Answer Explanations for: ACT Form 1267C (67C), from Preparing for the ACT

Mathematics 1) A) Multiply the $20 fee per vehicle by the number of vehicles, v, and the $10 fee per

person by the number of people, p. See equation building.

2) F) Substitute the numbers for the variables to get (9 + 5 – -6)(5 + -6) = (20)(-1) = -20. If you answered G, you likely forgot that when subtracting the -6, you are really adding 6, not subtracting 6.

3) E) According to the information in the problem, the first machine was working for 8 minutes, and the second machine was working for 8 – 2 = 6 minutes, since it started 2 minutes later. To find the number of copies made by the first machine, multiply the number of copies it makes per minute, 60, by the number of minutes it was working for, 8: 8 • 60 = 480. To find the number of copies made by the second machine, multiply the number of copies it makes per minute, 80, by the number of minutes it was working for, 6: 6 • 80 = 480. Since each machine made 480 copies, 480 + 480 = 960 copies were made in total.

4) J) To maintain his average, his score in his 6th game must be equal to his current average, since a higher or lower score would move his average up or down. An average is calculated by dividing the sum of the items in a set by the number of items in the set, so to find his current average, add up his scores and divide by 5, the number of scores: (210 + 225 + 254 + 231 + 280)/4 = 240. See averages.

5) C) According to the information provided, she earned $7.50 per hour for 40 of her 42 hours of work, so during those 40 hours she made $7.50 • 40 = $300. She made 1.5 times $7.50 for each of the two additional hours she worked. 1.5 • $7.50 = $11.25, so in the two hours she worked at this rate, she made $11.25 • 2 = $22.50. Therefore, she made a total of $300 + $22.50 = $322.50.

6) K) Translate this problem from English into an equation, keeping in mind that “squared” indicates a power of 2, “is” means equals, “more than” indicates addition, and “product” means multiplication. See equation building.

7) E) Distribute the 9 to both terms in the parentheses to get 9x – 81 = -11 9x = 70 x = 70/9. Another way of solving this problem is to begin by dividing both sides of the equation by 9, to get x – 9 = -11/9. Then add 9 to both sides to get x = -11/9 + 9. Because 9 is equal to 81/9, this equation can be rewritten as x = -11/9 + 81/9 = 70/9.

8) H) If he bought $60 worth of $4 tickets, then he bought a total of $60/$4 = 15 tickets. Add his total savings to the amount he spent to find the regular price of these 15 tickets: $60 + $37.50 = $97.50. Now determine the regular price per ticket by dividing the regular price of 15 tickets, $97.50 by 15: $97.50 / 15 = $6.50 per ticket.

9) A) You could expand this expression using the FOIL method, by multiplying the two First terms, the two Outer terms, the two Inner terms, and the two Last terms. However, the better way of solving is to recognize that this is an instance of the difference of two squares. The difference of two square states that (x + a)(x – a) = x2 – a2. Since the expression is of the form of the left side of this identity, it can be expanded to the form of the right side of this identity: (3x)2 – (4y2)2 = 9x2 – 16y4. Remember that the power of 2 must be distributed to both of the two terms that are multiplied together (the coefficient and the variable) inside the parentheses. If you did not know how to solve this problem, you could have gotten the correct answer by plugging in your own numbers for x and y, solving the equation in the question for a numeric value, and then plugging the same numbers into the answer choices for x and y to see which one or ones yield the same value. If more than one answer choice works initially, repeat using different numbers. See exponents and radicals and quadratics.

10) J) The area of a rectangle is equal to length times width, so lw = 32. The perimeter of a rectangle is equal to twice the sum of the length and the width, so 2(l + w) = 24 l + w = 12. At this point, you could use the two equations to solve via substitution, but this would be unnecessarily time consuming, since you would be left with a quadratic after performing the substitution. A better idea is to simply find two numbers that add to 12 and multiply to 32: 4 and 8. Therefore, the longer side is 8 feet in length and the shorter side is 4 feet in length. See equation building and plane geometry.

11) D) The sum of the three angles in a triangle is equal to 180˚, so if the sum of two angles is 47˚, then the third angle must be equal to 180˚ – 47˚ = 133˚. See plane geometry.

