By Brian Leaf — McGrawHill Professional
Updated on Sep 7, 2011
Answers
 E Rather than randomly trying numbers, just "Use the Answers"! Divide each choice by 4, 5, and 6. Only choice E divides evenly (no decimal or remander).
 G This "hard" algebra question seems impossible to many students. "Use the Answers" makes it easy! Just try the choices for x and see which one works. Choice G is correct, since 4^{2 + 1} = 8^{2} = 64.
 D "T in terms of D and R" means "solve for T, use algebra to get T alone." So, just divide both sides by R to get T alone, which gives = T.
 J Write out the earnings for the five days: 5, 6, 7, 8, 9. Then to find the "mean," which is just a fancy word for average, add them up and divide by 5.
 B The question tells us that BFE= 138° and the diagram shows us that BFE also equals 6m, so we know that 138 = 6m, and therefore m = 23. According to the diagram, BFD = 5m and must thusly (I always wanted to say thusly) be 5(23) = 115.
 K Use the strategies. Mark x = 25. Therefore, in the triangle, the missing angle is 180 – 108 – 25 = 47. And in the pair of parallel lines, there are two kinds of angles, big and little, and a big + a little = 180. Here 47 is a littler angle, and y is a bigger angle, and therefore equals 180 – 47 = 133.
 A When a picture is described, draw it. When you see a triangle with two sides equal, mark the angles opposite the equal sides as equal. So R = 34 also. Then since a triangle has 180 degrees, the third angle of the triangle must be 180 – 34 – 34 = 112.
 J (5x – 2)^{2} – (5x – 2)(5x – 2) = 25x^{2} + 10x – 10x + 4 = 25x^{2} – 20x + 4. Careless Error Buster: When you FOIL, remember the middle term!
 C You can make a list of the prime numbers less than 50, counting down from 50, or you can "Use the Answers." Just make sure to avoid the careless error of choosing 49 as a prime number (it's divisible by 7, so it's not prime). Either way, the three largest prime numbers less than 50 are 41, 43, and 47.
 J The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. You can use our pairs strategy to factor 36, or you can just "Use the Answers" and see which one works. Remember to underline vocab words to avoid a careless error. Choice G lists the prime factors and choice H lists some multiples, but we want factors.
 B Many kids flee this question, but with our Skills it's easy. Reread Skill 11 to see that the lowest common denominator is the least common multiple of 3, 8, and 12. So just "Use the Answers" and try each choice. Choices B, D, and E are multiples of 3, 8, and 12, but choice B is the lowest and is therefore correct.
 K "Use the Answers"! Simply test each point in the equation to see which one works. Point E has coordinates (2, 4), so 2 = 4 – 2 which works.
 A Parallel lines have equal slopes. The slope of line y = –2x – 3 is –2, so the slope of the two points given is "Use the Answers" or cross–multiply to solve for p: p = 11.
 H Solve the equation for y to get y = . The slope is – and a line parallel to this line will have the same slope.
 B The graph shows that Reisner won 25 tournaments. Remember that half a symbol is 5 tournaments, not half a tournament. There were a total of 100 tournaments, so Reisner won of them.
 K He has been to 10 classes, so simply plug in 10 for c to get 3(10) + 8 = 38. Remember that for a function question, P(10) does not mean P times 10—it means "P of (10)" or "the value of P when c is 10."
 C Use the distance formula:
 J Did you freak out and guess? It might look impossible—unless you use "Make It Real"! Then, it's easy! Just choose a number for x, say 3. When x = 3, y = . To see what happens when x gets larger, let's try x = 10: . And finally, let's try x = 1000 to see what happens when it gets really big: y = . So, clearly, as x gets very large, y approaches zero. Another case cracked by "Make It Real"!
 C Since the square requires 80 feet of fence, each side must be 20 feet long. Then the area equals (side)(side) = (20)(20) = 400.
 H Fun question. Remember that the area of a shaded region equals area big guy minus area donut hole. The area of the whole rectangle is 6 × 9 = 54, and the area of the unshaded triangle is (this is the cool part) Area = 0.5bh = 0.5(5)(6) = 15. It looks like the height wasn't given, but it was—it's 6, it's as tall as the rectangle. So the shaded region = 54 – 15 = 39.
