QUESTION EXPLANATIONS

1. (B) — We can solve this problem by plugging in the value given for y in order to solve for x. x = 2(x - 2) + 4. Simplify x = 2x - 4 + 4. Simplify further x = 0. The correct answer is (B).

1. (C) — (C) is the correct answer. The passage states that Mrs. Honeychurch “wanted to show people that her daughter was marrying a presentable man” (lines 4−5), indicating that she wants to show her neighbors that her daughter has made a respectable match. Furthermore, “it pleased her” (line 11) to be congratulated on this match; she likes feeling the approval of others. Together these reactions suggest that she is concerned with her image in her society, giving support to (C). Although Mrs. Honeychurch has positive feelings about her daughter’s engagement to Cecil, the passage does not give any indication that she feels particularly warmly toward him for his own sake; thus, (A) is incorrect. Although the passage refers briefly to Mrs. Honeychurch thinking back on a dress (lines 28−30), there is no specific indication that she sews dresses or anything else particularly well; thus, (B) is incorrect. Mrs. Honeychurch is the one who insists on their attendance at the garden-party (lines 1−3), so she is the opposite of disinterested in it; thus, (D) is incorrect.

2. (B) — The equation for a line can be represented by f(x) = mx + b where b is the y-intercept (where x equals zero and the line crosses the y-axis). From the table we know when x = 0, f(x) = 3. The y-intercept is 3 so we can eliminate answers (C) and (D). Plug in the next coordinates for answer (A) 4 does not equal 2 + 3, so knock out (A). The correct answer is (B). Double-check that the coordinates make the expression true.

3. (B) — We can set up an algebraic ratio of pigs/acres to solve the problem. Cross multiply 8 pigs/1.5 acres = x pigs/6 acres. 48 = 1.5x. x = 48/1.5 = 32, the answer is (B).

4. (B) — We can cross multiply to get 4(x - 1) = 3(2x - 6). Simplify 4x - 4 = 6x -18. Subtract 4x from both sides of the equations and add 18 to both sides of the equation. 14 = 2x, x = 7. The correct answer is (B).

5. (D) — We can solve for x first. Subtract 4 from both sides of the equation 8x + 4 = 48, 8x = 44, x = 5.5. Plug the value for x in to 2x + 1 = 2(5.5) + 1 = 12. The correct answer is (D).

6. (B) — The slope of a line is calculated by rise over run. Look carefully at the graph. The y-axis is marked by a 9 and -9 and delineated by units of 3. The x-axis is marked by -6 and 6 and delineated by units of 2. Look at the y intercept. The coordinates are 0, -3. The x intercept is 2,0. You can calculate the rise (3) over run (2) from these two points. The slope of the function is 3/2, the correct answer is (B).

7. (D) — The population(p) = current population x 2 every 5 years. So, the initial population is 300, in 5 years the population will be 600, in 5 more years the population will be 1200, in 5 more years the population will be 2400. The correct answer is (D).

8. (B) — If we write out an algebraic expression from the word problem we get $0.05v + $0.10t = $3.10, so we know (I) is true. The word problem also tells us that v + t = 54, so we know (II) is also true. v = 54 - t and t = 54 - v. We can see that (III) has misplaced the variables in the equation. $0.05t + $0.10v does not equal $3.10. Therefore, we know (III) is incorrect. The correct answer is (B); only (I) and (II) are true.

9. (C) — We know there are 180 degrees in a triangle. Therefore angle ACB = 180 degrees - 30 degrees - 40 degrees = 110 degrees. Angles on one side of a straight line will always add up to 180 degrees. We know angle BCE = 180 degrees - 110 degrees = 70 degrees. Because we know that line BC is parallel to line DE, we know that the value of angle BCE = the value of angle x. Answer (C) is correct.

10. (A) — We can write out 2 equations to represent the word problem: 10b + 7m = $50.95 and m = b - 0.25. Solve for b in terms of m in the second equation: b = m + $0.25. Plug this value of b back into the first equation in order to solve for price of a chocolate milkshake. Simplify 10(m + $0.25) + 7m = $50.95 = 10m + $2.50 + 7m = $50.95 = 17m + $2.50 = $50.95. Subtract $2.50 from both side of the equation. 17m = $48.45, divide each side of the equation by 17, m = $2.85, the correct answer is (A).

11. (B) — We know the 3 angles of a triangle add up to 180 degrees. A right triangle has an angle of 90 degrees so the remaining acute angles must add up to 90 degrees. The problem gives us the ratio of 12/3 so we can set up the equation 12/3 = (90 –x)/x to solve for the value of the smaller acute angle. Cross multiply and we get 12x = 270 – 3x = 15x = 270. Divide each side of the equation by 15 and we get x = 18 degrees which is the smaller acute angle. The larger acute angle = 90 – x = 90 – 18 = 72 degrees. The question is asking for the difference in the two angles measures. Subtract 18 from 72 and we get 54 degrees. The answer is (B).