12) K) Four events are occurring (a sandwich, a soup, a salad, and a drink are each chosen). Therefore, you should draw out four “slots” with multiplication signs in between them: __ • __ • __ • __. Then populate each slot with how many ways each event can occur: 3 • 3 • 4 • 2 = 72 different possible meals. See counting and probability.

13) B) Begin by translating the English into a mathematical equation. Let the smaller integer be x. The larger integer could then be written as (x + 1), since consecutive integers are always 1 apart. “Triple” means 3 times and “is” means equals, so you can set up the following equation: x + 3(x + 1) = 79 x + 3x + 3 = 79 4x = 76 x = 19. Since we designated x as the smaller number when setting up the equation, the answer is B. If you chose A, you may have done everything right but forgotten that x was the smaller, not the larger number. Likewise, if you chose C, you may have initially designated x as the larger number when setting up your equation, but then mistakenly confused it for the smaller number after solving the equation. See equation building.

14) F) To evaluate an equation written in function notation (f(x)), substitute whatever is in

the parentheses for x in the equation. Since -3 is in the parentheses, this gives you -8(-3)2 = -8(9) = -72. If you answered G, you likely made the mistake of thinking that (-3)2 = -9, when in reality (-3)2 = 9. This mistake was possibly the result of a calculator error, since you must put -3 in parentheses before squaring it to obtain the correct result. If you answered J or K, you might have mistakenly squared the -8 in addition to the -3. See function notation and exponents and radicals.

15) C) This problem could be solved using logarithms. Take log3 of both sides to get log33x = log354. Since log3 cancels the base of 3, you are left with x = log354. Evaluate the right side of this equation on your calculator using the change of base property: x = log54/log3 3.63, so B is the correct answer. Another way of solving this problem is simply by guess and check. The answers involve ranges into which the actual value of x will fall rather than specific values of x in order to prevent students from using guess and check, but it is possible to do so anyways. Begin by calculating 33, since 3 is in the middle of the numbers offered in the answer choices. Since 33 = 27, which is less than 54, x must be greater than 3. Next calculate 34. 34 = 81, which is greater than 54, so x must be less than 4. Therefore, you know that x must be between 3 and 4.

16) J) The quickest way to find a least common multiple on the ACT is to guess and check. Always start with the lowest answer choice offered and work your way up. That way, the first answer choice you try that is in fact a multiple of all three numbers must be the least common multiple. Using your calculator or in your head, check to see if the answer choice yields an integer when divided by 70, 60, and 50. F is eliminated because neither 70 nor 50 goes evenly into 60. G is eliminated because neither 70 nor 50 goes evenly into 180. H is eliminated because neither 60 nor 50 goes evenly into 210. J is the correct answer because 70, 60, and 50 all go evenly into 2100, since 2100/70 = 30, 2100/60 = 35, and 2100/50 = 42. Although 21,000 is also a common multiple of these three numbers, K is incorrect because it is not the least common multiple.

17) B) The volume of a rectangular prism can be found according to the following formula: v = lwh, where l, w, and h, are the length, width, and height, respectively. Since you know the length, width, and volume of the rectangular prism in the question, you can plug these values into the equation to solve for h, the height. (45)(30)h = 81,000 1350h = 81000 h = 81,000/1350 = 60.

18) J) This problem is much easier if you draw it. Paying close attention to the details from the problem, you get the following drawing. Note that the numbers labeled represent the arc lengths between consecutive labeled points:

This drawing reveals that J is the correct answer. If you answered G, you may have done everything correctly but put the points in order going counter-clockwise around the circle. This problem also could have been done without drawing a circle just by considering A both point 0 and point 15, since the circumference is 15. To find each point’s “coordinate” (with lower numbers being closer to A in the clockwise direction), add to 0 the units the point is clockwise from A, or subtract from 15 the units the point is counterclockwise from A. Hence, B = 15 – 2 = 13, C = 0 + 5 = 5, and D = 0 + 7 (or 15 – 8) = 7. The order these points moving clockwise from A is then the same as the increasing order of their “coordinate.” See circles.

19) D) Simply plug in 5 for t and solve for y: y = 16(2)5 = 16(32) = 512. See exponents and

radicals.

