic Percents
    Since 0.01 ounce of water evaporated each day for 20 days, a total of ounce evaporated. Then, to find the percent of the original amount of water that evaporated, divide the amount that evaporated by the original amount and multiply by 100 to convert the decimal to a percent. Thus, or 2%.
    The correct answer is D.
59. A glucose solution contains 15 grams of glucose per 100 cubic centimeters of solution. If 45 cubic centimeters of the solution were poured into an empty container, how many grams of glucose would be in the container?
    
    1. (A) 3.00
    2. (B) 5.00
    3. (C) 5.50
    4. (D) 6.50
    5. (E) 6.75
    Algebra Applied problems
    Let x be the number of grams of glucose in the 45 cubic centimeters of solution. The proportion comparing the glucose in the 45 cubic centimeters to the given information about the 15 grams of glucose in the entire 100 cubic centimeters of solution can be expressed as , and thus or .
    The correct answer is E.
60. On a certain day, orangeade was made by mixing a certain amount of orange juice with an equal amount of water. On the next day, orangeade was made by mixing the same amount of orange juice with twice the amount of water. On both days, all the orangeade that was made was sold. If the revenue from selling the orangeade was the same for both days and if the orangeade was sold at $0.60 per glass on the first day, what was the price per glass on the second day?
    
    1. (A) $0.15
    2. (B) $0.20
    3. (C) $0.30
    4. (D) $0.40
    5. (E) $0.45
    Arithmetic Applied problems
    The ratio of the amount of orangeade made and sold on the first day to amount of orangeade made and sold on the second day is 2:3, because the orangeade on the first day was 1 part orange juice and 1 part water, while on the second day it was 1 part orange juice and 2 parts water. Thus, the ratio of the number of glasses of orangeade made and sold on the first day to the number of glasses of orangeade made and sold on the second day is 2:3. Since the revenues for each day were equal and 2 glasses were sold on the first day for every 3 glasses that were sold on the second day, 2($0.60) = 3p, where p represents the price per glass at which the orangeade was sold on the second day. Therefore, .
    The correct answer is D.
61. In the xy-plane, what is the slope of the line with equation 3x + 7y = 9?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D) 3
    5. (E) 7
    Algebra Coordinate geometry
    Since the given equation of the line is equivalent to 7y = −3x + 9, or , the slope of the line is . Alternatively, choose 2 points lying on the line and then use the slope formula for these 2 points. For example, substitute x = 0 in 7y = −3x + 9 and solve for y to get , substitute y = 0 in 7y = −3x + 9 and solve for x to get (3,0), then use the slope formula to get .
    The correct answer is B.
    

    

62. In the figure above, if PQRS is a parallelogram, then
    
    1. (A) 30
    2. (B) 35
    3. (C) 40
    4. (D) 70
    5. (E) 100
    Geometry Polygons
    Since PQRS is a parallelogram, the following must be true:
    

    	corresponding angles are congruent
    	consecutive angles are supplementary

    Solving the first equation for y gives . Substituting this into the second equation gives
    
    
    
    Thus, .
    The correct answer is A.
63. If 1 kilometer is approximately 0.6 mile, which of the following best approximates the number of kilometers in 2 miles?
    
    1. (A)
    2. (B) 3
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Applied problems
    Since , divide to find that , or . Therefore, , or .
    The correct answer is A.
64. A certain fruit stand sold apples for $0.70 each and bananas for $0.50 each. If a customer purchased both apples and bananas from the stand for a total of $6.30, what total number of apples and bananas did the customer purchase?
    
    1. (A) 10
    2. (B) 11
    3. (C) 12
    4. (D) 13
    5. (E) 14
    Algebra First-degree equations; Operations with integers
    If each apple sold for $0.70, each banana sold for $0.50, and the total purchase price was $6.30, then , where x and y are positive integers representing the number of apples and bananas, respectively, the customer purchased.
    
    
    
    
    
    Since y must be an integer, must be divisible by 5. Furthermore, both x and y must be positive integers. For , 2, 3, 4, 5, 6, 7, 8, the corresponding values of are 8, 7, 6, 5, 4, 3, 2, and 1. Only one of these, 5, is divisible by 5.
    Therefore, and and the total number of apples and bananas the customer purchased is .
    The correct answer is B.
65. The average distance between the Sun and a certain planet is approximately inches. Which of the following is closest to the average distance between the Sun and the planet, in kilometers? (1 kilometer is approximately inches.)
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Measurement conversion
    Convert to kilometers and then estimate.
    