 D "Numerator" is just a superfancy word for the top of a fraction, and "denominator" is just the bottom. So if the numerator is 4 more than the denominator and the fraction equals , we can represent this as and crossmultiply to solve for d. Now (d + 4)(3) = (d)(5) which equals 3d + 12 = 5d. Collect like terms and 2d = 12 so d = 6. Notice that the question does not ask for d, but for the numerator which is d + 4; the numerator is 10. Careless Error Buster: Always check to make sure that you've finished the question and didn't stop in a preliminary step.
 J First multiply the 2 in the second matrix to get . Then subtract the matrices by simply subtracting the numbers in each corresponding spot:
 D When a picture is described, but not shown, draw it! Don't be a hero and try to do this one without a diagram. A diagram helps so much. It shows you what to do next, and it avoids careless errors. As soon as you draw a decent diagram, you immediately see that to be a right triangle, b must equal 5.
 F If the shortest side is x and the lengths of the sides are consecutive integers, then the longer sides can be called (x + 1) and (x + 2). Now, we just plug these into a^{2} + b^{2} = c^{2} to get (x)^{2} + (x + 1)^{2} = (x + 2)^{2}.
 B I love this question. It's rated "medium" only because most kids do not know special right triangles. If you take five minutes right now and memorize them, just the info on the page for Skill 25, then this "medium" becomes easy and you gain points! Once you know about special right triangles, choice B is obviously the answer. It follows the pattern x, x, x. If, for some crazy reason, you have not memorized the pattern, first of all shame on you, you silly slacker, but second of all, you could draw a diagram for each choice and see which look 45–45–90ish.
 H When a picture is described, draw it. This shows you that we have two triangles, which are similar. Set up a proportion: . When you see a proportion, cross–multiply: 5 = 3x. And divide by 3 to get 1.66 ≈ 1.7.
 B Translate English to math. "Is" means = and "what" means x, so 71 = x(1420). Divide both sides by 1420 to get x = 0.05, or 5%.
 F Translate English to math. "Of" means multiply, so (4.3)(1.10) + (0.5)(0.82) = 5.14.
 B Springboard! When you see x^{2} – y^{2}, factor it. So (x – y)(x + y) = 72, and we are told that x – y = 8, so 8(x + y) = 72 and x + y = 9.
 H Great exponents review! "Product" means multiply, and when you multiply, you add the exponents. But the bases must match. The bases, m and b, do not match, so we cannot combine them to get mb^{8}; instead, 3m^{4} × 2b^{4} = 6m^{4}b^{4} = 6(mb)^{4}.
 B Since both terms have an x^{2}, they are matching terms. To add them, we add their coefficients. "Coefficient" is just the fancy vocab term for the number in front of the variable. So 4x^{2} + 5x^{2} = 9x^{2}.
 J Brian spends 3 minutes per posture, so that number goes next to the p. Brian spends 7 minutes on warm–ups, and this number does not change with the number of yoga postures, so that number goes alone. y = 3x + 7.
 D When you see an arrangement question, draw a blank for each position, fill in the # of possibilities to fill each position, and multiply. So we have six seats to fill with six students: 6 × 5 × 4 × 3 × 2 × 1 = 720. Six students were available to fill the first seat, and once 1 sat down, there were only 5 available to fill the next seat, and so on.
 G SohCahToa! Sin L means opposite over hypotenuse, so sin L .
 A Great beyond SohCahToa question! All you have to do is to follow the directions. Even if you barely know trig, you can get this one if you follow the directions in the question! We are told that the law of cosines states that for any triangle with vertices A, B, and C and the sides opposite those vertices with lengths a, b, and c, respectively, c^{2} = a^{2} + b^{2} – 2ab cos C. So, to get the length of the side marked with the question mark, we call that side c, which makes its opposite angle C. We assign a and b to the other two sides. Then we plug in the appropriate measures to get c^{2} = a^{2} + b^{2} – 2ab cos C = 41^{2} + 32^{2} – 2(41)(32) cos 67.
