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1. (B) — You can find f (0) by plugging in 0 for x to get f (0) = 0 + 1 = 1. Similarly, g (0) = 0 – 4 = –4, so f (0) × g (0) = 1 × (–4) = –4x.
 
2. (D) — Since the ratio of novels written by Virginia Woolf to novels written by Ernest Hemingway is 1:2, if Hemingway wrote 18 novels, Virginia Woolf wrote 9. Since the ratio of novels written by James Joyce to novels written by Virginia Woolf is 1:3, James Joyce wrote 3 novels.

If you got (B), you found the number of novels Virginia Woolf wrote and stopped there.

 
3. (B) — Since 25 out of 100 people surveyed read 3 or more books per month, and the participants were chosen randomly, you know that the same proportion of people in the entire city read 3 or more books per month. You can set up the equation , where x represents the number of citizens of Omaha who read 3 or more books per month, and solve to find x = 105,000.
 
4. (B) — You can set up a conversion factor as follows:

The printer conveniently prints at a rate of 300 pages per hour, so it will take one hour to print 300 pages.

 
5. (D) — The mode of a data set is the most common entry, so the mode of this data set is 3, and (A) is true. The mean of this data set is , so (B) is true. A total of 12 of the 20 students surveyed use the internet 3 or 4 hours per day, which is 12⁄20 = 60% , so (C) is true. However, the median of this data set is 3; there are 1 + 7 + 8 + 4 = 20 students surveyed in total, so the middle value is the average of the 10th and 11th student, both of whom spent 3 hours a day on the internet. Therefore, (D) is false, so it is the correct answer.
 
6. (C) — Since the population of Dwarf Lop rabbits doubles every year, there will be 100 rabbits at the start of 2018, 200 rabbits at the start of 2019, 400 rabbits at the start of 2020, and 800 rabbits at the start of 2021.
 
7. (C) — The line in the graph goes up to the right, which means it has a positive slope (as xincreases, so does y). It crosses the y-axis below the x-axis, so it has a negative y-intercept. The line in option (C) has a slope of 2 and a y-intercept of –1, and it is the only answer option that has a positive slope and a negative y-intercept.
 
8. (C) — After the first day, the bird will have 5 pieces of string, then 8, then 11, then 14, then 17, and so on. Of the answer options, only 14 is a possible number of pieces of string that the bird could have.
 
9. (C) — You can tell from the chart that for every two additional milligrams of tannin dose per kilogram of body weight, a rat will produce 3 fewer milliliters of urine, so the slope of this linear relationship is –3⁄2. Since a rat in the control group produces 41 mL of urine per day, the y-intercept of this linear relationship is 41, so it can be modeled by the equation y = (–3⁄2)x + 41, where x is the amount of tannin given to a rat and y is the amount of urine produced. Plugging in x = 11 gives you y = 24.5 milliliters of urine in a day.
 
10. (C) — You can use the chart to find that for every 2 mg/kg increase in the amount of tannin given to a rat, the rat's urine will contain 0.2 more millimoles of sodium per liter and 0.4 more millimoles of potassium per liter; therefore, for every 1 mg/kg increase in tannin, the urine will contain 0.1 more millimoles of sodium and 0.2 more millimoles of potassium. If you let x equal the dose of tannin, then the amount of sodium can be modeled by the expression 0.1x + 1.6 and the amount of potassium can be modeled by the expression 0.2x + 0.3. These will be equal when 0.1x + 1.6 = 0.2x + 0.3, which you can solve to get x = 13.
 
11. (A) — The correct answer is (A). Standard deviation measures how much a data set varies from the mean. University A had a higher standard deviation than University B, which means that the number of hours of sleep per night varies more among students at University A.

(C) and (D) are incorrect because the question tells you nothing about the number of students at each university.

 
12. (A) — If x represents the store's original stock, you know that 36 = 0.4x, because 36 pieces was 40% of the stock. Dividing both sides by 0.4 gives you x = 90. Since the question is asking for the remaining stock, you can subtract 36 from 90 to get 90 – 36 = 54 pieces still in the store.