    
    
    
    
    
    The correct answer is B.
66. At a certain school, the ratio of the number of second graders to the number of fourth graders is 8 to 5, and the ratio of the number of first graders to the number of second graders is 3 to 4. If the ratio of the number of third graders to the number of fourth graders is 3 to 2, what is the ratio of the number of first graders to the number of third graders?
    
    1. (A) 16 to 15
    2. (B) 9 to 5
    3. (C) 5 to 16
    4. (D) 5 to 4
    5. (E) 4 to 5
    Arithmetic Ratio and proportion
    If F, S, T, and R represent the number of first, second, third, and fourth graders, respectively, then the given ratios are: (i) , (ii) , and (iii) . The desired ratio is . From (i), , and from (ii), . Combining these results, . From (iii), . Then . So, the ratio of the number of first graders to the number of third graders is 4 to 5.
    The correct answer is E.
67. If m is the average (arithmetic mean) of the first 10 positive multiples of 5 and if M is the median of the first 10 positive multiples of 5, what is the value of ?
    
    1. (A) −5
    2. (B) 0
    3. (C) 5
    4. (D) 25
    5. (E) 27.5
    Arithmetic Statistics
    The first 10 positive multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. From this, the average (arithmetic mean) of the 10 multiples, that is, , can be calculated:
    
    
    .
    Since there is an even number of multiples, the median, M, is the average of the middle two numbers, 25 and 30:
    
    .
    Therefore, the median minus the average is:
    
    .
    This problem can also be solved as follows. Since the values can be grouped in pairs (i.e., 5 and 50, 10 and 45, 15 and 40, etc.), each of which is symmetric with respect to the median, it follows that the average and median are equal.
    The correct answer is B.
    
    
    
68. Two integers will be randomly selected from the sets above, one integer from set A and one integer from set B. What is the probability that the sum of the two integers will equal 9?
    
    1. (A) 0.15
    2. (B) 0.20
    3. (C) 0.25
    4. (D) 0.30
    5. (E) 0.33
    Arithmetic; Algebra Probability; Concepts of sets
    The total number of different pairs of numbers, one from set A and one from set B is . Of these 20 pairs of numbers, there are 4 possible pairs that sum to 9: 2 and 7, 3 and 6, 4 and 5, and 5 and 4. Thus, the probability that the sum of the two integers will be 9 is equal to .
    The correct answer is B.
69. In the coordinate plane, a circle has center (2,–3) and passes through the point (5,0). What is the area of the circle?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Geometry Coordinate geometry; Circles; Area
    The area of a circle is given by , where r is the radius of the circle. The value of r ² is the square of the distance from the center to a point of the circle. Using the distance formula, r ² = (2 − 5)² + (–3 − 0)² = 9 + 9 = 18. Therefore, the area of the circle is .
    The correct answer is E.
70. At a certain instant in time, the number of cars, N, traveling on a portion of a certain highway can be estimated by the formula
    
    
    where L is the number of lanes in the same direction, d is the length of the portion of the highway, in feet, and s is the average speed of the cars, in miles per hour. Based on the formula, what is the estimated number of cars traveling on a mile portion of the highway if the highway has 2 lanes in the same direction and the average speed of the cars is 40 miles per hour?
    
    1. (A) 155
    2. (B) 96
    3. (C) 80
    4. (D) 48
    5. (E) 24
    Algebra Simplifying algebraic expressions
    Substitute , , and into the given formula and calculate the value for N.
    
    
    
    
    
    
    
    The correct answer is D.
71. Yesterday’s closing prices of 2,420 different stocks listed on a certain stock exchange were all different from today’s closing prices. The number of stocks that closed at a higher price today than yesterday was 20 percent greater than the number that closed at a lower price. How many of the stocks closed at a higher price today than yesterday?
    
