# ACT

**Published: **11-07-2016

**1,001 ACT questions with step-by-step solutions**

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Article / Updated 04-26-2017

There's no getting around it: you'll probably have to solve some questions on the ACT Math exam that deal with circles. To solve the following practice questions, you'll need to know the formulas for the area of a circle and the general equation for a circle. Practice questions The figure shows a portion of the graph of y = (1.5)x and a circle with center (3, 1). The two meet at the point indicated in the graph. Which is nearest to the area of the circle in square units? Which of the following represents the equation of a circle in the standard xy-coordinate plane that is tangent to the x-axis at 3 units and to the y-axis at 3 units? A. x2 + y2 = 9 B. (x + 3)2 + (y – 3)2 = 9 C. (x – 3)2 + (y – 3)2 = 9 D. (x – 3)2 + (y – 3)2 = 6 E. (x + 3)2 – (y + 3)2 = 6 Answers and explanations The correct answer is Choice (A). Don't let the equation with the x exponent throw you. Just use a couple of familiar formulas. The formula for the area of a circle is So find the radius of the circle, apply it to the formula, and you're done. The radius is the distance from (3, 1) to the point on the circle that intersects with the graph. Find the coordinates of that point, and you can use the handy dandy distance formula to discover the length of the radius. The point's x-coordinate is obvious. The dotted line on the figure indicates that it's 2. The y-coordinate, then, is what you get when you plug 2 in for x in the equation of the curve: The y-coordinate is 2.25. So the coordinates of the second point are (2, 2.25). With the coordinates of the two points, you can use the distance formula to find the radius. The radius of the circle is Plug that value into the area formula (Don't bother to find the square root of 2.5625 because you just square it again in the area formula): The correct answer is Choice (C). For this problem, you need to know the general equation for a circle: (x – h)2 + (y – k)2 =r2, where h and k are the x- and y-coordinates of the center of the circle and r is its radius. Because the circle is tangent to the x and y at 3 units, the radius of the circle is 3. Eliminate Choices (D) and (E) because they don't have 32 on the right side of the equation. Choice (A) is wrong because it's the equation for a circle with a center on the origin (0, 0). There's no way this circle's center is on the origin if it's touching both axes. Choice (B) adds rather than subtracts within the first parentheses, which would be true only if the center were (–3, 3). Choice (C) is the only equation in proper format for a circle with a center point of (3, 3).

View ArticleArticle / Updated 04-26-2017

In geometry, polygons cover a lot of ground, so you can bet that some questions on the ACT Math exam will involve polygons—specifically, finding the interior angles of a polygon. Fortunately, as you'll see in the following practice questions, there's a handy formula that you can use to find a missing interior angle in a polygon, whether it's a square, a hexagon, or whatever. Practice questions In the figure, the following is true about the value of the degree measurement of angles a and b: 70 < a + b < 150. Which of the following describes all possible values in degrees of c + d? A. 210 < c + d < 290 B. 30 < c + d < 110 C. 120 < c + d < 200 D. 390 < c + d < 470 E. 570 < c + d < 650 The regular polygon shown here has 6 congruent sides and 6 congruent interior angles. Two of the sides are extended until they meet at point A. What is the measure of angle A? A. 160 degrees B. 120 degrees C. 72 degrees D. 60 degrees E. 35 degrees Answers and explanations The correct answer is Choice (A). To find the sum of the interior angles of a polygon, you use this formula: where n is the number of sides in the polygon. If you can't remember that formula, simply divide the shape into triangles. The sum of the interior angles in each triangle measures 180 degrees, so for each triangle add 180 degrees and you get the sum of all the angles in the polygon. The polygon in this problem has 4 sides, so you know its interior angles add up to 360 degrees. The problem tells you that the sum of angles a and b is more than 70 degrees. The lowest possible value for a + b is 71 degrees. If at its lowest, then at its highest. That means that The answer that states c + d < 290 is Choice (A). Double-check the rest of the information to make sure Choice (A) is the right answer. The problem says that a + b < 150. If a + b < 150, then its highest value is 149 degrees; the lowest the sum of c and d can be is 360 – 149 = 211. Choice (A) works. The correct answer is Choice (D). The formula for finding the interior angle measures of a regular polygon is as follows, where n represents the number of sides in the polygon: Apply the formula to the figure in the question to find the measure of each interior angle in the polygon: So each angle in the hexagon measures 120 degrees. The angles in the triangle that's formed by the two extended sides must each be 60 degrees because two of those angles form a straight line with the two 120-degree angles in the hexagon and 180 – 120 = 60. If two angles in a triangle each measure 60 degrees, the third angle, angle A, must also measure 60 degrees.

