Act Math Test Practice

Integrating Essential Skills, Modeling, Properties of Plane Figures Emily is covering a 12 inch by 15 inch canvas with square tiles 3/4 inch long on each side. How many whole tiles can fit on the canvas? A 120 B 140 C 300 D 320 E 480

The correct answer is (D). To figure out how many inch 3/4 tiles can fit along each side of the canvas, divide the length of each of the canvas by the length of the tiles. Start with the 12-inch side. 12 / 3/4 Dividing a fraction is the same thing as multiplying by its reciprocal, so flip the fraction, cancel like terms, and simplify. 12 / 3/4 12 * 4/3 4 * 4 16 16 tiles fit along the edge of the shorter side of the board. Repeat the process for the 15-inch side. 15 / 3/4 15 * 4/3 5 * 4 20 20 tiles fit along the edge of the longer side of the board. 16 × 20 = 320, so 320 (D) tiles fit on the interior of the canvas.

Writing Expressions and Equations, Integrating Essential Skills, Modeling Matilda knits hats to sell at a local store. She spends $56.40 on the materials to make eight hats, and each hat sells for $15. Assuming there are no additional costs or fees, how much profit does Matilda earn per hat? A $7.05 B $7.85 C $7.90 D $7.95 E $8.00

Matilda spent a total of $56.40 on materials, so divide that by 8 to determine how much she spent on each hat. $56.40 ÷ 8 = $7.05 Each hat earns Matilda $15.00, so her profit is the difference between that value and the cost of materials. The word "difference" indicates we should subtract. $15.00 - $7.05 = $7.95 Matilda makes $7.95 profit on each hat that she knits (D). <---Answer $7.05 is the amount Matilda spends on each hat, not the amount she earns from it (A).

Integrating Essential Skills, Ratios, Proportions, and Percent A bag contains 13 blue marbles, 10 green marbles, and 5 red ones. How many red marbles must be added to make the proportion of red marbles exactly 1/4? F 1 G 2 H 3 J 4 K No number of marbles makes the proportion equal to 1/4

The correct answer is (K). Currently there are 13 + 10 + 5 = 28 marbles in the bag, and only 5 of them are red. 5 ÷ 28 ≈ 0.178, so currently about 17% of the marbles of red. We're trying to raise the percentage of red marbles to one-fourth, or exactly 25%, of the entire bag. Let's get adding. If we add 1 more red marble (F), the proportion becomes 6 out of 29, or 20.7%. Notice how both the number of red marbles and the number of total marbles in the bag increases. If we add 2 red marbles (G), the proportion changes to 7 to 30, or 23.3%. Closer, but still not exact, so keep adding. Three extra red marbles changes the ratio to 8:31, or 25.8%. 25.8% is almost 25%, but its not exactly 25%. On top of that, 8/31 != 1/4 If adding 3 red marbles makes the percentage too high, then adding 4 red marbles (J) isn't going to help either. There's no way to fill the sock exactly 25% full of red marbles (K) if we only change the number of red marbles in the bag.

Basic Operations and Place Value, Number and Quantity, Modeling For the most accurate assessment of the sailboat's value at any given year, Emil should round to the: A Hundredths place B Tenths place C One's place D Tens place E Hundreds place

he correct answer is (A). Nothing can be worth less than a penny, and a penny is worth one-hundredth of a dollar, or $0.01. Rounding to any place value greater than that would give Emil an estimate, but it wouldn't be the most accurate estimate (B), (C), (D), (E). Hundredths place—sold (A).

Understanding Simple Descriptive Statistics, Modeling, Statistics and Probability After taking four 100-point tests in her math class, Ashley has an average of 85%. If only test scores count toward her final grade, what must Ashley earn on her fifth test to have an overall average of 90% in the class? F 90% G 96% H 98% J 99% K It is impossible for Ashley to earn a 90%.

The average of the scores on the four lost tests is 85, which means that the total sum of the points on those tests divided by 4 equals 85. To find the sum, multiply 85 by 4. 85 × 4 = 340 For Ashley to bump her grade up, the sum of the first four test scores plus the fifth test score divided by 5 must be equal to 90. 340 + x/5 = 90 340 + x = 450 x = 110 The maximum possible percentage for anything, including test scores, is 100%, so it's impossible for Ashley to earn a 110%. In other words, nothing Ashley does on her final test will get her to a 90% (K).

