QUESTION EXPLANATIONS

1. (B) — Subtracting 4 from both sides, we get a = 8. Multiplying by 4, we get 4a = (4)(8) = 32.

2. (A) — Dividing the cost of the 2 hour session by the number of hours, you can find that the 2 hour session costs $95 an hour. You can subtract this from the cost of the 1 hour session to find that the 2 hour session is cheaper by $100 - $95 = $5 per hour.

3. (B) — First notice f(c) = 2c, and then plug this expression into g to get g(f(c)) = 5(2c) + 1 = 10c + 1.

4. (B) — From the question, we know that U = 0.75T and V = 0.05U. You can substitute the first equation into the second, giving you V = (0.05)(0.75T) = 0.0375 = 3.75%.

5. (B) — We can see that the pie chart is divided into thirds, with Homework taking up a third, Miscellaneous and Meals taking up another third, and Sleep and Classes taking up the last third. Therefore, the student spends 1/3 of his day sleeping and going to classes.

6. (A) — You can multiply all of f(x) by 4, you get 4f(x) = 8x + 8. You can see that a slope of 8 is 4 times as large as a slope of 2, and the line is therefore 4 times steeper.

(B) is false because the new slope is steeper, not less steep. (C) is false because x-values are not affected by this transformation. (D) is false because the y-intercept of the new line is 8.

7. (B) — By looking at the graph, you can see that the function has a negative slope and a positive y-intercept. The only equation that satisfies both these conditions is equation (B).

8. (D) — The volume of the first object is 3 cm Χ 9 cm Χ 4 cm = 108 cm³. The volume of the second object is 6 cm Χ 2 cm Χ 3 cm = 36 cm³. 108/36 = 3, so the first object is 3 times larger than the second object, which also means the second object is 3 times lighter than the first object, since they are made of the same material.

9. (C) — The only way for a new addition to be made to a set of numbers and not change the mean is if that number is equal to the mean. Therefore we can calculate the mean of the current set of numbers to find the value of the sixth integer: (30 + 45 + 75 + 75 + 100) χ 5 = 65.

10. (A) — From looking at the graph, we can see that the number of businesses increases by about 10 every 2 years. By 2015, there will be about 20 more businesses than in 2011. Since there were 100 businesses in 2011, there will be 120 businesses in 2015.

11. (A) — Substituting the values for x, y, and z into x – y + 2z, you get (a + 2b) – (2a – b) + 2(–2b) = a + 2b – 2a + b – 4b = –a – b.

12. (D) — We expect the Dungess population (solid line) to increase non-linearly because their numbers increase by a percentage of the previous year’s population, so their population growth will increase year by year. We expect the Horseshoe crab population (dotted line) to increase linearly because their numbers grow by a set amount each year. The option with a linear dotted line and a non-linear solid line is option (D).

13. (B) — Since Tom and Isabella are driving in opposite directions, the speed at which they are moving apart from each other is 65 + 77 = 142 km/h. At this rate, to find the time it takes them to be 639 km apart, simply divide 639 by 142 to get 4.5 hours. They started driving at 9:46 AM, and 4.5 hours after that is 2:16 PM.

14. (B) — We can first solve for the variables with the system of equations given. Substituting the first equation into the second, we get L + (L + 11) = 93. Solving for L, we get L = 41. When we plug this value into either equation, we find that A = 30. The product of L and A is therefore 1230.

15. (A) — Since you know that f(0) = -3, you can plug 0 into each of the answer choices to see which ones satisfy that relationship -- in this case, all the answer choices do. Next, you can try plugging in x = 1 to see which answer choices give you -4 -- in this case, only (A) and (D) do. You can plug in either of the other given x-values and determine that (A) is the only answer option satisfying all of the given points.

16. (A) — There are 3 sides with length x and 2 sides with length y², so the perimeter of this figure is given by 3x + 2y² = 333. Substituting the given value for x, you can find that y² = 144, so y = 12.

17. (A) — You can see from the chart that as p increases by 1, N(p) doubles. This means that the relationship is exponential with 2 as the base and p as the exponent. The only option that satisfies this is (A).

18. (C) — To find the density of the final solution, we first have to find the mass of the amount of ethanol and water used to make it. Because density is mass divided by volume, this means that mass = density Χ volume. The mass of ethanol added is (0.789 g/cm³)(8 cm³) = 6.312 grams, and the mass of water added = (1 g/cm³)(4 cm³) = 4 grams. The density of the final solution then is equal to total mass divided by total volume, which is (6.312 g + 4 g)/(8 cm³ + 4 cm³) = 0.859 g/cm³.

19. (C) — You can cross multiply the equation to get (x – 1)(x + 1) = 1(x + 5). Since the left side is the factored form of a difference of squares, you can simplify it to x² – 1, giving you x² – 1 = x + 5. You can now move all the terms to one side and factor the equation to get 0 = (x – 3)(x + 2), which means the roots are 3 and –2.

20. (D) — The area of a square is the square of its side, so A = s². This means each side of the original square has a length of . Looking at the rearranged figure, you can see that there are 3 full sides of the square and 4 half sides of the square making up its perimeter (the one long side on the right is a full side plus a half side). This gives you the expression 3 + 0.5(4) for the perimeter. Simplifying that equation, you get 3 + 2 = 5.

21. (C) — First, we can set everything on the left side of the equation to base 3. We know that so becomes. This means that . We now have the expression we are looking for as the exponent of 3, so we can now set 3 to the exponent of the numbers in the answers until it equals 2189.