View ArticleArticle / Updated 04-26-2017

A big part of geometry involves working with angles, so it shouldn't be a surprise that the ACT Math exam contains a number of questions involving them. You may want to brush up on the properties of angles before you take on the following practice questions (and definitely before you tackle the ACT!) Practice questions In the figure, A, B, and C are collinear. The measure of angle ABD is 4 times that of angle DBC. What is the measure of angle ABD? A. 36 degrees B. 45 degrees C. 72 degrees D. 108 degrees E. 144 degrees The next figure shows three straight lines that intersect at point M. If angle EMF measures 47 degrees and angle AMB measures 29 degrees, what is the degree measure of angle CME? A. 72 degrees B. 76 degrees C. 104 degrees D. 133 degrees E. 151 degrees Answers and explanations The correct answer is Choice (E). Because the value of angle ABD measures 4 times that of angle DBC, angle ABD = 4x and angle DBC = x. Because A, B, and C are collinear, the sum of angle DBC and angle ABD is 180 degrees. To find the measure of angle ABD, set up an equation and solve for x: Don't stop there and pick Choice (A), though. The value of x is the measure of angle DBC. Multiply 36 by 4 to get 144 degrees, which is 4x and the measure of angle ABD. The correct answer is Choice (D). Scan the figure to determine which angles are equal. Angle AMB and angle DME are vertical angles, so they're equal. Angle DME also measures 29 degrees. The same goes for angle EMF and angle BMC. They're equal, so angle BMC also measures 47 degrees. The remaining two angles, angle CMD and angle AMF, are also vertical angles and equal. The degree measures of the 6 angles in the figure total to 360 degrees because they circle around the center point M. Create an equation to solve for the degree measure of the remaining two angles: Hang on, though. This is the degree measure of angle CMD, but the question asks for the measure of angle CME. You need to add 29 degrees to 104 degrees for a degree measure of angle CME, which equals 133 degrees.

View ArticleArticle / Updated 04-26-2017

Okay, so "combinations and permutations" sounds like the name of a class you would take at wizards' college, but these are actually topics that you would cover in a statistics class. They're also something you'll probably need to know for the ACT Math exam. Fortunately, the following practice questions will help you brush up on your skills: first, you'll need to calculate the total number of possible license plate designations for a community, and then you'll be asked to find the total possible number of combinations for a secret code. Practice questions License plate designations in Tinytown consist of three characters. The first is either the letter M or F depending on the gender of the car's owner, the second is a single digit between 0 and 9, and the last is a single letter of the entire alphabet from A to Z. How many license plate designations are possible? A. 38 B. 468 C. 520 D. 780 E. 6,760 A secret code is created by combining any 2 letters from the English alphabet and any 2 one-digit numbers between and including 0 and 9. How many different code combinations are possible if numeric digits can be repeated but letters cannot? A. 71 B. 72 C. 60,840 D. 65,000 E. 67,600 Answers and explanations The correct answer is Choice (C). There are 2 choices for the first character (M or F), 10 choices for the second character (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), and the 26 letters of the alphabet for the third character. All you do is multiply the possibilities: If you picked Choice (B), you thought there were 9 numbers from 0 to 9 instead of 10. Choice (A) results from incorrectly adding the numbers instead of multiplying them. The correct answer is Choice (D). Determine the total pool of elements you have to create code combinations. You can repeat numbers, and there are 10 separate digits from 0 to 9. Letters cannot be repeated, and there are 26 possibilities in the English alphabet. Apply the multiplication principle by multiplying the total possibilities for each element of the code. There are 10 for the first position, 10 for the second, 26 for the third, and 25 for the fourth (because you can't repeat the letter in the third position). The product of 10, 10, 26, and 25 is 65,000.