Integrating Essential Skills, Sequences and Patterns An arithmetic sequence has a second term of 9 and a fourth term of 21. What is the 40th term in the sequence? A 240 B 237 C 234 D 12 E 6

The correct answer is (B). An arithmetic sequence is a set of numbers where each term is found by adding a constant value to the term before it. For example, 3, 6, 9, 12 is an arithmetic sequence where each number is 3 greater than the number before it. Here we're given the second and fourth numbers (it would just be too easy to be given the first two!). That means to get from 9 to 21, the same number has been added to 9 twice. The numbers are 12 apart, so that means that 6 has been added to 9 to get the third term, 15, then the fourth term, which is 21. This number is known as the common difference and is going be crucial to our next step. Considering you only have 60 minutes, no one wants to keep adding 6 until you get to the 40th term, although this would give you the correct answer. But let's save some time! Let's find the first term, then use some quick multiplication to get us to that 40th term. Since the second term is 9, the first term has to be 3. See why? Its 6 less than the second term. So we have the first term and we know that we add 6 39 more times to get to the 40th term. So let's take 6 × 39, which gives us 234. That looks like choice (C), but don't be fooled—we're not done yet. We need to add that 234 to the first term, which gives us 237. (Now we're done.)

Integrating Essential Skills, Ratios, Proportions, and Percent If -4 ≤ x < 2 and x is an integer, how many different values can x have? A 4 B 5 C 6 D 7 E Infinitely many

The correct answer is (C). Integers are numerical values with no decimal part, and there are six integers greater than or equal to -4 and less than 2 (C). There's no substitute for visual proof, so here they are. -4, -3, -2, -1, 0, 1

Algebra, Rational, Radical, and Logarithmic Expressions What value of x makes 3^0.5x = 92 true? A 4 B 5 C 6 D 7 E 8

The correct answer is (E). The x is in the exponent, so we could solve for it using logarithms. Another option, and one that doesn't require remembering log rules, is to try to write the bases of the expressions as the same number. 9 is the same as 32, so 92 is the same as (32)2. To simplify a power raised to a power, multiply the exponents. 30.5x = 9230.5x = (32)230.5x = 34 Because the bases are equivalent, the exponents must also be equivalent. Set 4 to equal 0.5x. 0.5x = 4x = 8 The value of x is 8 (E). Forgetting to change the base of the exponent before making them equal could result in 4 (A).

Sequences and Patterns, Functions What is the sum of the first 40 odd integers greater than zero? F 40 G 400 H 820 J 1600 K 1640

The correct answer is (J). Every number is 2 greater than the one that came before it. The common difference indicates we're actually trying to find the sum of an arithmetic sequence whose first term a1 is 1 and whose common difference d is 2. There's a formula for situations like this. S(Smalln)n = n(a(small1)1 + a(Smalln)n)/2 The piece of information we're missing is the last term an. There's a formula for finding an as well. an = a1 + (n - 1)d n is the number of terms, which is 40. Plug all three knowns into the equation to find the unknown an. an = a1 + (n - 1)d an = 1 + (40 - 1)(2) an = 1 + (39)(2) an = 79 Now plug all of the knowns into the first formula. sn = 40(1 + 79)/2 sn = 40(80)/2 sn = 1600 J IS ANSWER

Algebra, Polynomials The expression (3x - 2y)2 is equivalent to: F 9x2 + 4y2 G 6x2 - 4y2 H 9x2 - 12xy - 4y2 J 9x2 - 12xy + 4y2 K 6x2 - 10xy - 4y2

The correct answer is (J). Squaring anything, even a quantity, is the same thing as multiplying it by itself. Because we're squaring a binomial, use a method like FOIL to multiply the two expressions, then combine like terms. (3x - 2y)2(3x − 2y)(3x − 2y)9x2 − 6xy − 6xy + 4y29x2 − 12xy + 4y2 (3x - 2y)2 is equivalent to 9x2 − 12xy + 4y2 (J). We like those terms even more.

Using Functions to Model, Functions, Modeling Which of the following expressions can be used to calculate the boat's value when it is eight years old? F 45,000(0.04)^1 G 45,000(0.12)^8 H 45,000(0.88)^7 J 45,000(0.88)^8 K 45,000(0.94)^1

The correct answer is (J). We've only just come up with the value of the boat in 2012, when it was 2012 - 2008 = 4 years old, and now we need to figure out its worth four years later. All of the answer choices have exponents, so there's probably some sort of pattern afoot.

Integrating Essential Skills, Evaluating Expressions If m and n are both odd integers, which of the following must also be an odd integer? I. 2m + nII. (m + n)2 III. mn F I G II H III J I and II K I and III

The correct answer is (K). The variable was a great invention. It's ability to stand for unknowns saves us from having to write out extremely long words and phrases like "hypotenuse" all. the. time. Sometimes, though, there's no replacement for true-blue solid numbers, and now is one of those times. The question says that both m and n are odd integers, so let m = 3 and n = 5. Any odd integers would work, even 1,347,349 and -189, but 3 and 5 seem much easier to work with. Plug those numbers into all three expressions. I: 2m + n = 2(3) + 5 = 6 + 5 = 11 II: (m + n)2 = (3 + 5)2 = 82 = 64 III: mn = 3(5) = 15 11 and 15 are both odd, but 64 isn't, so I and III (K) must always be odd integers too. I (F) and III (H) are both true, but neither one is a good enough answer on its own. mn always works out to an even integer, not an odd one (G), (J).