    1. (A) 484
    2. (B) 726
    3. (C) 1,100
    4. (D) 1,320
    5. (E) 1,694
    Arithmetic Percents
    Let n be the number of stocks that closed at a lower price today than yesterday. Then 1.2n is the number of stocks that closed at a higher price today than yesterday, and 1.2n is the value asked for. Because the total number of stocks is 2,420, it follows that n + 1.2n = 2,420, or 2.2n = 2,420.
    Therefore, , and hence 1.2n = (1.2)(1,100) = 1,320.
    The correct answer is D.
72. If and , then
    
    1. (A)
    2. (B)
    3. (C)
    4. (D) 1
    5. (E) 4
    Algebra First-degree equations
    Since , it is possible to simplify this equation and solve for x as follows:
    

    	divide both sides by y
    	multiply both sides by 2
    	solve for x

    The correct answer is C.
73. If and , the value of x must be between which of the following pairs of numbers?
    
    1. (A) −3 and 10
    2. (B) −3 and 4
    3. (C) 2 and 7
    4. (D) 3 and 4
    5. (E) 3 and 10
    Algebra Inequalities
    Isolate x in each given inequality. Since , then . Since , then . Thus, , which means the value of x must be between −3 and 10.
    The correct answer is A.
74. A gym class can be divided into 8 teams with an equal number of players on each team or into 12 teams with an equal number of players on each team. What is the lowest possible number of students in the class?
    
    1. (A) 20
    2. (B) 24
    3. (C) 36
    4. (D) 48
    5. (E) 96
    Arithmetic Properties of numbers
    The lowest value that can be divided evenly by 8 and 12 is their least common multiple (LCM). Since and , the LCM is .
    The correct answer is B.
    

    

75. In the figure above, triangle ABC is equilateral, and point P is equidistant from vertices A, B, and C. If triangle ABC is rotated clockwise about point P, what is the minimum number of degrees the triangle must be rotated so that point B will be in the position where point A is now?
    
    1. (A) 60
    2. (B) 120
    3. (C) 180
    4. (D) 240
    5. (E) 270
    Geometry Angles
    

    

    Since ΔABC is equilateral, the measure of ∠ACB is 60°. Therefore, the measure of ∠BCD is 180° − 60° = 120°. Rotating the figure clockwise about point P through an angle of 120° will produce the figure shown below.
    

    

    Then rotating this figure clockwise about point P through an angle of 120° will produce the figure shown below.
    

    

    In this figure, point B is in the position where point A was in the original figure. The triangle was rotated clockwise about point P through 120° + 120° = 240°.
    The correct answer is D.
76. At least of the 40 members of a committee must vote in favor of a resolution for it to pass. What is the greatest number of members who could vote against the resolution and still have it pass?
    
    1. (A) 19
    2. (B) 17
    3. (C) 16
    4. (D) 14
    5. (E) 13
    Arithmetic Operations on rational numbers
    If at least of the members must vote in favor of a resolution, then no more than of the members can be voting against it. On this 40-member committee, , which means that no more than 13 members can vote against the resolution and still have it pass.
    The correct answer is E.
77. If n = 20! + 17, then n is divisible by which of the following?
    I. 15
    II. 17
    III. 19
    
    1. (A) None
    2. (B) I only
    3. (C) II only
    4. (D) I and II
    5. (E) II and III
    Arithmetic Properties of numbers
    Because 20! is the product of all integers from 1 through 20, it follows that 20! is divisible by each integer from 1 through 20. In particular, 20! is divisible by each of the integers 15, 17, and 19. Since 20! and 17 are both divisible by 17, their sum is divisible by 17, and hence the correct answer will include II. If n were divisible by 15, then n − 20! would be divisible by 15. But, n − 20! = 17 and 17 is not divisible by 15. Therefore, the correct answer does not include I. If n were divisible by 19, then n − 20! would be divisible by 19. But, n − 20! = 17 and 17 is not divisible by 19. Therefore, the correct answer does not include III.
    The correct answer is C.
    

    

78. In the rectangular solid above, the three sides shown have areas 12, 15, and 20, respectively. What is the volume of the solid?
    