 J Probability = , so the probability of NOT selecting a white button is . This question requires that we reduce the .
 A Factor x^{2} + 2x – 24 = (x – 4)(x + 6) = 0. Either (x – 4) or (x + 6) must equal zero, so the solutions are 4 and –6, and the product (multiply) is (4)(–6) = –24. This question is a nice review of factoring, of "anything times zero is zero," and of the vocab word "product."
 F Rewrite y – 5 = (x – 4)^{2} so it matches the form y = (x – h)^{2} + k, by just adding 5 to both sides to get y = (x – 4)^{2} + 5. In this form, the vertex or minimum/maximum point is (h, k), so (4, 5) is the minimum point.
 D We just need to apply the info we are given to the equation for a circle, which is (x – h)^{2} + (y – k)^{2} = r^{2}. Since the center is (4, 9) and the radius is 9, h = 4, k = 9, and r = 9. So D is the answer. Notice that we can easily get this question by the process of elimination. For example, since r = 9, the number to the right of the equals sign, r^{2}, must equal 81, not 9. Also, watch out for choice C, which looks good except that we need a plus sign, not a minus sign, between the parentheses. Moral of the story, "Memorize the equation of a circle."
 G Since the ratio of their radii is 2 : 3, we can just use these as potential radii. So their areas are 4π and 9π, and the ratio of their areas is 4π:9π, which reduces to 4: 9.
 E Great "Use the Answers" review! Try each answer choice in the equation. Choice E is correct, since 15 – 5 = 10 = 10 and –5 – 5 = –10 = 10.
 J A geometric sequence is just a superfancy term for a list of numbers where you multiply each member by the same number to arrive at the next member on the list. To find that number, just notice what it is or divide a term by the one before it. –6/3 = –2. So –2 is the number you multiply each term by. The next term is 12 × –2 = –24.
 A You can do the algebra or just "Use the Answers," whichever you prefer. To use the answers, plug in 137 for F, and then try each answer for C, to see which choice works. To do the algebra, plug in 137 for F and solve for C:
 G Careless Error Buster: Remember to distribute the negative sign! (5x + 2) – (8x – 5) = 5x + 2 – 8x + 5 = –3x + 7
 D When x = –3, then y = 2(–3)^{2} + 4 = 22. Careless Error Buster: PEMDAS, remember to square –3 and then multiply by 2.
 H "Anything times zero is zero." To multiply to get zero, either (x – 2) = 0 or (x + 3) = 0. So x = 2 or –3. You could also just "Use the Answers" and try each choice to see which one(s) work(s).
 D log_{6} 216 = x means 6^{x} = 216. To solve, just "Use the Answers." Using your calculator, it takes only a few seconds; just try each answer in the equation to see which one works. Choice D is correct since 6^{3} = 216.
 G "The product of 3 and (2i^{2} + x) equals –9" translates to 3(2i^{2} + x) = –9. Distribute the 3 to get 6i^{2} + 3x= –9. And since i^{2} = –1, we can plug that into the equation for i^{2}, to get 6(–1) + 3x = –9, which simplifies to 3x = –3, and x = –1.
 C The line through the points (–6, 3) and (5, 8) has slope . Since parallel lines have equal slopes, the slope of the line parallel will also be .
 J Great review of weird number behavior and "Make It Real." Plug m = –2 and a value for n, say n – 3, into each answer choice, and look for the greatest result. Choice J is correct because the even exponent cancels out the negative sign, whereas the odd exponent in choice K keeps the negative.
5 + 6 + 7 + 8 + 9 = 35 ÷ 5 = 7
No problem. Stop looking at me like that, reducing is easy; just divide the top and bottom of the fraction by the same number. Or, if you really don't like reducing, use your calculator. Divide 12 by 16 to get 0.75. Then divide each answer choice (top by the bottom) to get decimals and see which one matches 0.75. Nice.
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From McGrawHill's Top 50 Skills for a Top Score: ACT Math. Copyright © 2010 by The McGrawHill Companies. All Rights Reserved.
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