Integrating Essential Skills, Ratios, Proportions, and Percent On a map, 1/2 inch represents 16 miles. How Many miles long is a road that is 4-3/4 inches on the map? A 76 B 112 C 128 D 139 E 152

0.5/16 = 4.75/x (Cross Multiply to find X) -----> x = 152 Miles (E)

Modeling, Understanding Simple Descriptive Statistics, Statistics and Probability The following table shows a summary of Janna's usage of call minutes over a six month period. Month-Minutes July-251 August-312 September-705 October-120 November-429 December-301 What is the sum of the mean and range of the data? A 938 B 892 C 585 D 353 E 232

The correct answer is (A). There are three calculations required to answer this question completely, but we'll take them one at a time, thanks. The mean is what we typically call average. To find the average, add all the values and divide by 6, the number of values. Average = 353 The mean is 353 minutes (D), but we're only one computation into the solving process. Find the range next. Range is the highest value minus the lowest. 705 - 120 = 585 The range is 585 (C), but that's only the second piece to the puzzle that is the correct answer. We're down to the last step, and it's really sumthing. Ahem, it's a sum. Add the mean and the range to find the final answer. 353 + 585 = 938 The sum of the mean and the range of the data set is 938 (A). 232 (E) is the difference between the mean and range, not the sum, and 892 (B) is the sum of the range and the median of the data. Similar words, totally different meanings.

Integrating Essential Skills, Understanding Simple Descriptive Statistics Given that the average of the set {2x, 8x, 14x} is 24, what is the value of x? A 2 B 3 C 8 D 10 E 15

The correct answer is (B). At first glance, the variables can make this question send a shiver of fear up your spine. But stand strong! This is just a question of averages. We know what the average is and we know that to find an average, we add up the three numbers and divide by 3. So let's just use that formula. 2x + 8x + 14x/3 = 24 24x/3 = 24 24x = 72 x = 3 Note one of the distracting answers is the average of (2, 8, 14). We know that won't catch you, though. Variables are nothing to fear when you follow the guidelines of a formula.

Geometry, Relations of Plane Figures What is the measure of each exterior angle in a regular pentagon? A 36° B 72° C 88° D 108° E 144°

The correct answer is (B). The difference between exterior and interior angles is like the difference between outdoor and indoor cats. The sum of the interior angles of a polygon changes according to how many sides the shape has, but the sum of the exterior angles is always 360°. Triangles, pentagons, dodecagons, the sum of a polygon's exterior angles is always 360°. As an added bonus, the number of exterior angles is always equal to the number of sides. In a regular polygon, all the exterior angles have the same measure. To find what that measure is for a regular pentagon, divide 360 by 5, the number of sides in a pentagon, to find that an exterior angle of a regular pentagon has a measure of 72° (B). 108° is the measure of an interior angle of a pentagon (D). There's one major difference between exterior angles and outdoor cats: Cats are cute, so we usually break the rules and let Fluffy come snuggle with us anyway.

Functions What is the equation of a circle that has a center at the origin and passes through the point (-6, 8)? A x2 + y2 = 64 B x2 + y2 = 100 C (x + 6)2 + (y - 8)2 = 85 D (x - 6)2 + (y - 8)2 = 100 E (x - 6)2 + (y + 8)2 = 28

The correct answer is (B). The general equation of a circle is (x - h)2 + (y - k)2 = r2, where (h, k) is the center of the circle and r is the radius. This is different from the equation of a Circle K, which is lots of snacks + open 24 hours = $$. Plot the center and the point (-6, 8) on a coordinate grid, then connect them to draw a triangle. Use the Pythagorean theorem to find the radius of the circle, which also happens to be the hypotenuse of the right triangle. The length of the horizontal leg is the distance between -6 and the origin, or 6 units, and the length of vertical leg is the distance between 8 and the origin, which is 8 units. a2 + b2 = c262 + 82 = r236 + 64 = r2100 = r2r = 10 The radius of the circle is 10. The origin is the point (0, 0), so h is 0 and k is also 0. Plug those values into the equation for a circle and simplify. (x - h)2 + (y - k)2 = r2(x - 0)2 + (y - 0)2 = (10)2x2 + y2 = 100 x2 + y2 = 100 (B) is the equation of the circle described in the question. Brain freeze caused by slurping a Polar Pop too quickly could result in the equation x2 + y2 = 64 (B), which has the correct center but the wrong radius. (x + 6)2 + (y - 8)2 = 85 (C), (x - 6)2 + (y- 8)2 = 100 (D), and (x -6)2 + (y + 8)2 = 28 (E) all have centers that aren't located at the origin.