20) J) The area of a rectangle is equal to its length multiplied by its width. Call the length and width of the smaller rectangle l and w, respectively. Therefore, the area of the smaller rectangle is equal to lw. If the larger rectangle has the same width and a length 3 times that of the smaller one, then its area is equal to 3lw, which is 3 times lw, the area of the smaller rectangle. Therefore, k = 3. If you answered G, you may have done everything correctly but mistakenly thought kA was the area of the smaller rectangle and A the area of the larger rectangle. If you answered K, you may have mistakenly thought that the length and the width of the larger rectangle were both 3 times larger than the length and width of the smaller rectangle. See plane geometry.

21) E) Combine like terms, making sure that you properly distribute the negative to each term in the second parentheses. a – 4a = -3a, 2b – 6b = -4b, and 3c – -5c = 3c + 5c = 8c, so E is the correct answer. If you answered D, you forgot to distribute the negative to the last term and subtracted 5c instead of adding 5c.

22) G) Sin = opposite/hypotenuse, so sin = a/c. Note that the opposite side is always the leg directly across from the angle, while the hypotenuse is always the longest side of a right triangle. See basic trigonometry.

23) B) According to these rules, the player who begins with the ball will end up getting the ball back on the 5th pass, after each of the other players has touched the ball exactly once. It is useful to conceptualize this problem visually by drawing it out. Begin by drawing the 5 players (depicted as points arranged roughly in a circle). If you connect

these points according to the rules in the question, it will create a 5 pointed star, and the player who began with the ball will get the ball back on the 5th pass.

24) H) The slope of a line is equal to m in y = mx + b (slope intercept) form. Hence, the slope of p is 0.12, and the slope of n is 0.12 + .1 = 0.22. See coordinate geometry.

25) A) Distribute the -8x3 to each of the two terms inside the parentheses. Remember to distribute the negative, and also remember that when you multiply the bases, you add the exponents. -8x3(7x6) = -56x9 and -8x3(-3x5) = 24x8, so A is the correct answer. If you answered B, you forgot to distribute the negative to the second term. If you answered C or D, you multiplied the exponents instead of adding them. If you answered E, you tried to combine all the terms into a single term, which cannot be done. See exponents and radicals.

26) G) Add the terms within the absolute value bars to get 2. Since the absolute value of 2 is equal to 2, multiply 2 by -3 to get the answer to this problem: -6. If you answered J, you may have made your answer positive at the end due to the presence of the absolute value. Although an absolute value itself cannot be negative, the result of the absolute value is multiplied by -3 in this question, which does indeed yield a negative number. If you answered F, you probably made the -6 into a positive 6 before adding it to the 8. Remember that an absolute value does not require you to make all the terms inside of it positive; instead, you must perform the mathematical operations within the absolute value and only after doing so make the result positive if it is negative. See absolute values.

27) B) Since triangles ACE and BCD have the same angle measurements, they are similar. The corresponding parts of similar figures are in proportion, so you can solve this problem by setting up a proportion. Before doing so, however, you must find the hypotenuse of triangle BCD, line segment BC. Although you could use the Pythagorean Theorem to find this hypotenuse (32 + 42 = x2), you can avoid doing so if you recognize that you have a 3-4-5 Pythagorean Triple. Either way, you get that BC = 5. Now you are ready to set up your proportion. When doing so, make sure your units line up on both sides. In other words, this/that = this/that. In this case, (short leg)/(hypotenuse) = (short leg)/(hypotenuse), so 3/5 = x/20. Multiply both sides by 20 to get x = 12. Even if you did not know how to do this problem, you may have been able to get the correct answer just by estimating the length of AE visually. AE certainly appears to be greater than half the length of AC, so it must be greater than 10, and it appears to be less than 3/4 of AC, so it should be less than 15. Remember that despite what is stated in the directions, all illustrative figures are in fact drawn to scale. See ratios and proportions and plane geometry.

28) H) In a linear equation of the form y = mx + b, m is the slope and b is the y-intercept. The y-intercept is, by definition, the y value when x (or t in this case) is equal to 0. Since y = 14 when t = 0, the y-intercept of this line is equal to 14, which narrows your options

down to F and H. The slope of this line is equal to 5, since the y-value increases by 5 for every 1 the x-value increases by. Therefore, H is the correct answer. If you were not comfortable enough with the concepts of slope and y-intercept to reason through this problem in this manner, you also could have solved it by choosing a value of t from the table, plugging it into the equation in every answer choice, and seeing which equation or equations yield the corresponding y-value from the table. If more than one answer choice works, decide between them by trying another value of t. See coordinate geometry.