If you got (D), you found the number of pieces total in the store before any were bought.

 
13. (C) — You can tell from the table that the number of fire ant colonies is increasing at a constant rate (2,381 per year), which means that it can be represented by a linear relationship. Furthermore, the number of colonies is increasing, so the line will have a positive slope, and since Greenville does not start out with 0 colonies in 1992, the y-intercept of the line will be positive. Option (C) is the only choice that satisfies these conditions.
 
14. (A) — If the dietician ate 90 grams of protein, she would need to eat 90⁄2 = 45 grams of fat, which would be a total of 135 grams. However, she needs to eat more than 135 grams of these combined per day, so this is not possible.
 
15. (D) — The easiest way to solve this problem is to test each answer choice—(D) works because in one year, Lola would be 32. Since Lola is 9 years older than Maggie, Maggie would be 23, which has the same digits as 32 but in reverse order.
 
16. (B) — You can expand the right side of the equation to get x2 – 81, so a = 0 and b = –81. Therefore, ab = 0 × –81 = 0.
 
17. (B) — You can multiply the expression by xy⁄xy and cancel out so that the denominators of each of the terms are the same:


 
18. (C) — You can use a weighted average to find that the percentage of people surveyed who thought the streets were safe for pedestrians is , so I is true. Since the survey was conducted randomly and 589 respondents preferred to walk while only 372 preferred to bike, II is also true. However, since the survey was conducted in New York only, the results will not be representative of the entire country, so III is not true.
 
19. (A) — Since f (2) = 3, you know that (2, 3) is a point on the line. The line also passes through the point (–6, –13), so you can find the slope of the line by dividing the difference in y-values by the difference in x-values: . You can then solve for the y-intercept, b, by plugging either point into the equation y = 2x + b to get 3 = 2(2) + b, so b = –1.
 
20. (C) — Since the two right triangles on the bottom have two sides (the sides marked with dashes and the shared side) and an angle (the right angle) in common, they are congruent, so each of them is a right-angled triangle with legs of length 3x and 4x, meaning their hypotenuses must each have length 5x, using the Pythagorean Theorem. The semicircle at the top has diameter 6x, so it has a circumference of 6πx⁄2 = 3πx. In total, the perimeter of the shape is 5x + 5x + 3πx = 10x + 3πx.
 
21. (A) — Use exponent rules to solve for x:


 
22. (A) — Since the odds of a male being born are 2:1, the probability is 2⁄(2+1) = 2⁄3. Since the chance that the female will reproduce asexually is 75%, the overall chance that a male will be born is 2⁄3 × 0.75 = 0.5
 
23. (D) — Since 16-ounce cans made up 20% of the total number of espresso cans sold, you can set up the equation s = 0.2(4,500 + s) and solve to get s = 1,125. Next, since you know the total number of 16-ounce cans sold, you can set 1,525 + m + 1,125 = 3,000 and find that m = 350.
 
24. (A) — Since (x + 2)2 = 4, you know that x + 2 = 2 or x + 2 = –2. Solving these equations gives you x = 0 or x = –4, and of these, only –4 is an answer option.
 
25. (B) — The correct answer is (B). You can find the number of interceptions that Brett Favre threw, b, by using the given TD:INT ratio for him: 1.53 = 508⁄b, so b = 508⁄1.53 = 332. You can use the same process to find the number of interceptions thrown by Peyton Manning, p: 2.13 = 539⁄p, so p = 539⁄2.13 = 253, which is a difference of 332 - 253 = 79.
 
26. (C) — Using the table, you can see that Joe Montana’s TD:INT ratio is 1.96, so his dot on the graph must be one of the ones near 2.00. To find out which, you can notice that the other TD:INT ratio close to 2 is Donovan McNabb’s, which is exactly 2, so you know that Joe Montana’s dot must be the one farther to the left. You can read the y-coordinate of this dot to find that Joe Montana passed just over 40,000 yards in his career. The only answer choice that is close to this is 40,500, so (C) is correct.
 
