QUESTION EXPLANATIONS

1. (D) — You can pick any two points on the graph and divide the rise (change in y-values) by the run (change in x-values) to find the slope. For example, you can see that at 4 days the sprout’s height is 6 cm, giving you the point (4,6). At 8 days the sprout’s height is 7 cm, giving you the point (8,7). You can find that the slope is , meaning that 1cm of growth happens every 4 days.

2. (D) — The line y = 2x – 7 has a slope of 2, so you need to decide which of the answer choices also has a slope of 2. By rearranging the answer options into the form y = mx + b, where m is the slope, you can see that (A) has a slope of 1, (B) has a slope of -2, (C) has a slope of ½, and (D) has a slope of 2. Therefore, (D) is correct.

3. (C) — To find the y-intercept, simply set x = 0 and solve for y. This gives you 3y + 5(0) - 3 = 0, so y = 1.

4. (C) — Since half of the 4,600 languages have writing systems, there are 2,300 languages with writing systems. Of these, 23 use Arabic script. As a percentage, this is 23/2,300 = 0.01 = 1%.

5. (A) — Each year, the price increases by a factor of 1.10, so after one year it will be ($3.50)(1.10), after two years it will be ($3.50)(1.10)(1.10), and so on. Since this price can be rewritten as ($3.50)(1.10)², it follows that after t years, the price will be (3.50)(1.10)^t.

6. (C) — You can add up the numbers in the top row to find that the total number of monolingual residents is 495. You can add up the numbers in the bottom row to find that the total number of bilingual residents is 359. Therefore, the ratio of monolingual residents to bilingual residents in 495:359.

7. (B) — Since the plumber uses 20 meters of copper pipe per week and the pipe costs $5 per meter, she spends 20 × $5 = $100 per week on copper pipe. Reducing her weekly expenditure by $4 means that she would like to reduce it to $96. Therefore, since copper pipe costs $5 per meter, she should buy 96/5 meters of copper pipe.

8. (C) — Looking at the equation, you can see that the slope (otherwise known as the rate of change) is 3.97. That means for every 1cm increase in the femur length m, a person’s height will increase by 3.97cm. So, someone with a femur 1cm longer will be 3.97cm taller.

9. (C) — Between the points (1,0) and (3,8), the data points increase by 2 along the x-axis and 8 along the y-axis, meaning that the line of best fit will have a slope of 4. You can also use the graph to see that the line will continue down to intersect the y-axis at about (0,-4), meaning that the y-intercept of the line of best fit is -4. Instead, you could have plugged any point (x,y) on the line into the equation y = 4x + b and solved for b.

10. (C) — Since the buyer must purchase at least 40 cows and sheep, that means that the total of the cows and sheep, c + s, must be greater than or equal to 40. So, c + s ≥ 40. The weight of all the cows, 700 kilograms × c, plus the weight of all the sheep, 85 kilograms × s, must be no more than 20,000 kilograms in total. So, 700c + 85s ≤ 20,000.

11. (C) — When t = 0, in the year 1986, there were 293 thousand farms, so you know that the constant term must be 293. This eliminates (A) and (B). Also, since the number of farms is decreasing every year, the slope must be negative, meaning that (C) is correct.

12. (A) — In 1991, the area covered by sod was 26,797 hectares. In 2006, the area covered by sod was 27,960 hectares. You can find the percentage increase by dividing = 4%.

13. (D) — You can see by the shape of the scatterplot that China’s production of greenhouse items is growing exponentially, since it is increasing by more each year. Using the table, you can see that the area of greenhouse products in Canada is growing linearly, since it is increasing by the same amount (4,129,466) every year.

14. (C) — You can use the points given in the figure to find that the slope of the line is and the y-intercept is 2, meaning that the line represents the equation , which is option I. You can rearrange this equation to get the equation in II, so II is also an equation of the given line. However, the slope of the line in option III is -3/2, so III is not an equation of the given line. Since I and II are the correct options, the answer is (C).