29) E) Begin by distributing the 6 and the 7 to each of the terms within the two sets of parentheses: 6x + 12 > 7x – 35. Now solve just like you would solve an equation containing an equals sign, but remember that if you multiply or divide both sides by a negative number, you must change the direction of the inequality sign. -x > -47 x < 47. See inequalities and the number line.

30) K) Any point that is 3 units from (2, 0) could be the location of another vertex of this square. Since (5, 0) has the same y-coordinate as (2, 0), the distance between the two points is simply equal to the difference of their x-coordinates, which is 3. Although there are infinite points that are 3 cm from (2, 0), the first ones you should be looking for are ones that share one of the coordinates with (2, 0) and have the other coordinate 3 off from the corresponding coordinate of (2, 0). See coordinate geometry.

31) E) Even if you did not immediately recognize that you have to use the Pythagorean Theorem to solve this problem, the answer choices provide a significant hint that you

do. x2 + 42 = y2 y = x2 + 16 x = √ . See plane geometry.

32) G) Probability is defined as (number of desired outcomes)/(number of possible outcomes). Therefore, (number of red marbles)/(total number of marbles) = 3/5. Because there are currently 12 red marbles and 32 marbles and the addition of a red marble increases both the number of red marbles and the total number of marbles, you can set up the following equation: (12 + x)/(32 + x) = 3/5. Cross multiply to get 5(12 + x) = 3(32 + x) 60 + 5x = 96 + 3x 2x = 36 x = 18. Alternatively, you could have solved this problem by guess and check, adding the different answer choices to both the numerator and the denominator of 12/32 until you get a fraction that is equal to 3/5. See counting and probability.

33) D) Begin by putting this equation into slope intercept form (y = mx + b, where m is the slope and b is the y-intercept). -2y = -4x + 8 y = 2x – 4. Based on this equation, the line has a slope of 2 and a y-intercept of -4. Any line with a negative y-intercept and a positive slope (meaning that it goes from lower left to upper right) will pass through quadrants I, III, and IV only. Once you got the equation of the line into y = mx + b form, you also could have found the answer by graphing it on your graphing calculator. See coordinate geometry.

34) F) Since (1, 2a) is a coordinate point on the function given in the question, you can plug in 1 for x and 2a for y in the equation: 2a = -5(1)2 + 9 2a = -5 + 9 2a = 4 a = 2. See coordinate geometry.

35) D) Begin by determining the fraction of the sandwich eaten by Seth. Since the entire sandwich should be considered 1 (since it is 1 whole sandwich or 100% of a sandwich), Seth ate 1 – 1/2 – 1/3 = 1 – 3/6 – 2/6 = 1/6. Although you could have done this on your calculator rather than by finding a common denominator of 6, it would be necessary to find the common denominator at this point in the problem anyway in order to compare the fractions. Since Jerome ate 3/6, Kevin ate 2/6, and Seth ate 1/6, the ratio of their shares in the order specified by the problem is 3:2:1. See ratios and proportions.

36) F) Standard form for a circle in the coordinate plane is (x – h)2 + (y – k)2 = r2, where (h, k) is the center point and r is the radius. (This is an equation you definitely want to memorize for the ACT.) If you know this equation, it should be apparent that F is the answer. If you answered H, you forgot that 38 is equal to r2, not r. If you answered J, you forgot that the h and k are subtracted from the x and y, not added. (Note that the absence of a k value in the equation simply means that k = 0.) See circles.

37) B) The two semi-circles, put together, have a circumference that is equal to the circumference of one circle of the same diameter. Since the diameter of the semicircles is 8, the sum of their semi-circular arcs is equal to 8π, since the circumference of a circle is equal to πd, where d is its diameter. The perimeter of the figure is equal to the sum of these two semi-circular arcs (8π) and the two 8cm straight lengths, so the correct answer is B. See plane geometry.

38) G) This problem is best done visually. It should be apparent that each of the shaded triangular regions are equal in area and that the unshaded quadrilateral EHFG has an area equal to two such triangles. Although the shaded triangles come in two shapes, we know that their area is the same because both shapes are equivalent to half that of the unshaded quadrilateral EGFH: one by dividing the quadrilateral with vertical line FE and the other by dividing the quadrilateral with horizontal line GH. By dividing quadrilateral EGFH into two triangles in either of the above ways, you can see that it is 1/3 the area of the shaded region, since the quadrilateral consists of 2 triangles, and the shaded region consists of 6 triangles of the same area. Therefore, the answer is G. If you answered H, you likely understood the problem correctly, but found the ratio of the area of quadrilateral EGFH to the area of rectangle ABCD rather than to the area of the shaded region. If you answered K, remember that the “cannot be determined” option is rarely correct and should only be chosen when you are certain. See plane geometry.