27. (C) — Since Dan Marino’s TD:INT ratio is 1.67, you know that he is represented by the dot on the graph that is just above 60,000 on the vertical axis, meaning that he passed just over 60,000 yards in his career. (Since Kurt Warner has a TD:INT ratio of 1.63, you know that the dot just above 30,000 must be his and not Dan Marino’s). To find Dan Marino’s expected number of passing yards according to the line of best fit, trace down from his dot to the line of best fit and then trace across horizontally to find that he was predicted to have thrown around 42,000 passing yards in his career. The difference between roughly 61,000 and roughly 42,000 is roughly 19,000 yards.
 
28. (C) — You can find the median number of hours for Class A by crossing off the four highest and four lowest entries and averaging the two remaining middle entries, leaving you with a median of 2 hours. Using the same process, you can find that the median of Class B is also 2 hours. To find the mean of Class A, add up all the data points and divide by the number of entries: . Similarly, you can find the mean of Class B to equal . Therefore, Classes A and B have the same median but Class B has a lower mean.
 
29. (B) — You can set up a right-angled triangle with Nathan at one acute angle, the beach at the right angle, and the vulture at the other angle. Since the angle at Nathan is 30°, this is a 30-60-90 triangle, so the ratio of the side opposite the 60° angle to the side opposite the 30° angle is √3 : 1. Since the base of the triangle is 3, the vertical distance between the beach and the vulture is 3⁄√3 = √3.
 
30. (D) — Since the parabola opens upward, you know that a is positive. Since the y-intercept of the parabola is positive, you know that c is positive. Therefore, ac must be positive since it is the product of two positive numbers.

(A) is incorrect since b must be negative, because at some positive x-value the function is negative, but ax2 and c are both still positive, so bx must be negative, meaning b is negative.

(B) is incorrect because b – a is a negative number minus a positive number, which will always be negative.

(C) is incorrect because c is positive, so -c is negative.

 
31. (15) — Linda worked a total of 6 + 8 + 6 + 8 + 5 = 33 hours, so you can divide 495 by 33 to get $15, her hourly wage.
 
32. (672) — Each molecule of nitrogen trifluoride contains 3 atoms of hydrogen, so 2,016 atoms of hydrogen can make 2016⁄3 = 672 molecules of nitrogen trifluoride.
 
33. (3) — You can add the top two equations together to find that A + B + C + D = 15. You can then add the bottom two equations together to find that A + B + C + D = 5x, so you know 5x = 15, meaning x = 3.
 
34. (0) — You can add the fractions on the right side of the equation using a common denominator, then expand this common denominator using FOIL:

Now that both sides have the same denominator of , you can set the numerators equal to each other to get . Since the question gives you the value of A, you can start by plugging in 2.5 where A appears. It may help to then multiply each term by 2 to eliminate the decimal in the coefficient 2.5:

From here, distribute the 5 and move the resulting terms to the left side of the equation:

You should now notice that also shares the factor , so you can factor the left side of the equation and divide both sides by :

You now know that . Therefore,

 
35. (6.5) — One quick way to solve this problem is by using the speed-time graph and finding the area underneath the line.

The total distance she ran in an hour is M + N, which will be 1(6) + 0.5(1 – 0.5)(2) = 6.5 miles.

Alternately, you can recognize that her speed in mph is equal to the distance, miles, divided by time, hours. So, the distance can be found by multiplying the speed by the time. You can see that she ran 3 miles in the first half hour, because .5 hours × 6 mph = 3 miles. In the second half hour, she starts at 6mph and ends at 8mph, making her average speed 7mph. So, .5 hours × 7 mph = 3.5 miles. Finally, 3 miles + 3.5 miles = 6.5 miles.

 
36. (12) — Since the sector has a central angle of 60°, you know it makes up 60°⁄360° = 1⁄6 of the circle’s area, meaning that the circle has an area of 24π × 6 = 144π . Since A = πr2, the radius of the circle is √144 = 12.
 
37. (50) — The correct answer is 50. Each trial lasts 5 minutes, which is 5 × 60 = 300 seconds. If a new problem is presented every 6 seconds, there are 300⁄6 = 50 problems presented in one trial.
 