    1. (A) 60
    2. (B) 120
    3. (C) 450
    4. (D) 1,800
    5. (E) 3,600
    Geometry Volume
    

    

    In the figure above, the areas of the three sides are given by HW, HL, and LW. Assuming HW = 12 = (3)(4), HL = 15 = (3)(5), and LW = 20 = (5)(4), it is clear that possible choices for the edge lengths are H = 3, W = 4, and L = 5. Therefore, the volume of the rectangular solid is (3)(4)(5) = 60.
    Alternatively, the product of the given areas, in some order, is (HW)(HL)(LW) = (12)(15)(20). Then,
    
    
    The correct answer is A.
79. After driving to a riverfront parking lot, Bob plans to run south along the river, turn around, and return to the parking lot, running north along the same path. After running 3.25 miles south, he decides to run for only 50 minutes more. If Bob runs at a constant rate of 8 minutes per mile, how many miles farther south can he run and still be able to return to the parking lot in 50 minutes?
    
    1. (A) 1.5
    2. (B) 2.25
    3. (C) 3.0
    4. (D) 3.25
    5. (E) 4.75
    Algebra Applied problems
    After running 3.25 miles south, Bob has been running for .
    Thus, if t is the number of additional minutes that Bob can run south before turning around, then the number of minutes that Bob will run north, after turning around, will be t + 26. Since Bob will be running a total of 50 minutes after the initial 26 minutes of running, it follows that t + (t + 26) = 50, or t = 12. Therefore, Bob can run south an additional before turning around.
    The correct answer is A.
80. M is the sum of the reciprocals of the consecutive integers from 201 to 300, inclusive. Which of the following is true?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Estimation
    Because is less than each of the 99 numbers , , . . . , , it follows that (the sum of 99 identical values) is less than .
    Therefore, adding to both sides of this last inequality, it follows that (the sum of 100 identical values) is less than . Hence, or . Also, because is greater than each of the 100 numbers , , . . . , , it follows that (the sum of 100 identical values) is greater than . Hence, or . From and , it follows that .
    The correct answer is A.
81. Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours. Working alone at its constant rate, Machine A produces 800 nails in y hours. In terms of x and y, how many hours does it take Machine B, working alone at its constant rate, to produce 800 nails?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Algebra Applied problems
    Let R_A and R_B be the constant rates, in nails per hour, at which Machines A and B work, respectively. Then it follows from the given information that and .
    Hence, , or .
    Therefore, the time, in hours, it would take Machine B to produce 800 nails is given by .
    The correct answer is E.
82. In the Johnsons’ monthly budget, the dollar amounts allocated to household expenses, food, and miscellaneous items are in the ratio 5:2:1, respectively. If the total amount allocated to these three categories is $1,800, what is the amount allocated to food?
    
    1. (A) $900
    2. (B) $720
    3. (C) $675
    4. (D) $450
    5. (E) $225
    Algebra Applied problems
    Since the ratio is 5:2:1, let 5x be the money allocated to household expenses, 2x be the money allocated to food, and 1x be the money allocated to miscellaneous items. The given information can then be expressed in the following equation and solved for x.
    

    	combine like terms
    	divide both sides by 8

    The money allocated to food is .
    The correct answer is D.
83. There are 4 more women than men on Centerville’s board of education. If there are 10 members on the board, how many are women?
    
    1. (A) 3
    2. (B) 4
    3. (C) 6
    4. (D) 7
    5. (E) 8
    Algebra Simultaneous equations; Applied problems
    Let m be the number of men on the board and w be the number of women on the board. According to the problem,
    

    	because there are 4 more women than men and
    	because the board has a total of 10 members.

    Substituting for w in the second equation gives:
    

    	combine like terms
    	subtract 4 from both sides
    	divide both sides by 2

    Using the first equation, women on the board.
    This problem can also be solved without algebra by listing the (m,w) possibilities for . These possibilities are (0,4), (1,5), (2,6), (3,7), etc., and hence the pair in which is (3,7).
    The correct answer is D.
84. Leona bought a 1-year, $10,000 certificate of deposit that paid interest at an annual rate of 8 percent compounded semiannually. What was the total amount of interest paid on this certificate at maturity?
    
    1. (A) $10,464
    2. (B) $ 864
    3. (C) $ 816
    4. (D) $ 800
    5. (E) $ 480
    Arithmetic Operations with rational numbers
    Using the formula , where A is the amount of money after t (1 year), P is the principal amount invested ($10,000), r is the annual interest rate (0.08), and n is the number of times compounding occurs annually (2), the given information can be expressed as follows and solved for A :
    
    
    
    
    
    Thus, since A is the final value of the certificate, the amount of interest paid at maturity is .
    The correct answer is C.
85. 
    