39) C) To find the x-coordinate of the midpoint, average the x-coordinates of the two endpoints: (-4 + 14)/2 = 5. If you answered D, you may have subtracted the x-coordinates of the endpoints instead of adding them. If you answered A, you found the

y-coordinate of the midpoint. If you answered E, you forgot to divide the sum of the x-coordinates of the endpoints by 2. See coordinate geometry.

40) G) Each face of an 8-inch cube has an area of 82 = 64. Since a cube has 6 faces, the total surface area would be 6 • 64 = 384. If you answered F, you found the volume of the cube instead of its surface area.

41) B) Because everything about the two equations is identical except for sign of the constant attached to the x, it should be apparent that the two equations have different slopes, which means that they must be intersecting lines. The two equations would form parallel lines if they had the same slope and different y-intercepts, and they would form a single line if they had the same slope and the same y-intercept. If it is not apparent to you that these two lines would have different slopes, you could put the equations into slope intercept form (y = mx + b, where m is the slope and b is the y-intercept). The first equation can be manipulated into the form y = (-b/a)x + c/a, while the second can be manipulated into the form y = (b/a)x + c/a. Because the m values (-b/a and b/a) are unequal, the two lines have different slopes and therefore intersect. See coordinate geometry.

42) F) You are given the side adjacent to the angle and are trying to find the side opposite the angle, so you should set up an equation using tangent, since tan = opposite/adjacent. tan 52 = x/30 x = 30 tan 52. See basic trigonometry.

43) D) Use the definition of odds given in the problem. Odds are different than probability, since odds are a measure of part to part while probability is a measure of part to whole. Since 42% of registered voters are in the 25-35 year old age rage, 58% are not in this range, so the odds of a person in this group being selected first would be 42:58, which reduces to 21:29. See counting and probability and ratios and proportions.

44) H) A line of symmetry is a line across which the figure is a mirror image of itself. For instance, a butterfly has 1 line of symmetry, a starfish has several, and a circle has infinite. This figure has 8 lines of symmetry: one from each of the points where the red pieces intersect the outside of the circle and one from each of the points where the red pieces intersect each other. Although there are 16 such points, you must keep in mind that a line of symmetry stretches from one such point to another (the one directly across from it), so there are 16/2 = 8 lines of symmetry.

45) A) The area of a circle is equal to πr2. Since the diameter of this circle is 2, its radius is 1, so its area is equal to π12 = π, which rounds to 3.1. If you answered C, you found the circle’s circumference, not its area. If you answered E, you used 2 for the circle’s radius in the area formula, when 2 is actually the circle’s diameter. See circles.

46) J) To increase the diameter by 75%, multiply it by 1.75 (the 1 representing the original length, and the .75 representing the 75% increase): 2 • 1.75 = 3.50. Alternatively, you

could have found 75% of 2 and then added this number to 2 to increase 2 by 75%, but this method is not as efficient. If you answered F, you simply found 75% of 2 rather than increasing 2 by 75%. See percentages.

47) C) The measure of BAC is actually completely unnecessary. Since AB and CD are parallel, CAB andACD are supplementary, meaning that they add to 180˚. You know that these two angles are supplementary because when two parallel lines are cut by a transversal, it forms two types of angles: obtuse ones and acute ones. All the acute angles are congruent and all the obtuse angles are congruent and the acute angles are supplementary to the obtuse angles. Since AE and CE bisect angles CAB and ACD, respectively, angles CAE andACE are equal to half the measures of angles CAB andACD, respectively. Therefore, the sum of CAE andACE is equal to half the sum of CAB andACD, which we know add to 180˚, so the sum of CAE and ACE is 90˚. Since the sum of the three angles of a triangle is 180˚, AEC, the remaining angle of triangle ACE is equal to 180˚ – 90˚ = 90˚. Note that even though the measure of BAC is unnecessary, as demonstrated above, you could have solved the problem using this as a starting point. This method would have involved the same reasoning as above, but would have used concrete numbers along the way. Although such an approach would be less abstract than the method demonstrated above, it would involve unnecessary computation, so it is not the preferable method. If you did not know how to solve this problem, C would have been a great guess, as AEC sure does look like a right angle. If you answered E, remember that the “cannot be determined” option is rarely correct, and it should be avoided unless you are certain. See plane geometry.