Solving Quadratic Equations by Factoring, Algebra What is the sum of the solutions for x^2 + 5x - 36 = 0? A -9 B -5 C 0 D 4 E 5

The correct answer is (B). The solutions of a polynomial can be found by setting it equal to zero and factoring. The first part is already done for us, but the second part is all ours. To factor the polynomial, find two numbers that multiply to -36 and add to 5. x^2 + 5x - 36 = 0 (x + 9)(x - 4) = 0 The product of two numbers can only equal 0 if one or both of the numbers is 0. Set each factor equal to 0 to find the value of x that make the expression zero. x + 9 = 0 x = -9 x - 4 = 0 x = 4 x = -9 (A) and x = 4 (D) are the two solutions to the equation, but neither is the solution to the question. The solution to the question is the sum of the solutions to the equation, which turns out to be -9 + 4 = -5 (B). (x + 9) and (x - 4) are factors, not solutions, to the equation, so 5 (E) isn't the sum of the solutions. 0 is the Sam Gamgee of this problem. The entire equation equals 0 and the Zero Property of Multiplication is indispensable in the quest for the correct answer, but 0 (C) still only plays a supporting role.

Values and Properties of Trigonometric Functions, Functions The tangent of ∠G is 1. What is the measure of ∠E? A 30° B 45° C 60° D 75° E Cannot be determined from the given information

The correct answer is (B). The tangent of an angle in a right triangle is equal to the ratio of the opposite leg to the adjacent leg. The fact that tan G = 1 means that the opposite side and adjacent side are exactly the same length. △GEF isn't just a right triangle; it's a special right triangle. EF and FG are congruent, so ∠E and ∠G are also congruent because they're the angles opposite those sides. Given that △GEF is a right triangle, both of the remaining angles must be equal to 45° (B) for the sum of the angles to be 180°. 30-60-90 right triangles are another form of special right triangle, but they aren't isosceles (A), (C). There's not a lot of information provided in the question, but there's enough to cook up a correct answer (E).

Geometry, Surface Area The length of the side of a cube is x inches. What is the surface area of the cube in inches squared? A x^2 B 6x^2 C x^3 D 3x^3 E 6x^3

The correct answer is (B). There are six congruent surfaces on a cube. Each is a square with a length of x and a width of x. The area of a square is length times width, so the area of each square face is x^2 (A). Six of those surfaces would have an area of 6x^2 (B). That's how to calculate the surface area of a cube. x^3 (C) represents the volume, not the surface area, of the cube, and 6x^3 (E) is the result of thinking that the area of each of the six sides is x3.

Integrating Essential Skills, Ratios, Proportions, and Percent Dezmond presented a scale drawing of a rectangular room that was 8 inches long by 12 inches wide. If Dezmond knows the actual length of the room is 14 feet, what is the actual width of the room? A 14 feet B 21 feet C 28 feet D 96 feet E 112 feet

The correct answer is (B). When we're setting up a proportion, look for a scale factor that shows the relationship between the drawing to the actual scenario. A scale factor is a fraction that relates two objects. In this case, we have a scale factor of 8 inches: 14 feet. We can use this as a fraction to set equal to the other measurement, 12 inches to x feet, where x is the actual width of the room. Here's that proportion: 8in/14ft = 12in/xft When solving a proportion, we cross multiply on both sides to solve the equation. That yields the equation 8x = 168. By dividing on both sides, x = 14.

Distance, Geometry What is the distance, in coordinate units, between the points (4, -2) and (1, -6) in the standard (x, y) coordinate plane? A 3 B 4 C 5 D 6 E 7

The correct answer is (C). (Create Right triangle on graph) The length of the horizontal leg is the absolute value of the difference of the x-coordinates of the endpoints, or |4 - 1| = 3, and the length of the vertical leg can be found by doing the same calculation on the y-coordinates of the endpoints. |-2 - (-6)| = 4, so the lengths of the legs of the triangle are 3 and 4. The hypotenuse is the square root of sum of the lengths of the legs squared. Ahem—all of that technical language was a result of a bean in our throat. We meant to say "Use the Pythagorean theorem to find the length of the hypotenuse." (a^2 + b^2 = c^2) The distance between the points is 5 units (C). 3 (A) is the distance of the horizontal leg, and 4 (B) is the distance of the vertical leg, so this triangle is another type of special triangle: A Pythagorean triple. Triangles, by the way, are the most privileged of polygons. They're always finding new ways to be "special."

Modeling, Writing Expressions and Equations, Algebra A car's ten-gallon gas tank is one-fourth full. If Stacey fills up the tank with gas priced at $2.98 a gallon and pays with $25.00, how much change will she receive? (Note: Assume there is no sales tax.) A $2.35 B $2.45 C $2.65 D $4.80 E It costs more than $25.00 to fill the tank.