    1. (A) 840.0
    2. (B) 84.0
    3. (C) 8.4
    4. (D) 0.84
    5. (E) 0.084
    Arithmetic Operations with rational numbers
    To make the calculations less tedious, convert the decimals to whole numbers times powers of 10 as follows:
    
    
    
    
    
    
    
    
    
    
    The correct answer is A.
86. Machine A produces bolts at a uniform rate of 120 every 40 seconds, and Machine B produces bolts at a uniform rate of 100 every 20 seconds. If the two machines run simultaneously, how many seconds will it take for them to produce a total of 200 bolts?
    
    1. (A) 22
    2. (B) 25
    3. (C) 28
    4. (D) 32
    5. (E) 56
    Algebra Applied problems
    Determine the production rates for each machine separately, and then calculate their production rate together.
    
    Rate of Machine bolts per second
    Rate of Machine bolts per second
    
    Build an equation with the number of seconds it takes to produce 200 bolts.
    

    	(rate)(time) amount produced
    	solve for s

    The correct answer is B.
87. If n is an integer greater than 6, which of the following must be divisible by 3?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Properties of numbers
    The easiest and quickest way to do this problem is to choose an integer greater than 6, such as 7, and eliminate answer choices in which the value of the expression is not divisible by 3:
    
    1. A , which is divisible by 3, so A cannot be eliminated.
    2. B , which is divisible by 3, so B cannot be eliminated.
    3. C , which is not divisible by 3, so C can be eliminated.
    4. D , which is not divisible by 3, so D can be eliminated.
    5. E , which is divisible by 3, so E cannot be eliminated.
    Choose another integer greater than 6, such as 8, and test the remaining answer choices:
    
    1. A , which is divisible by 3, so A cannot be eliminated.
    2. B , which is not divisible by 3, so B can be eliminated.
    3. E , which is not divisible by 3, so E can be eliminated.
    Thus, A is the only answer choice that has not been eliminated.
    For the more mathematically inclined, if n is divisible by 3, then the expression in each answer choice is divisible by 3. Assume, then, that n is not divisible by 3. If the remainder when n is divided by 3 is 1, then for some integer q. All of the expressions , , , and are divisible by 3 [i.e., , , , ], and none of the expressions , , , , , and is divisible by 3. Therefore, if the remainder when n is divided by 3 is 1, only the expressions in answer choices A, B, and E are divisible by 3. On the other hand, if the remainder when n is divided by 3 is 2, then for some integer q. All of the expressions , , , and are divisible by 3 [i.e., , , , ], and none of the expressions , , , , , and is divisible by 3. Therefore, if the remainder when n is divided by 3 is 2, only the expressions in answer choices A, C, and D are divisible by 3. Only the expression in answer choice A is divisible by 3 regardless of whether n is divisible by 3, has a remainder of 1 when divided by 3, or has a remainder of 2 when divided by 3.
    The correct answer is A.
88. The total cost for Company X to produce a batch of tools is $10,000 plus $3 per tool. Each tool sells for $8. The gross profit earned from producing and selling these tools is the total income from sales minus the total production cost. If a batch of 20,000 tools is produced and sold, then Company X’s gross profit per tool is
    
    1. (A) $3.00
    2. (B) $3.75
    3. (C) $4.50
    4. (D) $5.00
    5. (E) $5.50
    Arithmetic Applied problems
    The total cost to produce 20,000 tools is . The revenue resulting from the sale of 20,000 tools is . The gross profit is , and the gross profit per tool is .
    The correct answer is C.
89. A dealer originally bought 100 identical batteries at a total cost of q dollars. If each battery was sold at 50 percent above the original cost per battery, then, in terms of q, for how many dollars was each battery sold?
    
    1. (A)
    2. (B)
    3. (C) 150q
    4. (D)
    5. (E)
    Algebra Factoring and Simplifying algebraic expressions
    Since 100 batteries cost q dollars, division by 100 shows that 1 battery costs dollars. Then, since the selling price is 50 percent above the original cost per battery, the selling price of each battery can be expressed as .
    The correct answer is A.
90. In an increasing sequence of 10 consecutive integers, the sum of the first 5 integers is 560. What is the sum of the last 5 integers in the sequence?
    