12. (D) — The fastest way to solve this problem is to plug the answers in to the first three 4 digit palindromes (1001, 1111, 1221) and knock out answers. 2 does not divide evenly into 1001 so knock out (A). 3 does not divide evenly into 1001 so knock out (B). 7 divides into 1001, so now test 1111. 1111/7 does not divide evenly, so knock out (C). You are left with the correct answer (D). You can double-check that 11 divides into 1001, 1111, 1221, 1331, and any other 4 digit palindrome.

13. (C) — We can calculate the average daily number of books read in terms of x by adding the total number of books and dividing by 5 days. (x + 2x + 0.5x + x + 3.5x)/5. Combine like terms and we get 8x/5. The correct answer is (C).

14. (D) — The fastest way to solve this problem is by plugging in the answer choices to the inequality. Plugging in -3, -2, or -1 to the inequality makes it false. Plugging 0 into the inequality we get 0 times 0 – 1 < 0 times 0 times 0. 0 multiplied by any number is 0. This inequality is true -1 < 0.

15. (D) — From the graph, we can see that Bacteria A (solid line) is growing exponentially as this line would be expressed by an exponential function. Bacteria B (dotted line) is growing linearly or in a straight line. Answers (A) and (B) are false because Bacteria A is growing at an exponential rate and Bacteria B is growing at a linear rate. (C) is false because we know Bacteria B is growing at a linear rate. Choice (D) is the correct answer.

16. (C) — If we look at the equation y = - (x – 2)^2 + 4, we know that the largest possible value for y will be determined by the expression –(x – 2)^2 and any value plugged in for x will be squared and become negative. Thus, if we plug 2 in for x, -(2 – 2)^2 = 0 and y = 4. This is the largest possible value for y. The correct answer is (C). The other answer choices either produce a value of 0 or negative numbers for y.

17. (C) — We can solve for y by plugging in 12 as the value for x in the second equation. 3(12) = 4y^2. Multiply 3 and 12 and we get 36 = 4y^2. Divide both sides of the equation by 4 and we get 9 = y^2. Take the square root of each side of the equation and we get 3 = y. The question is asking for the value of x^2y. Plug in x =12 and y =3 and we get 12 times 12 times 3 = 432. The correct answer is (C).

18. (D) — We can write an algebraic equation from the word problem using x to represent the smaller positive integer. The consecutive positive integer can be represented as x + 2. We know from the problem that (x)(x + 2) = 168. Multiply the left expressions on the left side of the equation and we get x^2 + 2x = 168. Subtract 168 from each side of the equation and we get x^2 +2x - 168 = 0. We can now determine the possible values of x by factoring the equation. (x + 14)(x -12) = 0 We now know that x = -14 or x = 12. Since the question is asking for two positive consecutive integers we know that x = 12 is our answer.

19. (C) — The function g(x) = f(x) - 1 is graphed by taking the function f(x) and moving it down one unit on the y axis. The vertex for g(x) would be 0, -2. g(x) is greater than or equal to -2. The correct answer is C.

20. (B) — The likelihood of choosing any one of the three numbers on the first selection is 3/5. When choosing the second number, 2 out of 4 of the remaining numbers are the correct ones, so the likelihood of selecting 1 of the 2 is 2/4. The likelihood of selecting the last number from the remaining 3 numbers is then 1/3. We can calculate the likelihood of all 3 of these events occurring to be (3/5)(2/4)(1/3)= 6/60 = 1/10.

21. (B) — The ratio of d:c can be rewritten as d/c = 3/1. The second sentence tells us that d + c = s. We are looking for d in terms of s, so we are looking to get rid of the variable c. We can isolate c in the first equation and substitute its value into the second equation. Substituting c=d/3, we get d + d/3 = s. Adding the terms on the left side, we get (4d/3)= s, and dividing both sides by 4/3, we get d= (3/4)s.

22. (A) — The group that’s least likely to drink any cups of coffee in a day would be the group with the smallest percentage of people drinking 1 or 2 or more cups of coffee a day. We can find this group more easily by looking for the group with the largest percentage of people drinking 0 cups of coffee in a day. We can see from the graph that half of the freshman students drink 0 cups of coffee a day, so this makes them least likely to drink any cups of coffee during the day.

23. (B) — (A) is supported because we can see without calculating that 50/66 (juniors) is a higher percentage than 26/50 (sophomores). (B), on the other hand, is not supported because we can see that 50/66 (juniors) is a lower percentage than 32/34 (seniors). To confirm, (C) is supported because 40/200 = 20%, and (D) is supported because 25/50 is 50%.

24. (C) — When we model this like a linear function, we see that the y-intercept is 1500. We can then eliminate (A) and (B), which do not have this y-intercept. The number of passengers increase by 100 with each month, so we know m is multiplied by 100 in the function, which gives us (C) as the correct answer.

25. (C) — First, we can find how many of each fruit Claire bought: 4 dollars gets her 6 applies, 2 dollars gets get 2 peaches, and 3 dollars gets her 4 oranges. She therefore has 12 fruits in total. Because 6 of these 12 fruits are apples, there is a 1/2 probability of Claire randomly selecting an apple.