38. (20) — You can use the graph to see that in Trial 8, Participant 2 answered 50% of the questions correctly, and Participant 1 answered 60% of the questions correctly, so Participant 1 answered 60⁄50 = 1.20 times as many questions correctly as Participant 2, which is a percentage growth of 20%.


You can find f (0) by plugging in 0 for x to get f (0) = 0 + 1 = 1. Similarly, g (0) = 0 – 4 = –4, so f (0) × g (0) = 1 × (–4) = –4x.
Since the ratio of novels written by Virginia Woolf to novels written by Ernest Hemingway is 1:2, if Hemingway wrote 18 novels, Virginia Woolf wrote 9. Since the ratio of novels written by James Joyce to novels written by Virginia Woolf is 1:3, James Joyce wrote 3 novels.

If you got (B), you found the number of novels Virginia Woolf wrote and stopped there.

Since 25 out of 100 people surveyed read 3 or more books per month, and the participants were chosen randomly, you know that the same proportion of people in the entire city read 3 or more books per month. You can set up the equation , where x represents the number of citizens of Omaha who read 3 or more books per month, and solve to find x = 105,000.
You can set up a conversion factor as follows:

The printer conveniently prints at a rate of 300 pages per hour, so it will take one hour to print 300 pages.

The mode of a data set is the most common entry, so the mode of this data set is 3, and (A) is true. The mean of this data set is , so (B) is true. A total of 12 of the 20 students surveyed use the internet 3 or 4 hours per day, which is 12⁄20 = 60% , so (C) is true. However, the median of this data set is 3; there are 1 + 7 + 8 + 4 = 20 students surveyed in total, so the middle value is the average of the 10th and 11th student, both of whom spent 3 hours a day on the internet. Therefore, (D) is false, so it is the correct answer.
Since the population of Dwarf Lop rabbits doubles every year, there will be 100 rabbits at the start of 2018, 200 rabbits at the start of 2019, 400 rabbits at the start of 2020, and 800 rabbits at the start of 2021.
The line in the graph goes up to the right, which means it has a positive slope (as xincreases, so does y). It crosses the y-axis below the x-axis, so it has a negative y-intercept. The line in option (C) has a slope of 2 and a y-intercept of –1, and it is the only answer option that has a positive slope and a negative y-intercept.
After the first day, the bird will have 5 pieces of string, then 8, then 11, then 14, then 17, and so on. Of the answer options, only 14 is a possible number of pieces of string that the bird could have.
You can tell from the chart that for every two additional milligrams of tannin dose per kilogram of body weight, a rat will produce 3 fewer milliliters of urine, so the slope of this linear relationship is –3⁄2. Since a rat in the control group produces 41 mL of urine per day, the y-intercept of this linear relationship is 41, so it can be modeled by the equation y = (–3⁄2)x + 41, where x is the amount of tannin given to a rat and y is the amount of urine produced. Plugging in x = 11 gives you y = 24.5 milliliters of urine in a day.
You can use the chart to find that for every 2 mg/kg increase in the amount of tannin given to a rat, the rat's urine will contain 0.2 more millimoles of sodium per liter and 0.4 more millimoles of potassium per liter; therefore, for every 1 mg/kg increase in tannin, the urine will contain 0.1 more millimoles of sodium and 0.2 more millimoles of potassium. If you let x equal the dose of tannin, then the amount of sodium can be modeled by the expression 0.1x + 1.6 and the amount of potassium can be modeled by the expression 0.2x + 0.3. These will be equal when 0.1x + 1.6 = 0.2x + 0.3, which you can solve to get x = 13.
The correct answer is (A). Standard deviation measures how much a data set varies from the mean. University A had a higher standard deviation than University B, which means that the number of hours of sleep per night varies more among students at University A.

(C) and (D) are incorrect because the question tells you nothing about the number of students at each university.