View ArticleArticle / Updated 04-26-2017

There's a good chance that the ACT Math exam will contain one or more questions that deal with probability. There's also a good chance that the odds of your answering those questions correctly will improve if you tackle the following practice questions. Practice questions Sheila has 4 black socks and 2 navy socks in her laundry pile. If she randomly selects one sock from the pile, sets it aside, and then picks another sock from the pile, what is the chance that she will select the navy pair? The next question is based on the following information. The discard pile for a children's card game contains 6 green cards, 8 yellow cards, 10 blue cards, and several red cards. There are no other cards in the discard pile. The probability of randomly choosing a yellow card from the pile is 1/4. Armando shuffles the discard pile and places the whole pile face down in front of the players to ready it for further play. To put a card into play, the players flip over the top card of the newly shuffled discard pile. What is the probability that the first player will flip over a green card? Answers and explanations The correct answer is Choice (A). Before you find the chance Sheila will pick two navy socks, you first have to find the chance that the first sock she picks will be navy. That chance is 2/6 or 1/3 because 2 out of 6 socks are navy. Multiply that fraction by the probability that the second sock she chooses will also be navy. The second probability isn't also 1/3 because Sheila has already picked a navy sock from the pile and set it aside. After she chooses the first navy sock, 5 socks remain in the pile and only 1 is navy. So the chance that the second sock she picks will be navy is actually 1/5. The chance that both socks Sheila picks will be a navy pair results from multiplying 1/3 by 1/5, which is 1/15. Pick Choice (A). The correct answer is Choice (A). First, you need to know the number of cards in the shuffled discard pile. If the discard pile contains 8 yellow cards and the probability of choosing a yellow card is 1 out of 4, there must be a total of 32 cards in the discard pile because is the same as The pile has a total of 6 green cards, so the probability that a green card is the top card is

View ArticleArticle / Updated 04-26-2017

If you're good at finding patterns, then you'll probably enjoy tackling the geometric sequence questions on the ACT Math exam. In the meantime, you can enjoy working on the following practice questions, one that deals with a fairly simple sequence and the other requiring some algebra. Practice questions What is the fourth term of the geometric sequence whose second term is –6 and whose fifth term is 0.75? A. –3 B. –1.5 C. –0.5 D. 1.5 E. 3 Which of the following would express the 21st term of the geometric sequence represented by 3, 9b, 27b2…? A. (3b)21 B. 321b20 C. 320b21 D. 3b20 E. 9b21 Answers and explanations The correct answer is Choice (B). Create the number sequence with the information you're given: Because the sequence is geometric, you multiply by the same value to find each term. The second term is negative and the fifth is positive, so you must be multiplying by a negative value. Therefore, the fourth term must be negative and you can eliminate Choices (D) and (E). You could spend time determining the common ratio between each term, but it's likely faster to try out the remaining answer choices. If the fourth term is Choice (C), –0.5, the common ratio would be –15 because 0.75 divided by –0.5 is –15. When you multiply –6 by –15, you get 90 for the third term and –1,350 for the fourth term, so Choice (C) doesn't work. When you apply Choice (B), you have a fourth term of –1.5. Divide 0.75 by –1.5 to find the common ratio: If –0.5 is the common multiplier, the third term would be 3: The fourth term would be –1.5: That works, so the fourth term must be –1.5, Choice (B). The correct answer is Choice (B). Because the first number in this series is 3 and the next values are 9b and 27b2, the common multiplier in the geometric series is 3b. The first term in the series is 3. To reach the 21st term, you need to find 20 additional terms multiplied by 3b: 3(3b)20. Expand (3b)20 to and combine terms by adding the exponents: The correct answer is Choice (B). Choice (A) would make sense if the first term were 3b instead of 3.

View ArticleArticle / Updated 04-26-2017

If you think a logarithm is a tree that can do the Macarena, you may want to do some studying before you take the ACT Math exam. Then, you can come back and tackle the following practice questions, where you have to use the properties of logarithms to solve two different equations. Practice questions If 3x = 4y and 5y = 6z, then For which of following values for x is log64 + log6x = 2? Answers and explanations The correct answer is Choice (B). Looking at the answer choices tells you you're dealing with a logarithm problem. Because you're solving for and none of the answers has a y variable, you need to get rid of the y variables in the original equations. You can do this by solving both equations for y and them setting them equal to each other. Then, you can manipulate the equation to solve for First, take the log of both sides of the first equation and solve for y: Then, take the log of both sides of the other equation and solve for y: Set the equations equal to each other and move terms around until you've solved for x/z: The correct answer is Choice (D). When you add logs with the same base, you solve by multiplying: log6(4x) = 2. Then you can plug in the options for x. Choice (D) is correct. The product of 4 and 9 is 36, which is the value you get when you multiply 6 by itself two times. Choice (E) incorrectly adds 4 and x.