26. (C) — We can solve this question by setting up a system of equations with the information given in the question. A ratio of (j + n):k = 1:3 can be written as the fraction (j+n)/k=1/3. The ratio of n:(j +k) can also be written as a fraction, but this time we can set the proportion of (j + n) to 100, giving us n/(j+n)=x/100, where x is n expressed as a percentage of j + n. Plugging in our given values for j and k into equation 1, we get (925+n)/5550=1/3. We can then multiply both sides by 5550 and subtract both sides by 925 to solve for n. This gives us n = 925. Plugging in the values of n and j into equation 2, we get 925/(925+925)=x/100. Multiplying both sides by 100 to solve for x, we get x = 50, which means n is 50% of (j + n).

27. (C) — When Amelia hits the water, her height will be f(t) = 0. We can then set 0 = –2t^2 + 4t + 30. Factoring out the –2, we get 0 = –2(t^2 – 2t – 15). Factoring the rest of the equation, we get 0 = –2(t + 3)(t – 5), which means t = –3 or 5. Of these two answers, 5 seconds makes sense because the answer cannot be a negative number.

28. (A) — For the average of 5 positive numbers to be 85, n/5 = 85, where n is the sum of these 5 numbers. Solving for n, we find that n = 425. A states that the lowest score is 20, and we know that the highest a number can be is 100. When we look at the sum of these numbers, we see that we would need 4 scores of 100 and 1 score of at least 25 to get an average of 85, making option (A) impossible. (B) is possible because we can see the range of our scores can be from 25 to 100, which is 75. (C) is possible because the median would have to be greater than the lowest possible score at 25. (D) is possible because 85 could be the most frequently occurring score without affecting the mean.

29. (A) — We can first find the average number of languages offered across the 20 schools, which is ((1×0)+(3×1)+(5×2)+(8×3)+(2×4)+(1×5))/20 = 2.5. This means that any school offering 2 languages or less offer fewer than the average. We can see from the graph that 1 school offers 0 languages, 3 offer 1, and 5 offer 2. This makes it a total of 1 + 3 + 5 = 9 schools that offer fewer languages than the average.

30. (C) — 10% of 180 buses is (0.10)(180) = 18 buses. If each bus costs $200,000 to replace, it will cost the city (18)($200,000) = $3,600,000 to replace 18 buses.

31. (18) — Setting up the equation as described in the question, we get 2x = 11 + 12 + 13. Adding the right side and dividing both sides by 2, we get x = 18.

32. (142) — We can first expand the brackets to get –30 – 15n = –16n + 112. Isolating all the terms containing n to one side, we get –15n + 16n = 112 + 30. Simplifying this, we get n = 142.

33. (18) — To make this statement easier to understand, we can set up equations that describe the relationships. 60% of y is x, so x = 0.6y. 30% of z is y, so y = 0.3z. To find x as a percentage of z, we want to get rid of y. We can substitute the second equation into the first and solve for x, giving us x = 0.6(0.3z) = 0.18z. This means that x is 18% of z.

34. (3) — We can start by multiplying both sides by , which gives us . We can then rewrite all the bases in terms of base 2, giving us . Then, applying exponent rules, we can simplify the left side to , then . We can then take only the exponents from both sides to solve for x, giving us the equation 15x – 45 = 0. Adding 45 to both sides and dividing both sides by 15, we get x = 3.

35. (16) — We know that the angles within the rectangle are all right angles, so the larger triangles created by the diagonal lines are right-angle triangles. If 2 of the sides are 3 and 4, this makes the triangle with the diagonal line as its hypotenuse a 3:4:5 triangle. We can extrapolate from this that the length of the diagonal line is 5. There are 2 solid lines with a length of 3 and 2 solid lines with a length of 5. Therefore, the total length of the solid lines is 2 × 3 + 2 × 5 = 16.

36. (4) — We can see that the equation given is similar to the standard form for the equation of a circle, which is x^2 + y^2 = r^2 . Adding 7 to both sides, we get x^2 + y^2 = 16. We can see now that 16 should equal r^2. This means that r^2 = 16 and r = 4.

37. (12) — We see that the question asks for us to compare times for “each mile,” so we can set our calculations up as seconds per mile instead of miles per second. In Week 3, she took 68 min × 60 s/min = 4080 seconds to run 8 miles, meaning she took 4080/8 = 510 seconds to run each mile. In Week 4, she took 5220 seconds to run 10 miles, so it took her 522 seconds to run each mile. Compared to Week 3, it took her 12 more seconds to run each mile.

38. (8) — Again this question asks for her speed in terms of minutes per mile, so we’ll set up our questions likewise. We can start by getting her desired speed in terms of her current speed. 3 hours and 54 minutes translates into 234 minutes, so that’s a rate of 234 minutes/26 miles = 9 minutes/mile. She currently runs 11 minutes/mile, so she will have to improve by 2 minutes/mile in total, or 120 seconds/mile. If she improves at 15 seconds/mile every week, and she needs to improve by a total of 120 seconds/mile. That means it will take her (120 seconds/mile )/(15 seconds/mile)=8 weeks to get to her desired speed.