Tip #50 to Get a Top ACT Math Score

Example Problems

Name the skill(s) that you can use, and then solve each question.

Easy

1. Which of the following is equivalent to 2(3x⁴)³ ?
   1. 6x7
   2. 27x⁷
   3. 27x¹²
   4. 54x⁷
   5. 54x¹²
2. What is the length of the hypotenuse of a right triangle with legs that are 7 and 24 inches long, respectively?
   6. 
   7. 2
   8. 7
   10. 25
   11. 31
3. Kayla purchased a new phone at 31% off its original price. If she paid $34 for the phone, which of the following is an equation that could be used to find x, the original price of the phone?
   1. 34 = 1.31x
   2. 31 = x – 0.34x
   3. 31 = x – 0.66x
   4. 34 = x – 0.31x
   5. x = x – 0.31
4. What are the values for x that satisfy the equation x² – 5x – 14 = 0 ?
   6. –2 and 9
   7. –2 and –9
   8. 2
   10. 2 and –7
   11. –2 and 7

Medium

5. In the standard (x, y) coordinate plane, the midpoint of is (–3, 5). If point M is located at (2, 3), what are the coordinates of point N ?
   1. (–8, 7)
   2. (7, 1)
   3. (–1,8)
   4. (–6, 15)
   5. (0, 0)
6. For right triangle ΔABC, the sides measure 6, 8, and 10 inches. Which of the following equations could be used to solve for smallest angle, x, which is opposite the smallest side?
   6. sin x° = 0.6
   7. sin x° = 0.8
   8. tan x° = 0.8
   10. sin x° = 0.75
   11. tan x° = 0.6
7. If m < 0, which of the following is greatest?
   1. m
   2. 3 m
   3. 6 m
   4. 9 m
   5. It cannot be determined from the information given.
8. If triangle MNO is similar to triangle PQR shown below, which of the following could be the measures of the three sides of triangle MNO ?



9. If 4 times a number p is subtracted from 20, the result is negative. Which of the following gives the possible value(s) for p ?
   1. 0 only
   2. 5 only
   3. 20 only
   4. All p < 5
   5. All p > 5
10. Which of the following could be the equation of the graph below?



6. y = x – 1
7. y = x² –1
8. y = (x – 1)²
10. y = (x +1)²
11. y = x²
11. The inequality 3(3x – 2) < 6 – (5x – 2) is equivalent to
    1. x < 7
    2. x < 5
    3. x < 3
    4. x < 1
    5. x < –1

Hard

12. formula for the surface area of a rectangular solid can be SA = 2lw + 2lh + 2wh where l represents the length of the base, w represents the width of the base, and h represents the height of the solid. By halving each of the dimensions (l, w, and h) of a certain geometric solid, the surface area will be multiplied by what number?
    6. 0.1
    7. 0.25
    8. 0.33
    10. 0.50
    11. 0.75
13. The dartboard shown below has a small circle, with radius 3.5, inside a larger circle, with radius 5. When Matty randomly throws a dart at the board, which of the following is closest to the probability that it will land in the shaded region?



14. If the width of a little league baseball must satisfy the inequality |w – 3| ≥ 0.12, what is the widest that a ball can be?
    6. 2.12
    7. 2.88
    8. 3.00
    10. 3.12
    11. 3.15

Answers

1. E   Exponents (Skill 30) and avoiding careless errors (Skill 44). 2(3x⁴)³ = 2(27x¹²) = 54x¹². Careless Error Buster: Remember to apply the exponent to the 3 as well as the x⁴, and remember to multiply by 2 only after USING the exponent.
2. J   When you have a right triangle, try a² + b² = c² (Skill 24). The legs are 7 and 24, so (7)² + (24)² = c², and 49 + 576 = c². So 625 = c², and 25 = c. Remember to plug in the legs (the shorter sides) for a and b and the hypotenuse (the longest side, always opposite the right angle) for c.
3. D   Translation (Skill 27). $34 equals the original price minus 31% of the original price, so 34 = x – 0.31x.
4. K   Zero Times Anything Equals Zero (Skill 37). You can just "Use the Answers" on this type of question; try the choices and use the process of elimination, eliminating the ones that don't work. That's a quick, reliable, and easy way to get this one. Or you can factor x² – 5x – 14 = 0 to (x – 7)(x + 2) - 0. Since zero times anything is zero, either x – 7 = 0 or x + 2 = 0. Solve each of these to get x = 7 or –2.
5. A   Midpoint formula (Skill 17). Midpoint is really just the average of the x's and the average of the y's. Just plug in what you know and solve for the variable:



In other words, . When you see a proportion, cross–multiply, and solve to get x = –8 and y = 7. (–8, 7). You could also get this question, of course, by "Using the Answers," just trying the choices to see which one works.

6. F   Trig (Skill 34). When a picture is described, draw it. That always helps. Use SohCahToa; the smallest side is 6 and the hypotenuse is 10, so sin
7. A   Weird Number Behavior (Skill 46) and "Make It Real" (Skill 18). Without our Skills many kids would get this one wrong. I almost did, just now, as I was writing this solution. It's tempting to say, "Oh, easy, 9m must be the biggest." But remember our Weird Number Behavior Skill, the higher the digit of a negative number, the smaller the number actually is. Let's "Make It Real," and you'll see what I mean. Let's say m = –3. Now –3 is greater than 9(–3), which equals –27.
8. G   Similar triangles (Skill 26) and special right triangles (Skill 25). As soon as you see a right triangle with a 30 or 60 degree angle, consider special right triangles. A 30–60–90 triangle has sides that fit the pattern x, x , 2x. So since triangle PQR has a 60 degree angle and a side of 3, it must have sides of 3, 3 6. And triangle MNO, since it is similar, will have sides that are proportional to 3, 3, 6. Choice G works since 6, 6 and 12 is proportional (double) to 3, 3, 6
9. E   Translation (Skill 27). The question translates to: 20 – 4p = a negative. You could also say 20 – 4p < 0. Solve this to get



Remember to flip the > sign when you divide by a negative! As is so often the case, you could also just "Use the Answers" to find which one works. Choices A, B, and D do not work. Choice C works, but 20 is not the only solution; when you test choice E, with any number more than 5, it also works, so choice E is correct.