    1. (A) 585
    2. (B) 580
    3. (C) 575
    4. (D) 570
    5. (E) 565
    Algebra First-degree equations
    Let the first 5 consecutive integers be represented by x, , , , and . Then, since the sum of the integers is 560, . Thus,
    

    	solve for x

    The first integer in the sequence is 110, so the next integers are 111, 112, 113, and 114. From this, the last 5 integers in the sequence, and thus their sum, can be determined. The sum of the 6th, 7th, 8th, 9th, and 10th integers is .
    This problem can also be solved without algebra: The sum of the last 5 integers exceeds the sum of the first 5 integers by because the 6th integer exceeds the 5th integer by 1, the 7th integer exceeds the 4th integer by 3, etc.
    The correct answer is A.
91. If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers?
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E)
    Arithmetic Statistics
    For an odd number of data values, the median is the middle number. Thus, 120 is the middle number, and so half of the Q − 1 remaining values are at most 120 and the other half of the Q − 1 remaining values are at least 120. In particular, data values lie to the right of 120 when the data values are listed in increasing order from left to right, and so the largest data value is . Alternatively, it is immediate that (B), (C), or (E) cannot be correct since these expressions do not have an integer value when Q is odd. For the list consisting of the single number 120 (i.e., if Q = 1), (D) fails because and (A) does not fail because .
    The correct answer is A.
92. A ladder of a fire truck is elevated to an angle of 60° and extended to a length of 70 feet. If the base of the ladder is 7 feet above the ground, how many feet above the ground does the ladder reach?
    
    1. (A) 35
    2. (B) 42
    3. (C)
    4. (D)
    5. (E)
    Geometry Triangles
    

    

    Given the figure above, determine BC. Then add 7 to determine how far above the ground the ladder reaches.
    Triangle ΔABC is a 30°-60°-90° triangle with hypotenuse of length 70 feet. Since the lengths of the sides of a 30°-60°-90° triangle are in the ratio , and so , and and so .
    Therefore, the ladder reaches feet above the ground.
    The correct answer is D.
93. If Jake loses 8 pounds, he will weigh twice as much as his sister. Together they now weigh 278 pounds. What is Jake’s present weight, in pounds?
    
    1. (A) 131
    2. (B) 135
    3. (C) 139
    4. (D) 147
    5. (E) 188
    Algebra Systems of equations
    Let J represent Jake’s weight and S represent his sister’s weight. Then and . Solve the second equation for S and get . Substituting the expression for S into the first equation gives
    
    
    
    
    
    
    The correct answer is E.
94. A store reported total sales of $385 million for February of this year. If the total sales for the same month last year was $320 million, approximately what was the percent increase in sales?
    
    1. (A) 2%
    2. (B) 17%
    3. (C) 20%
    4. (D) 65%
    5. (E) 83%
    Arithmetic Percents
    The percent increase in sales from last year to this year is 100 times the quotient of the difference in sales for the two years divided by the sales last year. Thus, the percent increase is
    
    
    
    
    
    
    The correct answer is C.
95. When positive integer x is divided by positive integer y, the remainder is 9. If , what is the value of y?
    
    1. (A) 96
    2. (B) 75
    3. (C) 48
    4. (D) 25
    5. (E) 12
    Arithmetic Properties of numbers
    The remainder is 9 when x is divided by y, so for some positive integer q. Dividing both sides by y gives . But, . Equating the two expressions for gives .
    Thus, and
    
    
    
    
    The correct answer is B.
96. In a certain city, 60 percent of the registered voters are Democrats and the rest are Republicans. In a mayoral race, if 75 percent of the registered voters who are Democrats and 20 percent of the registered voters who are Republicans are expected to vote for Candidate A, what percent of the registered voters are expected to vote for Candidate A?
    
    1. (A) 50%
    2. (B) 53%
    3. (C) 54%
    4. (D) 55%
    5. (E) 57%
    Arithmetic; Algebra Percents; Applied problems
    Letting v be the number of registered voters in the city, then the information that 60% of the registered voters are Democrats can be expressed as 0.60v. From this, it can be stated that are Republicans. The percentage of Democrats and the percentage of Republicans who are expected to vote for Candidate A can then be expressed as . Simplify the expression to determine the total percentage of voters expected to vote for Candidate A.
    
    
    
    
    The correct answer is B.
97. 
    