The correct answer is (C). 3/4 of her 10-gallon tank must be filled with gas, and 3/4 of 10 gallons is 7.5 gallons. Multiply 7.5 by the cost of a gallon of gas to find the total cost of filling the tank. 7.5 gallons × $2.98 = $22.35 Unlike some people who shall remain nameless, Stacey brought her wallet. She has $25 in cash and there is no sales tax, so her change is the money that remains after she pays $22.35. $25 - $22.35 = $2.65 (C)

Integrating Essential Skills, Modeling, Ratios, Proportions, and Percent To the nearest dollar, how much was the boat worth in 2012?*** A $21,600 B $23,400 C $26,986 D $30,666 E $30,826

The correct answer is (C). Depreciation is the amount of value that the sailboat loses each year, but it's easier to calculate the amount of value it keeps. 100% - 12% = 88%, so each year the sailboat is worth about 88% of its previous value. The sailboat will never be completely worthless, no matter how far in the future we go. To find the value of the sailboat in 2009, multiply the initial value by 0.88, which is 88% written as a decimal. $45,000 × 0.88 = $39,600 Keep multiplying the previous year's worth by 0.88 until the value in 2012 is found. In 2010 the sailboat was worth $39,600 × 0.88 = $34,848, in 2011 it was worth $34,848 × 0.88 = $30,666 (D), and in 2012 it was worth $30,666.24 × 0.88 = $26,986 (C). Almost half the sailboat's value was lost after only four years. Because the value of the sailboat changes every year, the total depreciation isn't equal to 48% or 52% of the initial value (A), (B).

Integrating Essential Skills, Absolute Value, and Ordering Numbers by Value Which of the following numbers is the greatest in value? A 2/5 B 41/100 C 12/25 D 23/50 E 9/20

The correct answer is (C). To do this, we want to find the least common multiple and convert our fractions so that number becomes their common denominator. This is kind of like making all our fractions wear the same skirt, while we meanwhile let them choose their own top. Our denominators are 5, 100, 25, 50, and 20. Every single one of these numbers (including 100) go evenly into 100, so we know that 100 is the least common multiple of the group. To convert our fractions so that they each have a denominator of 100, we need to multiply the numerator and denominator of the fraction by the equivalent of 1. This helps us preserve the special individuality of each of our fractions, even as we dress them up for easy comparison. 2/5 * 20/20 = 40/100 12/25 * 4/4 = 48/100 23/50 * 2/2 = 46/100 9/20 * 5/5 = 45/100

Geometry, Volume Integrating Essential Skills, Basic Operations, and Place Value Which of the following simplifies to a negative, even integer? A (-2)^4 B 2^6/4 C (2-8)^2 * 4/6 D 4^3 * 2/-2 E (2 — 7)^2

The correct answer is (D). A negative integer that is raised to an even power will always result in a positive number, which means that (-2)^4 (A), 2^6/4(B), (2 - 8)^2 (C), and (2 - 7)^2 (E) won't be negative. Knowing this saves us some time: we only need to calculate one expression. 43 × 2 = 128, and when we divide that by -2, we get -64 (D). That number meets all of our requirements: it is both negative and an even integer.

Integrating Essential Skills, Modeling Kelsey is planning to leave the house at 8:00 a.m. on the morning of her ACT test. The testing center is located 20 miles from her house, and the test starts at 9:00 a.m. If Kelsey can average 40 miles per hour on the route she plans to take to the testing center, which statement below is true? A If traffic causes Kelsey to average 20 miles per hour, she will arrive 15 minutes after the test begins. B If all goes as planned, Kelsey will arrive at the testing center at 8:40 a.m. C It will take Kelsey two hours to travel to the testing center. D It will take Kelsey half an hour to travel to the testing center. E No conclusion about Kelsey's arrival time can be determined from the given information.

The correct answer is (D). This question is very open-ended, but the answer choices all talk about Kelsey's arrival time, so we should begin by determining how long it would take Kelsey to travel 20 miles at an average of 40 miles per hour. If we think logically about the question, we can get the answer without using any formulas. Kelsey can drive 40 miles in one hour, so it should take her half an hour to travel 20 miles. Alternatively, we could rely on the fact that distance = rate × time. We know the distance and the rate, so rearrange the formula to solve for time. time = distance/rate time = 20 miles/40miles time = 1/2 hour It'll take Kelsey half an hour to get to the testing center, so she'll have half an hour to get settled (D). Talk about good planning Traveling 20 miles at 20 miles per hour takes one hour—Kelsey would arrive at 9:00 a.m., not 9:15 a.m. (A). Kelsey should arrive at 8:30 am if she travels 20 miles at 40 miles per hour (B). Dividing rate by distance results in the conclusion that the trip would take two hours (C). The idea that no conclusion can be made is tempting, but unfortunately we still have to think carefully when this possibility appears among the answer choices (E).