10. G   y = x² + bx + c (Skill 38). For the equation y = x² + bx + c, the sign in front of the x² term tells us if the U– shaped graph opens up or down, and the c term tells us the y intercept of the graph. The graph shown in the question opens up, so it should not have a negative in front of the x²; and it has a negative y intercept, so c should be negative. It also must have an x² since it is a parabola (a U–shaped graph). That eliminates choices F, J, and K. Choices G and H are left. Choice G is correct since the –1 outside the parenthesis represents the y intercept, whereas the –1 inside the parenthesis relates to a shift of the x values, not y values. If that last part is confusing, you could also just graph the two choices on your calculator and see which one matches the diagram shown in the question.
11. D   Algebra (Skill 2) and avoid careless errors (Skill 44). Simplify the inequality:

3(3x – 2) < 6 – (5x – 2)

9x – 6 < 6 – 5x + 2

14x < 14

x < 1

Careless Error Buster: Remember to distribute the negative sign.

Once again, you could also just "Use the Answers" and see which choice works.

12. G   Make It Real" (Skill 18) and area/volume (Skill 19). Let's say l = 4, h = 6, and w = 8; then the surface area of the solid is SA = 2lw + 2lh + 2wh = 2(4)(8) + 2(4)(6) + 2(8)(6) = 64 + 48 + 96 = 208. When we halve each dimension, we get l = 2, h = 3, and w = 4, so the new surface area is SA = 2lw + 2lh + 2wh = 2(2)(4) + 2(2)(3) + 2(4)(3) = 16 + 12 + 24 = 52. So the new surface area is onequarter the original; the original times 0.25 equals the new surface area. Without "Make It Real" many people would incorrectly pick choice J, since we halved the dimensions.
13. C   Probability (Skill 36), donut area (Skill 20), or use the diagram (Skill 23). Probability = , so the probability of hitting the shaded region is

You could also use the diagram and realize that the shaded region being half the area of the larger circle is the only answer choice that makes sense with the diagram.

14. J   Absolute value (Skill 41) and "Use the Answers" (Skill 1). You could solve the inequality statement or just try the choices and see which is the greatest one that works. Choices F, G, H and J work, but choice J is the largest one that works.

Brian's Friday Night Spiel Recommendations for the Days Preceding the Test

Studies show that sleeping and eating healthfully two days before the test (or any important event) is as important as sleeping and eating healthfully the night before. So Thursday night eat a healthy dinner and go to bed early—not so early that you're lying in bed at 7:00 p.m. tense, hungry, and staring at the cracks in the ceiling, but normal early, maybe 10:00 p.m.

Friday, have a normal day, no need to cram or stress. If you have completed this book and one or more timed practice tests, you are ready. Go to school, play sports or whatever you do after school, have a healthy dinner, and do something fun and relaxing. Don't hang out with anyone who stresses you out or obsesses over the test. Read, play a game, or watch a funny movie—I recommend Fletch, Wedding Crashers, or 40 Year–old Virgin—and go to bed at a sensible time. If you live in a household where, in the morning, everyone roams the house screaming for a clean shirt and their car keys, then gather your snack, drink, admission ticket, ID, pencils, watch, and calculator in the evening.

You should eat breakfast and pack a snack because it's a long day and you have to feed the brain. I recommend a cheese sandwich or two Luna bars; they are high in protein and not too high in sugar, good brain food. If you need an extra special boost, in India some people take a few drops of almond oil with breakfast on the morning of a test.

Solution: Relax and sleep well, you are prepared. Now, go get'em!!

Correct answer: E

Brian's Friday Night Spiel Drills

Here is your last drill section. Your last assignment is to be able to stay relaxed, even under pressure. So here is a little tool that you can use anytime, even during the test.

In the 1970s Herbert Benson, a researcher at Harvard Medical School, published work on what he called the relaxation response, a physiological response where the body and mind relax. Benson reported that the relaxation response was triggered by practicing 20 minutes of a concentration exercise, basically meditation. Apparently, Yale, always in competition with Harvard, decided to one–up them. "We need a way to trigger the relaxation response but in less than Benson's 20 minutes!" they might have bemoaned. They researched, and they tried as hard as they could to relax; it was quite stressful. Finally, someone came up with the following goofy exercise. And it is goofy, but the thing is, it works! It totally works. Do it and you'll see.

Follow these steps:

1. Breathing through your nose, become aware of your breath.
2. Relax your shoulders and face.
3. Allow your exhale to be longer than your inhale.
4. Now, drop your shoulders and head and smile, and then bring your head back up.
5. Repeat: drop your shoulders and head and smile, and then bring your head back up.
6. Notice how you feel.

That's it! Anytime you feel stressed, even during the test, try this very simple exercise to trigger your "relaxation response."

Score is tied: Yale, 1. Harvard, 1.

Go back to: Tip #1