48) H) Since PST = 30˚, PTS = 30˚ as well, since triangle PTS is an isosceles triangle (as PT and PS are both radii of the circle) and the base angles of an isosceles triangle are equal. Therefore, TPS = 180˚ – 30˚ – 30˚ = 120˚, since the angles of a triangle add up to 180˚. RPS = 180˚ – 120˚ = 60˚, since it forms a linear pair with TPS and a linear pair is supplementary. If RPS = 60˚, then arc RS is a 60˚ arc, since the measure of an arc is equal to the measure of the central angle that intercepts it. If you answered E, remember that the “cannot be determined” option is rarely correct, and it should be avoided unless you are certain. See circles.

49) B) If there are infinite solutions to this system of 2 linear equations, the two equations must in reality be the same equation, as 2 lines intersect in infinite points only if they are in fact the same line. To make these two equations the same, recognize that 6x is 3 • 2x and 3y is 3 • y. Therefore, if the equations are to be the same, 4a = 3 • 8 = 24. Therefore, a = 24/4 = 6. See linear systems.

50) F) The horizontal line segment that passes through the point (9, 2) is the horizontal line y = 2. Because this line represents 2 large frames and the shaded region representing the frames produced per week is greater than or equal to this line (since this line forms the bottom of the shaded region), this line represents the minimum number of large frames Marcia makes in a week, 2.

51) C) Begin by determining how many hours Marcia worked during the week. If she made

4 large frames, she spent 4 • 3 = 12 hours doing so, and if she made 2 small frames, she spent 2 • 2 = 4 hours doing so. Hence, she spent a total of 16 hours making frames. Since she donated $3/hour to charity, she donated a total of 16 • $3 = $48 to charity. Now find the total amount of profit she made during the week. Based on the equation given in the text above the graph, she made a profit of 4 • $70 = $280 from the large frames and 2 • $30 = $60 from the small frames for a total profit of $280 + $60 = $340. If she donated $48 out of a total profit of $340, then she donated 48/340 .14, or 14% of her profit to charity. See percentages.

52) J) Determine her profits at point (0, 8) and point (9, 2) on the graph, as these are the points that feature the maximum number of large and small frames, respectively. At point (0, 8), she makes 0 small frames and 8 large frames, for a total profit of $70 • 8 = $560. At point (9, 2), she makes 9 small frames and 2 large frames for a total profit of $30 • 9 + $70 • 2 = $410. Since the line segment connecting these two points forms the upper constraint of how many frames she can make in a week, you know that her maximum profit will be found along this line. Since her profit is higher at (0, 8) than at (9, 2), you know that her profit at (0, 8) must also be her highest profit anywhere along this line, since her profit will steadily decrease as you move from (0, 8) to (9, 2). Therefore, the $560 profit at point (0, 8) is the greatest profit she can make in a week.

53) E) Simply use the definition of a determinant that is provided in the problem. Using this definition, the determinant of the matrix in question is x2 – 8x, which you can then set equal to -16. x2 – 8x = -16 x2 – 8x + 16 = 0. It is easiest to solve this quadratic by factoring. Hopefully, you recognize that it is a perfect square, but if not, find two numbers that add to -8 and multiply to 16: -4 and -4. Therefore, the factorization of this quadratic is (x – 4)(x – 4) = 0, so x = 4. If you are not good at factoring, you also could have solved the quadratic by using the quadratic formula, or you could have plugged in the answer choices. See quadratics.

54) K) To isolate P, simply divide both sides of the equation by (1 + 0.01i)n, leaving you with K as the correct expression for P. This is a very easy problem for #54, so don’t overcomplicate things by expecting it to be more difficult than it is.

55) C) Based on the inequalities in the problem, you know that x > y. Answer choice C must be true because if you add 5 to both sides of the equation and then multiply both sides of the equation by 3, you get x > y. Answer choice A is never true, since it simplifies to x < y (since you must remember to flip the inequality sign when dividing both sides by a negative number), and answer choices B, D, and E are true only sometimes. See inequalities and the number line.