    1. (A)
    2. (B)
    3. (C)
    4. (D)
    5. (E) 0
    Arithmetic Operations on rational numbers
    Perform the operations in the correct order, using least common denominators when adding or subtracting fractions:
    
    
    
    
    
    
    
    
    The correct answer is E.
98. Water consists of hydrogen and oxygen, and the approximate ratio, by mass, of hydrogen to oxygen is 2:16. Approximately how many grams of oxygen are there in 144 grams of water?
    
    1. (A) 16
    2. (B) 72
    3. (C) 112
    4. (D) 128
    5. (E) 142
    Algebra Applied problems
    The mass ratio of oxygen to water is . Therefore, if x is the number of grams of oxygen in 144 grams of water, it follows that . Now solve for x: .
    The correct answer is D.
99. If and , then
    
    1. (A) −3
    2. (B)
    3. (C) 0
    4. (D)
    5. (E)
    Algebra Second-degree equations; Simultaneous equations
    Setting each factor equal to 0, it can be seen that the solution set to the first equation is and the solution set to the second equation is . Therefore, is the solution to both equations.
    The correct answer is B.
100. On a scale that measures the intensity of a certain phenomenon, a reading of corresponds to an intensity that is 10 times the intensity corresponding to a reading of n. On that scale, the intensity corresponding to a reading of 8 is how many times as great as the intensity corresponding to a reading of 3?
     
     1. (A) 5
     2. (B) 50
     3. (C) 10⁵
     4. (D) 5¹⁰
     5. (E)
     Arithmetic Operations on rational numbers
     Since 8 can be obtained from 3 by “adding 1” five times, the intensity reading is greater by a factor of .
     The correct answer is C.
101. For the positive numbers, n, , , , and , the mean is how much greater than the median?
     
     1. (A) 0
     2. (B) 1
     3. (C)
     4. (D)
     5. (E)
     Algebra Statistics
     Since the five positive numbers n, , , , and are in ascending order, the median is the third number, which is . The mean of the five numbers is
     
     
     
     
     Since , the mean is 1 greater than the median.
     The correct answer is B.
102. If , and if , then
     
     1. (A)
     2. (B) 322
     3. (C) 490
     4. (D) 554
     5. (E)
     Algebra First-degree equations
     Substitute 290 for T in the equation, and solve for K.
     
     
     
     
     
     
     
     
     The correct answer is D.
103. The water from one outlet, flowing at a constant rate, can fill a swimming pool in 9 hours. The water from a second outlet, flowing at a constant rate, can fill the same pool in 5 hours. If both outlets are used at the same time, approximately what is the number of hours required to fill the pool?
     
     1. (A) 0.22
     2. (B) 0.31
     3. (C) 2.50
     4. (D) 3.21
     5. (E) 4.56
     Arithmetic Operations on rational numbers
     The first outlet can fill the pool at a rate of of the pool per hour, and the second can fill the pool at a rate of of the pool per hour.
     Together, they can fill the pool at a rate of of the pool per hour.
     Thus, when both outlets are used at the same time, they fill the pool in hours.
     The correct answer is D.
104. If a square mirror has a 20-inch diagonal, what is the approximate perimeter of the mirror, in inches?
     
     1. (A) 40
     2. (B) 60
     3. (C) 80
     4. (D) 100
     5. (E) 120
     Geometry Perimeter; Pythagorean theorem
     Let x be the length of one of the sides of the square mirror.
     

     

     The triangles created by the diagonal are isosceles right triangles for which the Pythagorean theorem yields the following equation that can be solved for x.
     
     
     
     
     
     Therefore, the perimeter is . To avoid estimating a value for , note that , , and . The perimeter is closest to 60 because 3,200 is closer to 3,600 than it is to 1,600 or 6,400.
     The correct answer is B.
105. The present ratio of students to teachers at a certain school is 30 to 1. If the student enrollment were to increase by 50 students and the number of teachers were to increase by 5, the ratio of students to teachers would then be 25 to 1. What is the present number of teachers?
     
     1. (A) 5
     2. (B) 8
     3. (C) 10
     4. (D) 12
     5. (E) 15
     Algebra Applied problems
     Let s be the present number of students, and let t be the present number of teachers. According to the problem, the following two equations apply:
     

     	Current student to teacher ratio
     	Future student to teacher ratio

     Solving the first equation for s gives . Substitute this value of s into the second equation, and solve for t.
     

     	multiply both sides by
     	simplify by subtraction

     The correct answer is E.
106. What is the smallest integer n for which ?
     
