C H A P T E R ACT Math Test Practice Over view: About the ACT Math Test The 60-minute, 60-question ACT Math Test contains questions from six categories of subjects taught in most high schools up to the start of 12th grade The categories are listed below with the number of questions from each category: I I I I I I Pre-Algebra (14 questions) Elementary Algebra (10 questions) Intermediate Algebra (9 questions) Coordinate Geometry (9 questions) Plane Geometry (14 questions) Trigonometry (4 questions) Like the other tests of the ACT, the math test requires you to use your reasoning skills Believe it or not, this is good news, since it generally means that you not need to remember every formula you were ever 131 – ACT MATH TEST PRACTICE – taught in algebra class You will, however, need a strong foundation in all the subjects listed on the previous page in order to well on the math test You may use a calculator, but as you will be shown in the following lessons, many questions can be solved quickly and easily without a calculator Essentially, the ACT Math Test is designed to evaluate a student’s ability to reason through math problems Students need to be able to interpret data based on information given and on their existing knowledge of math The questions are meant to evaluate critical thinking ability by correctly interpreting the problem, analyzing the data, reasoning through possible conclusions, and determining the correct answer—the one supported by the data presented in the question Four scores are reported for the ACT Math Test: Pre-Algebra/Elementary Algebra, Intermediate Algebra/Coordinate Geometry, Plane Geometry/Trigonometry, and the total test score Pretest As you did with the English section, take the following pretest before you begin the math review in this chapter The questions are the same type you will find on the ACT When you are finished, check the answer key on page 138 to assess your results Your pretest score will help you determine in which areas you need the most careful review and practice For a glossary of math terms, refer to page 201 at the end of this chapter If a student got 95% of the questions on a 60-question test correct, how many questions did the student complete correctly? a 57 b 38 c 46 d 53 e 95 What is the smallest possible product for two integers whose sum is 26? f 25 g 15 h 154 i 144 j 26 132 – ACT MATH TEST PRACTICE – What is the value of x in the equation −2x + = 4(x + 3)? a − ᎏ6ᎏ 11 b 11 c − ᎏ6ᎏ d −9 e − ᎏ3ᎏ What is the y-intercept of the line 4y + 2x = 12? f 12 g −2 h i −6 j The height of the parallelogram below is 4.5 cm and the area is 36 sq cm Find the length of side QR in centimeters Q R 4.5 P a b c d e S 31.5 cm cm 15.75 cm cm cm Joey gave away half of his baseball card collection and sold one third of what remained What fraction of his original collection does he still have? f ᎏᎏ g ᎏᎏ h ᎏ1ᎏ i ᎏᎏ j ᎏᎏ 133 – ACT MATH TEST PRACTICE – Simplify ͙40 ෆ a 2͙10 ෆ b 4͙10 ෆ c 10͙4 ෆ d 5͙4 ෆ e 2͙20 ෆ What is the simplified form of −(3x + 5)2? f 9x2 + 30x + 25 g −9x2 − 25 h 9x2 + 25 i −9x2 − 30x − 25 j −39x2 − 25 Find the measure of ∠RST in the triangle below T 111° S a b c d e 2x° x° R 69 46 61 45 23 10 The area of a trapezoid is ᎏ1ᎏh(b1 + b2) where h is the altitude and b1 and b2 are the parallel bases The two parallel bases of a trapezoid are cm and cm and the area of the trapezoid is 28 sq cm Find the altitude of the trapezoid f 14 cm g cm h 19 cm i 1.9 cm j cm 134 – ACT MATH TEST PRACTICE – 11 If 9m − = −318, then 14m = ? a −28 b −504 c −329 d −584 e −490 