15. (C) — You can put both equations in y = mx + b form and compare them. The first equation is equivalent to . You can divide the second equation by 6 to find that it is also equivalent to . Therefore, the equations represent the same line.

16. (B) — No matter what brightness the star and the nebula start with, exponential growth will always overtake linear growth eventually, since their brightness will eventually grow to a point where a 1% increase is larger than a 1-gigawatt increase. Therefore, given enough time, the nebula will eventually be brighter than the star.

17. (B) — In California, 161 + 72 = 233 people were not born in-state, out of 863 total people surveyed. In Rhode Island, only 87 out of the 863 people surveyed were not born in-state. Therefore, you can conclude that residents of California are more likely than residents of Rhode Island not to be born in-state.

(A) is incorrect because the table says nothing about leaving the state. (C) is incorrect because the table says nothing about traveling. (D) is incorrect because residents of California are still much more likely to have been born in-state than not in-state, since 630 out of the 863 people surveyed were born in-state.

18. (D) — You can calculate the fraction of Rhode Island residents who were surveyed who were born in another U.S. state as 23/863. Similarly, the percentage of California residents that were born in another U.S. state is 161/863. Therefore, the percentage of residents born in another U.S. state is 7 times greater in California than Rhode Island. Since the population of California is 37 times the population of Rhode Island, the number of residents of California born in another U.S. state is approximately 28,000 × 7 × 37 = 7,252,000, or 7.25 million.

19. (A) — Let x represent the number of tickets that were sold individually and y represent the number of tickets that were sold as part of a pair. Since 33 games of whack-a-mole were played, you know that 66 tickets were sold in total, so x + y = 66. Also, since the vendor made $172, you know that 3x + 2.5y = 172, since each individual ticket costs $3 and each ticket that is part of a pair costs $2.50. You can multiply the first equation by 2.5 and subtract it from the second equation to find that 0.5x = 7, or x = 14, meaning that 14 tickets were sold for $3.

20. (B) — For a randomly selected toaster to overheat, it must be defective, which has a 20% chance of occurring. It must also be set to the bagel setting, which has a 1/8 chance of occurring. It then has a 3% chance of overheating, so in total its chance of overheating is 0.2 × 1/8 × 0.03 = 0.00075 = 0.075%.
If you picked (D), you may have missed a decimal place.

21. (A) — Since Type B packages have a surface area of 240 square centimeters and Type A packages’ surface area is ¼ of that, you know that Type A packages have a surface area of 60 square centimeters. Since Type C packages have 2/3 the surface area of Type A packages, you know that Type C packages have a surface area of 40 square centimeters. Therefore, the difference between the surface area of a Type A package and that of a Type C package is 60 - 40 = 20 square centimeters.

22. (D) — Most of the buttercups that were exposed to the chemical were resistant to the fly, and most of the buttercups that were not exposed to the chemical were not resistant to the fly. Since the buttercups were chosen at random, there is enough evidence for a causal relationship, so (D) is correct.
(A) is incorrect because a control group helps to confirm the accuracy of an experiment. (B) is incorrect because splitting the buttercups into equal groups is not enough to make the experiment yield valid results. (C) is incorrect because random assignment does provide evidence for a causal relationship.

23. (A) — There were 88 buttercups that were found to be resistant to the fly, and 78 of them were exposed to the chemical. Therefore, the probability that a resistant buttercup was exposed to the chemical is 78/88 = 39/44.

24. (B) — Since the farmer ties every third vine to a stake, you know that there is 1 stake per 3 tomato vines, or 1/3 of a stake per vine. In the equation, you can see that as the number of vines, t, increases by 3, the number of stakes, s, increases by 1, because of the constant 1/3.

25. (C) — On the dotted line, when x = 0, y = 3. Since the solid line crosses the y-axis above the dotted line, you know that a > 3, since when x = 0, y = a on the solid line. However, since the dotted line eventually is higher than the solid line, you know that it is growing more quickly, meaning that b < 5, since these are the rates of growth of the two curves.