In the figure below, △DEF is a right triangle. If the length of DE is 12 inches and tan F = 3/4 , what is the length, in inches, of the hypotenuse? A 8 B 16 C 18 D 20 E 22

The correct answer is (D). ;)

Which of the following expressions is √x^-3y^-7z^-2 equivalent to if x, y, and z are positive real numbers? F z/xy^3√xy

The correct answer is (F). Exponents are added when two variables with the same base are multiplied, we can do the reverse and "split" exponents into two terms. Squared terms inside a square root can be simplified and any variable with an even exponent is a perfect square, so rewrite the expression by turning all of the odd exponents into even ones. Now take the square root of the variables with even exponents. This is the same thing as dividing the exponent by 2. Keep in mind that taking the square root with a negative value in the exponent is not the same as taking the square root of a negative number, which is why the last step was perfectly allowable. A negative in the exponent indicates that the value is a fraction, not that the value is a negative number. None of our answer choices have negative exponents, so simplify them now. We have a match! The original expression is equivalent to (F). (G) ignores the negative signs on all of the exponents. x3y7z2 (H) is the result of wishful thinking, and (J) is the result of leaving in the numerator. The negative exponents still move the expression to the denominator even though the negative exponents were inside a radical. In (K), the squares were pulled out without the square roots being taken.

Graphing Trigonometric Functions, Functions As part of her project, Sandra creates the following trigonometric model to show the height of a specific point on the Laxey wheel over time. -IMAGE The amplitude of the trigonometric function shown above is defined as the average of the absolute values of the maximum value of f(x) and the minimum value of f(x). Which of the following aspects of the Laxey wheel does the amplitude represent? F The width of the wheel G The radius of the wheel H The diameter of the wheel J The number of revolutions the wheel makes in one minute K The time it takes for the wheel to make one revolution

The correct answer is (G). All the details needed to answer this question are given, but they're hidden in thick mathematical language. According to the problem, an amplitude is "the average of the absolute values of the maximum value of f(x) and the minimum value of f(x)." Taking the absolute value of a number makes it positive, but both the maximum and minimum values of the graph are above the y-axis, so they're already positive and we can disregard that step. In Sandra's model, the amplitude is the average of the maximum and minimum y-values of the graph. The graph repeats itself, but the maximum is always located about 73 and the minimum is always 0. The average value is found by adding the two numbers together and dividing by 2. 73 + 0 = 73, and 73 divided by 2 is 36.5. We're told at the beginning of the problem set that the wheel makes three revolutions in one minute (J), so the wheel makes one revolution in one-third of a minute, or 20 seconds (K). 36.5 must represent some physical aspect of the wheel. On the interval between x = 0 and x = 1, the graph returns to its maximum value a total of three times, so each repetition of the graph represents one full rotation of the wheel. The specific point Sandra is modeling starts at the top of the wheel—represented by the maximum point on the graph—descends about 73 units down the bottom, and then rises 73 units back to its starting location. The diameter of the wheel is its height, so the diameter must be 73. A radius is half a diameter, and 36.5 is half of 73, so the amplitude represents the radius of the wheel (G). Don't believe us? Seeing is believing.

In the figure below, AB is parallel to CD. If ∠A measures 47° and ∠C measures 65°, what is the measure of ∠AEC? F 68° G 112° H 115° J 133° K Cannot be determined from the given information

The correct answer is (G). Angles C and B are congruent because AB and CD are parallel. We've heard that "opposites attract," and the math version of that maxim is that opposite interior angles of parallel lines are congruent. If m∠C = 65°, then m∠B is also 65°. We already know that the measure of ∠A is 47° because that nugget of information was in the question, and the angles in △AEB, like the angles in all triangles, must add to 180°. m∠A + m∠B + m∠AEB = 180° 47° + 65° + m∠AEB = 180° 112° + ∠AEB = 180° ∠AEB measures 68° (F), but that's not the angle we're looking for. The angle we are looking for, ∠AEC, lies on a line with ∠AEB. All lines measures 180°, so the m∠AEC = 180° - 68° = 112° (G).

Integrating Essential Skills, Factors The prime factorization of 663 is 3 × 13 × 17. Knowing this, what is the prime factorization of 6630? F 3 × 10 × 13 × 17 G 2 × 3 × 5 × 13 × 17 H 1 × 2 × 13 × 17 J 3 × × 13; × 170 K 30 × 13 × 17

The correct answer is (G). Prime factors are the numbers that can be divided into something whose only factors are one and itself. Think of the number 3. How can you multiply two numbers to get to 3? Just 3 and 1, which makes 3 prime. A number's prime factorization is all of a number's prime factors, written as a product. For example, the prime factorization of 20 is 2 × 2 × 5. We're breaking a number down into its smallest prime parts. We're given this great hint that the prime factorization of 663 is 3 × 13 × 17. 6630 is ten times our original number, so we need to add in the prime factors of ten into our answer. Ten isn't prime (because 2 ×5 makes 10), so we have to include the 2 and 5 into our prime factorization. Not feeling confident? We can check this problem in two ways. First, we can eliminate (F), (J), and (K) because those solutions contain numbers that aren't prime. Not allowed. Then, let's multiply and find the product for (G) and (H). (G) is the only product that actually amounts to 6630, so we've found our winner.