22. (B) — For 85 to be the median of the five numbers, it must be the missing number x, because the median must be a value present in the set of data. We can see then that A is true. We can see without calculating anything that D is also true. Both B and C talk about the mean, so we can calculate the mean, which is 83.6. The median is therefore greater than the mean.

23. (D) — The first person has already been chosen, so we are looking for the probability that a person from England will be chosen from the people that are left. Originally, 7 out of 11 people were from England. After a person from England leaves the group, there are now 6 out of 10 people who are from England. This means that there is a 6 out of 10 or 60% chance that the second person will also be from England.

24. (A) — From the graph, we can see that 35 students enrolled in psychology in 2000, 40 students enrolled in 2001, and 35 students enrolled in 2002. This means a total of 35 + 40 + 35 = 110 students enrolled in psychology during the period from 2000 to 2002.

25. (D) — Both the mean and median of the number of students enrolled in biology is 25. The mean of the number of students enrolled in psychology is 33, while the median is 35. With this information, we can already identify (D) to be false without having to calculate for (C).

26. (B) — (A) is not supported by the graph because there were actually fewer students enrolled in biology than in psychology in 2001. (C) is not supported because the enrollment in biology only increased by 5 students per year from 2002 to 2004. (D) is not supported because again there were fewer students enrolled in biology than in psychology in 2003. (B) is supported by the graph because we can see that the number of students enrolled in psychology decreased by 5 students every year from 2001 to 2004.

27. (B) — From the question, we can see that x must be greater than or equal to 1, meaning –1 is not a possible value of x. This means statement I is incorrect. We can then eliminate all answer choices that contain statement I, which is everything except for (B).

28. (B) — The table gives us a system of 3 equations to work with. We can use these equations to find A, B, or C, then add its value to an equation containing the other 2 variables. As an example, we can find C and add it to the first equation (A + B) to find the value of A + B + C. Rearranging equations 1 and 2, we get A = 2.50 – B and C = 2.62 – A, respectively. Then, we substitute equation 1 into equation 2, getting C = 2.62 – (2.50 – B), which simplifies to C = 0.12 + B. Rearranging equation 3, we get B = 2.12 – C. We can plug this in to the previous equation, getting C = 0.12 + (2.12 – C), which we can then use to solve for C, giving us C = 1.12. Adding C to the original equation 1, we get A + B + C = 2.50 + 1.12 = 3.62.

29. (C) — You can start by drawing a line between A and O. This line segment has length 5, since it is a radius of the circle and the radius is half the diameter. Also, since the length of AC is 8, you know that the length of AB is half of that, which is 4. You can use the Pythagorean Theorem on the triangle ABO to find that the length of the missing side, BO, is 3.

30. (A) — Since h(t) is the function of the ball’s height, h(t) = 0 when the ball is on the ground. Setting h(t) to 0 and factoring out –2, you get 0 = –2(t² – 5t – 50). When you factor this, you get 0 = –2(t – 10)(t + 5), meaning t = –5 or 10. Since you are looking for the time after the ball was launched, you know the answer must be a positive amount of time, meaning it must be 10.

31. (8) — Subtracting the two equations gives you 2x + 7 – (2x – 1) = 2x + 7 – 2x + 1 = 8.

32. (32) — Expanding the brackets gives you 3x – 12 – 16 + 2x = 4x + 4. When we isolate all the terms containing x to one side, we get 3x + 2x – 4x = 4 + 12 + 16, which simplifies to x = 32.

33. (6) — The sentence "Four times b is equal to ten” can be expressed as 4b = 10. From this you can see that b = 10/4 = 5/2. The question then asks you to reduce b by 20%, then multiply it by 3. This can be expressed as (b – 0.2b) Χ 3, or more simply 0.8b Χ 3. Substituting the value of 0.8b, you can see that this expression is equal to (2)(3) = 6.

34. (15) — Taking the square of both sides, you get 2x + 10 = (x + 5)^2. You can then expand the equation to get 2x + 10 = x² + 10x + 25, and set one side to 0 in order to solve for x: 0 = x² + 8x + 15 = (x + 5)(x + 3). This means x = –5 or –3, making the product of the solutions (–5)(–3) = 15.

35. (1/29) — The probability of choosing one student wanting to enter finance is 6/30. The probability of choosing another student wanting to enter finance is then 5/29. The probability of both of these events occurring is therefore (6/30)(5/29) = 30/870, which can be reduced to 1/29.

36. (40) — Since the small triangle and the entire triangle share two angles (the one on the right, and the right angle), they must share all three angles, and therefore are similar triangles. You can use the Pythagorean Theorem on the bigger triangle to find that the length of the horizontal side is the square root of 100² - 60², which is 80. (This is a special 3-4-5 triangle with the side lengths multiplied by 20). Since the ratio of the vertical side to the horizontal side of the big triangle is 60/80 = 3/4, this must be the same as the corresponding ratio in the small triangle. Since the vertical side has length 30, the length marked x must have length 40.

37. (10) — There are 6 full cages of mice, which means there are 6 Χ 5 = 30 mice. If we let n be the number of male mice, then the number of female mice is twice that at 2n. There are 30 mice in total, so n + 2n = 30. Solving for n gives you n = 10. Since n is the number of male mice, there are 10 male mice.

38. (17) — The reduced cost of maintaining each cage is 0.5($1.25) = $0.625 per cage per day. The student needs 21 cages to house 102 mice, so her daily cost is $0.625 Χ 21 = $13.125. She has a budget of $225, so she can afford to maintain the cages for $225/$13.215 = 17.14 days. Rounding this to the nearest whole day gives you 17 days.