     1. (A) 6
     2. (B) 7
     3. (C) 8
     4. (D) 9
     5. (E) 10
     Arithmetic Operations with rational numbers
     Because , a common base is 5. Rewrite the left side with 5 as a base: . It follows that the desired integer is the least integer n for which . This will be the least integer n for which , or the least integer n for which , which is 7.
     The correct answer is B.
107. Sixty percent of the members of a study group are women, and 45 percent of those women are lawyers. If one member of the study group is to be selected at random, what is the probability that the member selected is a woman lawyer?
     
     1. (A) 0.10
     2. (B) 0.15
     3. (C) 0.27
     4. (D) 0.33
     5. (E) 0.45
     Arithmetic Probability
     For simplicity, suppose there are 100 members in the study group. Since 60 percent of the members are women, there are 60 women in the group. Also, 45 percent of the women are lawyers so there are women lawyers in the study group. Therefore the probability of selecting a woman lawyer is .
     The correct answer is C.
108. Each year for 4 years, a farmer increased the number of trees in a certain orchard by of the number of trees in the orchard the preceding year. If all of the trees thrived and there were 6,250 trees in the orchard at the end of the 4-year period, how many trees were in the orchard at the beginning of the 4-year period?
     
     1. (A) 1,250
     2. (B) 1,563
     3. (C) 2,250
     4. (D) 2,560
     5. (E) 2,752
     Arithmetic Operations on rational numbers
     Increasing the number of trees each year by of the number of trees in the orchard the preceding year is equivalent to making the number of trees increase 25% per year, compounded yearly. If there were n trees at the beginning of the 4-year period, then there will be 1.25n trees at the end of the first year, 1.25(1.25n) = (1.25)²n trees at the end of the second year, 1.25[(1.25)²n] = (1.25)³n trees at the end of the third year, and 1.25[(1.25)³n] = (1.25)⁴n trees at the end of the fourth year. Hence, 6,250 = (1.25)⁴n and . The arithmetic can be greatly simplified by rewriting (1.25)⁴ as and 6,250 as (625)(10) = (5⁴)(10). Then .
     The correct answer is D.
     

     

109. According to the chart shown, which of the following is closest to the median annual number of shipments of manufactured homes in the United States for the years from 1990 to 2000, inclusive?
     
     1. (A) 250,000
     2. (B) 280,000
     3. (C) 310,000
     4. (D) 325,000
     5. (E) 340,000
     Arithmetic Interpretation of graphs and tables; Statistics
     From the chart, the approximate numbers of shipments are as follows:
     

     Year	Number of shipments
     1990	190,000
     1991	180,000
     1992	210,000
     1993	270,000
     1994	310,000
     1995	350,000
     1996	380,000
     1997	370,000
     1998	390,000
     1999	360,000
     2000	270,000

     Since there are 11 entries in the table and 11 is an odd number, the median of the numbers of shipments is the 6th entry when the numbers of shipments are arranged in order from least to greatest. In order, from least to greatest, the first 6 entries are:
     

     Number of shipments

     180,000

     190,000

     210,000

     270,000

     270,000

     310,000

     The 6th entry is 310,000.
     The correct answer is C.
110. For the positive integers a, b, and k, means that a^k is a divisor of b, but a^{k + 1} is not a divisor of b. If k is a positive integer and , then k is equal to
     
     1. (A) 2
     2. (B) 3
     3. (C) 4
     4. (D) 8
     5. (E) 18
     Arithmetic Property of numbers
     Since 72 = (2³)(3²), it follows that 2³ is a divisor of 72 and 2⁴ is not a divisor of 72. Therefore, , and hence k = 3.
     The correct answer is B.
111. If is expressed as a terminating decimal, how many zeros will t have between the decimal point and the first nonzero digit to the right of the decimal point?
     
     1. (A) Three
     2. (B) Four
     3. (C) Five
     4. (D) Six
     5. (E) Nine
     Arithmetic Exponents; Operations with rational numbers
     Use properties of positive integer exponents to get
     
     
     
     
     
     
     
     
     So, .
     Since , then , so and t has four zeros between the decimal point and the first nonzero digit to the right of the decimal point.
     The correct answer is B.