12 What is the solution of the following equation? |x + 7| − = 14 f {−14, 14} g {−22, 22} h {15} i {−8, 8} j {−29, 15} 13 Which point lies on the same line as (2, −3) and (6, 1)? a (5, −6) b (2, 3) c (−1, 8) d (7, 2) e (4, 0) 14 In the figure below, MN = inches and ෆM = inches Find the area of triangle MNP ෆෆ Pෆ P in N in M f square inches g 15 square inches h 7.5 square inches i 12 square inches j 10 square inches 135 – ACT MATH TEST PRACTICE – 15 ෆC and ៮៮៮ are both radii of circle C and have a length of cm The measure of ∠ACB is 35° Find the Aෆ BC area of the shaded region B 35° 6c m A C a 79 ᎏᎏπ ᎏᎏπ b c 36π d 65 ᎏᎏπ e 4π 16 If f(x) = 3x + and g(x) = −2x − 1, find f(g(x)) f x + g −6x − h 5x + i 2x2 − j −6x2 − 7x − 17 What is the value of log464? a b 16 c d −4 e 644 136 – ACT MATH TEST PRACTICE – l y= mx m +b 18 The equation of line l is y = mx + b Which equation is line m? f y = −mx g y = −x + b h y = 2mx + b i y = ᎏ1ᎏmx − b j y = −mx + b 19 If Mark can mow the lawn in 40 minutes and Audrey can mow the lawn in 50 minutes, which equation can be used to determine how long it would take the two of them to mow the lawn together? a b c 40 50 ᎏᎏ + ᎏᎏ = x x x x ᎏᎏ + ᎏᎏ = 40 50 1 ᎏᎏ + ᎏᎏ = 90 x x d 50x + 40x = e 90x = ᎏ1ᎏ x 20 If sinθ = ᎏ2ᎏ, find cosθ f ᎏᎏ 21 g Ίᎏ5ᎏ ๶ 21 h ᎏ5ᎏ i ᎏᎏ j Ίᎏᎏ ๶ 21 137 – ACT MATH TEST PRACTICE – Pretest Answers and Explanations Choice a is correct Multiply 60 by the decimal equivalent of 95% (0.95) 60 × 0.95 = 57 Choice f is correct Look at the pattern below Product Sum + 25 25 + 24 48 + 23 69 + 22 88 + 21 105 The products continue to get larger as the pattern progresses The smallest possible product is × 25 = 25 Choice c is correct Distribute the 4, then isolate the variable −2x + = 4(x + 3) −2x + = 4x + 12 = 6x + 12 −11 = 6x 11 −ᎏ6ᎏ = x Choice j is correct Change the equation into y = mx + b format 4y + 2x = 12 4y = − 2x + 12 y = − ᎏ1ᎏx + The y-intercept is Choice b is correct To find the area of a parallelogram, multiply the base times the height A = bh Substitute in the given height and area: 36 = b(4.5) 8=b Then, solve for the base The base is cm Choice h is correct After Joey sold half of his collection, he still had half left He sold one third of the half that he had left (ᎏ1ᎏ × ᎏ1ᎏ = ᎏ1ᎏ), which is ᎏ1ᎏ of the original collection In total, he gave away ᎏ1ᎏ and sold 6 2 1 2 ᎏᎏ, which is a total of ᎏᎏ of the collection (ᎏᎏ + ᎏᎏ = ᎏᎏ + ᎏᎏ = ᎏᎏ = ᎏᎏ) Since he has gotten rid of ᎏᎏ of the col6 6 6 3 lection, ᎏ1ᎏ remains Choice a is correct Break up 40 into a pair of factors, one of which is a perfect square 40 = × 10 ͙40 = ͙4 ͙10 = 2͙10 ෆ ෆ ෆ ෆ 138 – ACT MATH TEST PRACTICE – Choice i is correct −(3x + 5)2 = −(3x + 5)(3x + 5) −(3x + 5)(3x + 5) −(9x2 + 15x + 15x + 25) −(9x2 + 30x + 25) −9x2 − 30x − 25 Choice b is correct Recall that the sum of the angles in a triangle is 180° 180 = 111 + 2x + x 180 = 111 + 3x 69 = 3x 23 = x The problem asked for the measure of ∠RST which is 2x Since x is 23, 2x is 46° 10 Choice j is correct Substitute the given values into the equation and solve for h A = ᎏ1ᎏh(b1 + b2) 28 = ᎏ1ᎏh(3 + 5) 28 = ᎏ1ᎏh(8) 28 = 4h h=7 The altitude is cm 11 Choice e is correct Solve the first equation for m 9m − = −318 9m = −315 m = −35 Then, substitute value of m in 14m 14(−35) = −490 12 Choice j is correct |x + 7| − = 14 |x +7| = 22 |22| and |−22| both equal 22 Therefore, x + can be 22 or −22 x + = 22 x = 15 {−29, 15} x + = −22 x = −29 139 – ACT MATH TEST PRACTICE – 13 Choice d is correct Find the equation of the line containing (2, −3) and (6, 1) First, find the