Integrating Essential Skills, Absolute Value and Ordering Numbers by Value Simplify the following: 2|4 - 6| + 6(3 + 1)^2 F 116 G 100 H 92 J 64 K 56

The correct answer is (G). Remember those vertical bars are absolute value symbols. Absolute value is the distance between two numbers on a number line. This means that the absolute value is never negative. In this case, | 4 - 6 | = 2, not -2, since 4 and 6 are 2 units apart on the number line. On the other side of the expression, we will add what's in parentheses before squaring anything. Off we go! 2|4 - 6| + 6(3 + 1)2 = 2(2) + 6(4) 2 = 4 + 6(16) = 4 + 96 = 100 Now, if you fell, bumped your head, and forgot about absolute value, then we would end up with this simplification: 2|4-6| + 6(3+1)2 = 2(-2) + 6(4)2 = -4 + 6(16) = -4 + 96 = 92, which is (H). If you decided to skip the parentheses and distribute the square first, we have this: 2|4-6| + 6(3+1)2 = 2(2) + 6(9 + 1) = 4 + 6(10) = 4 + 60 = 64, which is (J). If after you bumped your head, you took a nap and woke up groggy to do this problem, you might have made both mistakes! That would have led to this: 2|4-6| + 6(3+1)2 = 2(-2) + 6(9 + 1) = -4 + 6(10) =- 4 + 60 = 56, which is (K). Finally, what about answer (F)? That's for the silly goose that worked a little too fast. Instead of subtracting 4 and 6, they added. 2|4+6| + 6(3+1)2 = 2(10) + 6(4)2 = 20 + 6(16) = 20 + 96 = 116 Rookie mistake, eh?

Integrating Essential Skills, Ratios, Proportions, and Percent A city determines the property taxes of a given piece of property by deducting $25,000 from a property's value, then levying taxes on 2% of the property's assessed value over $25,000. To encourage new construction, the city agrees to not collect taxes on the property for the first 5 years. If a property is worth $250,000, what is the total amount of taxes that the city is agreeing to forfeit for those 5 years? F $250,000 G $25,000 H $22,500 J $5,000 K $4,500

The correct answer is (H). Phew. This question has a couple different parts, so make sure to read it carefully. First, let's find the value of the abatement for a single year. The property will be taxed 2% for its value over $25,000, so the value being taxed is $225,000. Make sure you catch this instead of taxing the whole thing—you know they'll put that wrong answer as an option. First, we need to find the value of 2% of 225,000. To find this value, we write 2% as a decimal (0.02), then multiply. 0.02% × 225,000 = 4,500 2% of $225,000 is $4,500. But that's just for a year. For a 5 year abatement, multiply the $4,500 by 5 years to obtain a total of $22,500. Notice that $4,500 is a distracting answer (K), waiting to tempt you in case you fail to complete all the steps of the problem.

Integrating Essential Skills, Systems of Equations The sum of two numbers is 42. The lesser number is 6 more than half the greater number. What is the product of the two numbers? F 6 G 42 H 432 J 441 K 540

The correct answer is (H). Since we don't know which numbers are behind all these words, we'll use variables to represent them. In the first sentence, the word "sum" tells us that two anonymous numbers are being added together, and the word "is" means that we ought to use an equal sign. "The sum of two numbers is 42" can be therefore translated into the equation x + y = 42. Moving on to the next sentence, let's agree that the lesser number is x and the greater number is y. The word "is" tells us again to use an equal sign, the phrase "6 more" indicates we ought to add 6, and half the great number (y) tells us we ought to divide the greater number by 2. The second sentence, "the lesser number is 6 more than half of the greater number" leads to the equation x = 1/2y + 6 . Now we have a system of equations with two variables. We need to combine them in a way we can solve for one variable at a time. Substitution looks like a good method because x is already isolated in one of our equations. This means we can rewrite the first equation as x = 42 − y and we have two equations that both are equal to x. This means we can set them equal to each other. Excellent! Our new equation is 42 - y = 1/2y + 6 . Let's combine the variables on the right side to yield 42 = 3/2y + 6. Then, subtract 6 on both sides so 36 = 3/2y . Finally, multiple both sides by 2/3 to isolate y, and we find that y is 24. That's our greater number. After working out your brain like that, luckily the smaller number is easier to find. Remember, the sum of the two numbers is 42. If the greater number is 24, than the lesser number must be 18. Here's the icing on the cake; to find "the product," multiply the two numbers together. After all of that work, we arrive at the correct answer of 432.

If the volume of a cone that is 9 inches high is 48π inches cubed, what is the diameter of the circular base in inches? (Note: The formula for the volume of a cone is V = 1/3πr^2h.) F 4 G 6 H 8 J 12 K 16

The correct answer is (H). There's no variable for diameter in the formula for the volume of a cone, but there's a variable that represents the radius. The diameter of a circle is twice the radius, so if we know the radius, we can find the diameter. Substitute 48π for the volume and 9 for the height in the formula V = 1/3πr^2h , then solve for r. 48π = 1/3πr^2(9) 48 = 3r^2 16 = r^2 4 = r The radius of the circular base is 4 (F), so the diameter must be 4 × 2 = 8 (H). Dividing by 2 instead of taking the square root when solving for r could have led to an answer of 16 (K).