56) J) Consider x (line segment AB) the base of triangle ABC. Now draw an altitude from point C to line segment AB. This altitude, by definition, is perpendicular to AB. Label

this line segment h. To solve for h, set up a trigonometric ratio using y and B, which is 70˚: sin 70 = h/y h = y sin 70. Since the area of a triangle is equal to ½ bh, and we know that the area of triangle ABC is 30, you can write the following equation: ½ x (y sin 70) = 30. Now look at the other triangle, triangle PQR. Again, consider x the base. Now draw an altitude from point R to a continuation of line segment PQ outside of point Q. This altitude will be external to the triangle, but will still, by definition, be perpendicular to PQ. This altitude forms a right triangle outside of and adjacent to triangle PQR, that contains a 70˚ angle adjacent to the 110˚ angle (since the two angles form a linear pair and are therefore supplementary). Use this 70˚ angle to solve for h, the altitude: sin 70 = h/y h = y sin 70. At this point, you do not even need to plug anything into the formula for the area of a triangle. Triangles ABC and PQR each have a base of x and a height of y sin 70, so their areas must be equal. Therefore the area of triangle PQR is also 30 cm2. See basic trigonometry.

57) E) On this problem, you are given both the law of sines and the law of cosines. The challenge is figuring out which one you need to use. Observing both equations, you can tell that law of sines can be used to solve for the side of a triangle when you are given two angles and one side. Law of cosines, on the other hand, can be used to solve for a side length when you are given the two other sides and the angle opposite the side you are trying to find, as you are here. Once you have determined that you will be solving using the law of cosines, you can quickly zero on E as your answer; answer choices A and B both represent incorrect attempts to use the law of sines, and C and D represent incorrect attempts to use the Pythagorean Theorem on a non-right triangle. E is the only option that even attempts to use the law of cosines. Indeed, E correctly uses the law of cosines, letting BC, the unknown side, be c in the equation, and plugging in the other pieces appropriately.

58) G) In an arithmetic sequence, a fixed amount, known as the common difference, is added to each term to produce the next term. Since the 6th and 10th terms, four terms apart, have values that are 13 – 8 = 5 apart, the common difference of this sequence is equal to 5/4 = 1.25. Now you can find terms 1 through 4 by subtracting 1.25 repeatedly from 8, the value of the 6th term. The 5th term is equal to 8 – 1.25 = 6.75, the 4th term is equal to 6.75 – 1.25 = 5.5, the 3rd term is equal to 5.5 – 1.25 = 4.25, the 2nd term is equal to 4.25 – 1.25 = 3, and the 1st term is equal to 3 – 1.25 = 1.75. Therefore, the sum of the first 4 terms is equal to 1.75 + 3 + 4.25 + 5.5 = 14.5. See series and sequences.

59) C) If there is only one value for x, then the quadratic must be a perfect square. If that only solution for x is -3, then the quadratic’s only factor is (x + 3). If (x + 3) is the quadratic’s only factor, than the quadratic’s complete factorization is (x + 3)2. Expand (x + 3)2 by FOILing (x + 3)(x + 3). To expand via FOIL, multiply the First two terms, the Outer two terms, the Inner two terms, and the Last two terms. Better yet, you can avoid having to use FOIL altogether if you remember that perfect square quadratics expand as follows: (x + a)2 = x2 + 2ax + a2. Therefore, (x + 3)2 = x2 + 6x + 9, so m = 6. See quadratics.

60) F) If you are not comfortable with absolute values, you could do this problem by coming

up with the only two numbers 5 units from -3: -8 and 2. Then plug these values into the answer choices for x and choose the only option that works for both -8 and 2, F. If you have a great conceptual grasp of absolute values, you could do this problem intuitively by thinking about it in terms of the definition of an absolute value, which is the distance from zero. Thinking about it this way, the equation in answer choice F would translate to “the difference between x and -3 (since x +3 is the same as x – -3) is 5 units from zero.” More loosely translated, you have “x and -3 are 5 units apart.” The one way of solving this problem that is not recommended, unless you have a surplus of time, is by solving the equations in each of the answer choices: doing so would be unnecessarily time-consuming. See absolute values. (Admittedly, the explanation of how to solve this intuitively is rather confusing; the absolute values Content Page provides plenty of examples to help get you thinking about absolute values in this type of intuitive manner.)