View ArticleArticle / Updated 04-26-2017

All things being equal, the chances that you'll have to solve some inequalities on the ACT Math exam are very good. Fortunately, the following practice questions will help you become greater than the challenge! Practice questions Given that x is an integer, for what value of x is and x + 4 < 16? A. 7 B. 8 C. 10 D. 12 E. 13 Which of the following represents all possible solutions for x in the inequality –2x – 7 < 3x + 5? Answers and explanations The correct answer is Choice (C). Instead of working out complex calculations, first examine the answer choices. Based on the second equation, you can eliminate Choices (D) and (E). Neither 13 + 4 nor 12 + 4 is less than 16. Then plug the remaining options into the first equation. Because you're determining which answer results in a value greater than 16, start with the greatest value, 10. If x is 10 in the first expression, then which is certainly more than 16. The first equation is valid. Try the second: 10 + 4 < 16. This inequality is also true, so the answer must be Choice (C). You can try the other options, but neither x = 7 nor x = 8 results in a value greater than 16. The correct answer is Choice (B). Solve the inequality just as you would an ordinary equation. Move all the constants to the right and all the x terms to the left: Choice (B) is the answer. Choice (A) is wrong because you must change the direction of the sign when you divide both sides of an inequality by a negative value.

View ArticleArticle / Updated 04-26-2017

It's nice to be color-coordinated and even physically coordinated, but, on the ACT Math exam, it's really useful to be skilled at coordinate geometry. To help you prepare, the following practice questions ask you to find a line that is perpendicular to a given line, and then find the distance between two given points. Practice questions Use the following graph to solve the first practice question. What would the slope be of any line in the standard xy-coordinate plane that is perpendicular to the line l in the graph? What is the distance in the standard xy-coordinate plane between the points (–2, 1) and (3, –2)? Answers and explanations The correct answer is Choice (A). The slope of a perpendicular line is the opposite reciprocal of the slope of the line it intersects. The figure shows line l with a positive slope, so the slope of a perpendicular line has to be negative. Eliminate Choices (D) and (E). To choose among the remaining answers, find the slope of the given line. The graph shows that the given line travels through the points (–5, 2) and (7, 3). Plug values into the slope equation to solve: Switch the sign to negative and flip the numerator and denominator. The slope is –12. The correct answer is Choice (C). You can solve this question is by plugging the given values into the distance formula and solving:

View ArticleArticle / Updated 04-26-2017

It's very useful to know functions for the ACT Math exam. As you'll see in the following practice questions, function problems can be obvious, showing the trademark f(x), or they can be hidden in a word problem. Practice questions A man threw a baseball from the top of a skyscraper at a height of 1,454 feet. The height of the baseball after the man threw it is a function of the time that has expired from the time he threw it. If t represents the time, in seconds, that has expired since the man threw the ball and h(t) represents the height, in feet, of the baseball, then I(t) = 1,454 – 16t2 + 8t + 4. What is the closest approximation, in feet, of the height of the baseball at 2 seconds? A. 450 B. 727 C. 1,410 D. 1,454 E. 1,538 If then f(g(20)) = Answers and explanations The correct answer is Choice (C). Consider this to be a function problem with h(t) as f(x). Functions are really just fancy substitution problems. To solve this function, simply substitute 2 for t on the right side of the equation, and then solve for h(t): The correct answer is Choice (A). Functions are really just fancy substitution problems. This one combines two functions, which means you have to perform two substitutions. Work from inside the parentheses out (that's easy because you're used to working this way under the standard order of operations). The problem asks you to find f(g(20)). In other words, solve g(x) for g(20) in the second equation, and then plug what you get in for x in the other equation. Substitute 20 for x in g(x): Now substitute 3 for x in the equation for f(x):

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