Integrating Essential Skills, Relations of Plane Figures Ray BD bisects ∠ABC, and ray BE bisects ∠DBC. What is the ratio of the measure of ∠ABE to the measure of ∠DBC? F 1:2 G 2:1 H 2:3 J 3:2 K Cannot be determined from the given information

The correct answer is (J). Ratios are comparisons of values, so it would be nice to have some values to work with. It doesn't matter what those values are so long as we make sure they fit the relationships described in the problem. Go ahead and say that the measure of the largest angle, ∠ABC, is equal to 64°. Any number would work, but whenever a figure is bisected, dissected, or otherwise cut into pieces, it's nice to pick a number with a lot of factors. Ray BD bisects ∠ABC, so the measures of both∠ABD and ∠DBC are half the measure of ∠ABC, or 64° ÷ 2 = 32°. Ray BE bisects ∠DBC, so the measures of ∠DBE and ∠EBC are half of 32°, or 16°. The ratio of the m∠ABE to m∠DBC is the former divided by the latter. We've already decided m∠DBC is 32°, but we still need to calculate m∠ABE, which is equal to m∠ABD plus m∠DBE. m∠ABD + m∠DBE = m∠ABE32° + 16° = m∠ABE48° = m∠ABE The ratio of m∠ABE to m∠DBC is 48:32, which can be reduced to 3:2 (H). 2:3 is the reversed ratio (G). When creating a ratio, the quantity that is listed first should be the first number in the comparison.

Statistics and Probability, Basic Probabilities What is the probability that a number randomly picked from the set {2, 3, 5, 8, 13, 21, 34, 45} will be prime? F 1/8 G 1/4 H 3/8 J 1/2 K 5/8

The correct answer is (J). The probability of randomly picking a prime number from the set is equal to the number of primes in the set divided by the total number of numbers in the set. A quick count shows that there are eight numbers in the set, but counting the primes takes a little more effort. The only factors of prime numbers are 1 and themselves. Every even number is divisible by 2, but 2 itself isn't divisible by any other number, which makes it the only even prime number. All the other prime numbers in the set—3, 5, and 13, to be specific—are odd.

Statistics and Probability, Permutations and Combinations The school lunch consists of a choice between the sandwich of the day, the daily hot item, and the salad bar. In addition, students have the choice of milk, juice, or water to drink, and the option of either an apple or a banana. If a student chooses one main course, one drink, and one piece of fruit, how many different lunch combinations can be made from these options? F 8 G 10 H 15 J 18 K 20

The correct answer is (J). To find the total number of school lunch combinations, multiply the number of options in each category. There are three different main courses, three drink options, and two fruit choices, for a total of 3 × 3 × 2 = 18 (J) possible lunch combinations Adding instead of multiplying leads to 8 (F) combo meals. Using a less mathematical method of calculation, like trying to list all of the options, isn't recommended because it's easy to forget some combinations (G), (H) or count one twice (K).

Modeling, Integrating Essential Skills Jessica is three years older than James. James' age is two less than twice the age of Jonathan. If the sum of all three ages is 59, how many years old is Jessica? F 12 G 20 H 22 J 24 K 25

The correct answer is (K). Jessica, James, and Jonathan must have been really bored at the family holiday party this year because, instead of simply telling Uncle Shmoop their ages, they came up with a riddle for him to solve. Uncle Shmoop doesn't like math too much, so we offered to solve it for him. Start by turning all the sentences into equations. Jessica, James, and Jonathan's names all start with J, so we'll use the last letters of their names as variables instead. a stands for Jessica's age, s represents James' age, and n is Jonathan's age. Jessica is three years older than James, so a = s + 3. James' age is two less than twice Jonathan's age, or 2n, making s = 2n - 2 The sum of everyone's age is 59, which means that a + s + n = 59. We have a system of three equations with three unknowns, so use substitution to get one of the equations in terms of only one of the variables. Start by replacing a with (s + 3) in the final equation, then simplify. a + s + n = 59(s + 3) + s + n = 592s + n + 3 = 59 Now replace s with 2n - 2 and solve for n. 2s + n + 3 = 592(2n - 2) + n + 3 = 594n − 4 + n + 3 = 595n − 1 = 595n = 60n = 12 Jonathan is 12 years old (F), but the question asks for Jessica's age. The expression that represents her age, a = s + 3, is in terms of s, so we need to find James' age first. Substitute n = 12 into the equation s = 2n - 2. s = 2n - 2s = 2(12) - 2s = 22 James is 22 years old (H), and Jessica is three years older, so Jessica is 22 + 3 = 25 (K).