If x represents the store's original stock, you know that 36 = 0.4x, because 36 pieces was 40% of the stock. Dividing both sides by 0.4 gives you x = 90. Since the question is asking for the remaining stock, you can subtract 36 from 90 to get 90 – 36 = 54 pieces still in the store.

If you got (D), you found the number of pieces total in the store before any were bought.

You can tell from the table that the number of fire ant colonies is increasing at a constant rate (2,381 per year), which means that it can be represented by a linear relationship. Furthermore, the number of colonies is increasing, so the line will have a positive slope, and since Greenville does not start out with 0 colonies in 1992, the y-intercept of the line will be positive. Option (C) is the only choice that satisfies these conditions.
If the dietician ate 90 grams of protein, she would need to eat 90⁄2 = 45 grams of fat, which would be a total of 135 grams. However, she needs to eat more than 135 grams of these combined per day, so this is not possible.
The easiest way to solve this problem is to test each answer choice—(D) works because in one year, Lola would be 32. Since Lola is 9 years older than Maggie, Maggie would be 23, which has the same digits as 32 but in reverse order.
You can expand the right side of the equation to get x2 – 81, so a = 0 and b = –81. Therefore, ab = 0 × –81 = 0.
You can multiply the expression by xy⁄xy and cancel out so that the denominators of each of the terms are the same:


You can use a weighted average to find that the percentage of people surveyed who thought the streets were safe for pedestrians is , so I is true. Since the survey was conducted randomly and 589 respondents preferred to walk while only 372 preferred to bike, II is also true. However, since the survey was conducted in New York only, the results will not be representative of the entire country, so III is not true.
Since f (2) = 3, you know that (2, 3) is a point on the line. The line also passes through the point (–6, –13), so you can find the slope of the line by dividing the difference in y-values by the difference in x-values: . You can then solve for the y-intercept, b, by plugging either point into the equation y = 2x + b to get 3 = 2(2) + b, so b = –1.
Since the two right triangles on the bottom have two sides (the sides marked with dashes and the shared side) and an angle (the right angle) in common, they are congruent, so each of them is a right-angled triangle with legs of length 3x and 4x, meaning their hypotenuses must each have length 5x, using the Pythagorean Theorem. The semicircle at the top has diameter 6x, so it has a circumference of 6πx⁄2 = 3πx. In total, the perimeter of the shape is 5x + 5x + 3πx = 10x + 3πx.
Use exponent rules to solve for x:


Since the odds of a male being born are 2:1, the probability is 2⁄(2+1) = 2⁄3. Since the chance that the female will reproduce asexually is 75%, the overall chance that a male will be born is 2⁄3 × 0.75 = 0.5
Since 16-ounce cans made up 20% of the total number of espresso cans sold, you can set up the equation s = 0.2(4,500 + s) and solve to get s = 1,125. Next, since you know the total number of 16-ounce cans sold, you can set 1,525 + m + 1,125 = 3,000 and find that m = 350.
Since (x + 2)2 = 4, you know that x + 2 = 2 or x + 2 = –2. Solving these equations gives you x = 0 or x = –4, and of these, only –4 is an answer option.
The correct answer is (B). You can find the number of interceptions that Brett Favre threw, b, by using the given TD:INT ratio for him: 1.53 = 508⁄b, so b = 508⁄1.53 = 332. You can use the same process to find the number of interceptions thrown by Peyton Manning, p: 2.13 = 539⁄p, so p = 539⁄2.13 = 253, which is a difference of 332 - 253 = 79.
Using the table, you can see that Joe Montana’s TD:INT ratio is 1.96, so his dot on the graph must be one of the ones near 2.00. To find out which, you can notice that the other TD:INT ratio close to 2 is Donovan McNabb’s, which is exactly 2, so you know that Joe Montana’s dot must be the one farther to the left. You can read the y-coordinate of this dot to find that Joe Montana passed just over 40,000 yards in his career. The only answer choice that is close to this is 40,500, so (C) is correct.
Since Dan Marino’s TD:INT ratio is 1.67, you know that he is represented by the dot on the graph that is just above 60,000 on the vertical axis, meaning that he passed just over 60,000 yards in his career. (Since Kurt Warner has a TD:INT ratio of 1.63, you know that the dot just above 30,000 must be his and not Dan Marino’s). To find Dan Marino’s expected number of passing yards according to the line of best fit, trace down from his dot to the line of best fit and then trace across horizontally to find that he was predicted to have thrown around 42,000 passing yards in his career. The difference between roughly 61,000 and roughly 42,000 is roughly 19,000 yards.
You can find the median number of hours for Class A by crossing off the four highest and four lowest entries and averaging the two remaining middle entries, leaving you with a median of 2 hours. Using the same process, you can find that the median of Class B is also 2 hours. To find the mean of Class A, add up all the data points and divide by the number of entries: . Similarly, you can find the mean of Class B to equal . Therefore, Classes A and B have the same median but Class B has a lower mean.
You can set up a right-angled triangle with Nathan at one acute angle, the beach at the right angle, and the vulture at the other angle. Since the angle at Nathan is 30°, this is a 30-60-90 triangle, so the ratio of the side opposite the 60° angle to the side opposite the 30° angle is √3 : 1. Since the base of the triangle is 3, the vertical distance between the beach and the vulture is 3⁄√3 = √3.
Since the parabola opens upward, you know that a is positive. Since the y-intercept of the parabola is positive, you know that c is positive. Therefore, ac must be positive since it is the product of two positive numbers.