26. (A) — You can set up the ratios as and then cross-multiply to get w(w + 2) = (w + 1)(w+1), or w² + 2w = w² + 2w + 1. This equation simplifies to 0 = 1, which is impossible. This means that the original equation has no solution.

27. (C) — Since π and r are constants, the volume is in a proportional relationship with d². If the volume increases from 12 to 192, it increases by a factor of 16. This means that d² also increases by a factor of 16, so d increased by a factor of 4, since (4d)² = 16d².

28. (C) — Since the wind speed is 17.6 meters per second, you can solve for B as follows:

Since B = 7.6, you can round it up to 8 and consult the table to find that it represents a Gale.

29. (C) — You can solve for the point(s) of intersection by setting x² + a = 4x, or x² - 4x + a = 0. Since you know that this equation only has one solution, the quadratic must factor as a perfect square, meaning that x = 2, so (x-2)(x-2) = 0 and therefore a = 4.

30. (D) — If you let x represent the number of $0.50 increases that the company makes to the price of their book, the price can be represented as (20 + 0.5x) and the amount sold can be represented as (50 - x), since one fewer book gets sold for each price increase. The revenue is then equal to (20 + 0.5x)(50 - x) = 1000 + 5x - 0.5x². To maximize the revenue, you should look for the vertex of this parabola, which is where the revenue is at its highest (since this parabola opens down). The x-coordinate of the vertex of a parabola in the form ax² + bx + c is equal to . Since x is the number of $0.50 increases to the book price, the new price should be 20 + 0.5(5) = 22.50 dollars.

31. (1/3) — You can use multiple conversion factors to make the units what you want, as follows:

32. (4) — If |x| = x + 8, either x = x + 8 or -x = x + 8. The first case reduces to 8 = 0, so it is impossible. The second case reduces to x = -4, so -x = 4.

33. (140) — Since a and c are opposite angles, they are equal. Since their sum is 80 degrees, they must each be 40 degrees. Finally, since L and S are parallel, you know that b and a are supplementary angles, meaning they add to 180°. This means that b = 180° - 40° = 140°.

34. (66) — You can model the number of bees that survive the winter with the expression 30,000 + 500h, where h is the number of ounces of honey the beekeeper adds to the hive. Setting 30,000 + 500h = 63,250, you can solve for h = 66.5. Since the question is asking for the maximum whole number of ounces that can be added, the answer is 66.

35. (5) — Since the proposed road would be 4 kilometers long and would reduce the total travel time between Elon and McCormack by half, you know that the distance along the current roads from Elon to McCormack is 8. You can notice that the roads form a 3-4-5 special triangle (since 3 + 5 = 8), so the distance from Springton to McCormack is 5.
If you don’t notice this, you can let x be the distance from Springton to McCormack and use the Pythagorean Theorem to find that (8 - x)² + 4² = x², which you can solve to get x = 5.

36. (2) — Since Lauren can fit 2 books in a box with volume x³ and 16 books in a box with volume (kx)³ = k³x³, you know that the larger box is 8 times as big as the smaller box, so k³ = 8, meaning that k = 2.

37. (9/2 or 4.5) — You can see on the dot plot that the fastest robot cleaned 100% of the floor in 30 minutes, while the slowest robot only cleaned 30% of the floor. The robots’ speed is proportional to the percentage of the floor they cleaned, so 100/30 = 15/x, where x is the speed, in square feet per minute, of the slowest robot. You can cross-multiply and solve this equation to get x = 4.5.

38. (75) — The first step is to find the median efficiency, which you can do by examining the dot plot and repeatedly crossing off the highest and lowest dots. After you have crossed 11 dots on each side, there will be one dot remaining, in the “70” column. This means that a robot with the median efficiency cleans 70% of the room in 30 minutes. To find out how much is cleaned in 75 minutes, simply multiply 70% of the room × (75 mins)/(30 mins) = 175% of the room. The question is asking for the percentage of the room that gets cleaned twice, so this is 175% - 100% = 75%.