(A) is incorrect since b must be negative, because at some positive x-value the function is negative, but ax2 and c are both still positive, so bx must be negative, meaning b is negative.

(B) is incorrect because b – a is a negative number minus a positive number, which will always be negative.

(C) is incorrect because c is positive, so -c is negative.

Linda worked a total of 6 + 8 + 6 + 8 + 5 = 33 hours, so you can divide 495 by 33 to get $15, her hourly wage.
Each molecule of nitrogen trifluoride contains 3 atoms of hydrogen, so 2,016 atoms of hydrogen can make 2016⁄3 = 672 molecules of nitrogen trifluoride.
You can add the top two equations together to find that A + B + C + D = 15. You can then add the bottom two equations together to find that A + B + C + D = 5x, so you know 5x = 15, meaning x = 3.
You can add the fractions on the right side of the equation using a common denominator, then expand this common denominator using FOIL:

Now that both sides have the same denominator of , you can set the numerators equal to each other to get . Since the question gives you the value of A, you can start by plugging in 2.5 where A appears. It may help to then multiply each term by 2 to eliminate the decimal in the coefficient 2.5:

From here, distribute the 5 and move the resulting terms to the left side of the equation:

You should now notice that also shares the factor , so you can factor the left side of the equation and divide both sides by :

You now know that . Therefore,

One quick way to solve this problem is by using the speed-time graph and finding the area underneath the line.

The total distance she ran in an hour is M + N, which will be 1(6) + 0.5(1 – 0.5)(2) = 6.5 miles.

Alternately, you can recognize that her speed in mph is equal to the distance, miles, divided by time, hours. So, the distance can be found by multiplying the speed by the time. You can see that she ran 3 miles in the first half hour, because .5 hours × 6 mph = 3 miles. In the second half hour, she starts at 6mph and ends at 8mph, making her average speed 7mph. So, .5 hours × 7 mph = 3.5 miles. Finally, 3 miles + 3.5 miles = 6.5 miles.

Since the sector has a central angle of 60°, you know it makes up 60°⁄360° = 1⁄6 of the circle’s area, meaning that the circle has an area of 24π × 6 = 144π . Since A = πr2, the radius of the circle is √144 = 12.
The correct answer is 50. Each trial lasts 5 minutes, which is 5 × 60 = 300 seconds. If a new problem is presented every 6 seconds, there are 300⁄6 = 50 problems presented in one trial.
You can use the graph to see that in Trial 8, Participant 2 answered 50% of the questions correctly, and Participant 1 answered 60% of the questions correctly, so Participant 1 answered 60⁄50 = 1.20 times as many questions correctly as Participant 2, which is a percentage growth of 20%.