slope y2 − y1 ᎏᎏ x2 − x1 1− ᎏ = ᎏ(−3) = ᎏ4ᎏ = 6−2 Next, find the equation of the line y − y1 = m(x − x1) y − = 1(x − 6) y−1=x−6 y=x−5 Substitute the ordered pairs into the equations The pair that makes the equation true is on the line When (7, 2) is substituted into y = x − 5, the equation is true = − is true 14 Choice f is correct Triangle MNP is a 3-4-5 right triangle The height of the triangle is and the base b ᎏ is To find the area use the formula A = ᎏ2h (3) 12 ᎏ A = ᎏ2(4) = ᎏ2ᎏ = The area of the triangle is square inches 15 Choice d is correct Find the total area of the circle using the formula A = πr A = π(6)2 = 36π A circle has a total of 360° In the circle shown, 35° are NOT shaded, so 325° ARE shaded The fraction of the circle that is shaded is ᎏ2ᎏ Multiply this fraction by the total area to find the shaded 360 area 36π ᎏᎏ 11,700π 65π × ᎏ2ᎏ = ᎏ6ᎏ = ᎏ2ᎏ 360 16 Choice g is correct f(g(x)) = f(−2x − 1) Replace every x in f(x) with (−2x − 1) f(g(x)) = 3(−2x − 1) + f(g(x)) = −6x − + f(g(x)) = −6x − 17 Choice a is correct; log464 means 4? = 64; 43 = 64 Therefore, log464 = 18 Choice j is correct The lines have the same y-intercept (b) Their slopes are opposites So, the slope of the first line is m, thus, the slope of the second line is −m Since the y-intercept is b and the slope is −m, the equation of the line is y = −mx + b 140 – ACT MATH TEST PRACTICE – 22 Choice h is correct The equation is quadratic, so there are two ways to solve it First, try to factor the left-hand side of the equation Since it is factorable, solve the equation using factoring x2 + 8x + 15 = (x + 5)(x + 3) = Set each of the factors equal to zero and solve for x x+5=0 x=−5 x+3=0 x = −3 The solution set is {−5, −3} The quadratic equation can also be used to solve the equation x= ෆc −b ± ͙b2 − 4aෆ ᎏᎏ 2a x= −8 ± ͙(8)2 − ෆ15) ෆ(4)(1)(ෆ ᎏᎏᎏ x= −8 ± ͙64 − 60 ෆ ᎏᎏ −8 ± x = ᎏ2ᎏ −8 + x = ᎏ2ᎏ −8 − x = ᎏ2ᎏ −6 x = ᎏ2ᎏ = −3 −10 x = ᎏ2ᎏ = −5 The solution set is {−5, −3} 23 Choice b is correct Solve the equation for m using inverse operations 5k = 9m − 18 5k + 18 = 9m 5k + 18 ᎏᎏ =m Since this answer does not appear as one of the choices, you must determine if any of the choices are equivalent to it If you divide each of the numerator terms by you get ᎏ5ᎏk + = m, which is choice b 24 Choice h is correct Solve the equation by moving all x terms to one side 5x − 5x − = 5x − 5x + 10 − = 10 − ≠ 10 Ø The x’s cancel, leaving −7 = 10, which is not true Since −7 never equals 10, there is no solution 25 Choice e is correct Factor the numerator (4x − 1)(x + 3) ᎏᎏ x+3 Use the denominator as a clue when factoring the numerator Most likely, the denominator will be one of the factors in the numerator Cancel the x + in the numerator with the x + in the denominator This leaves 4x − 188 – ACT MATH TEST PRACTICE – 26 Choice f is correct Subtract the numbers in y from the corresponding numbers in x [35 − (( − 2) − − 1) 4−4 = 6−0 6 ] [ ] 27 Choice b is correct log 3x = is equivalent to 32 = x Therefore, x = 28 Choice h is correct Factor the numerator Use the denominator as a clue Most likely, one of the factors in the numerator will be the same as the denominator Also, notice that the numerator is the difference of two squares (x − 3)(x + 3) ᎏᎏ x−3 The x − in the numerator cancels with the x − in the denominator leaving an answer of x + 29 Choice b is correct Use the distance formula or the Pythagorean theorem to find the distance The distance formula is d = ͙(x2 − x1)2 + (y2 − y1)2 Substitute the x and y values for points A and C and ෆෆෆ solve d = ͙(−2 − −ෆ−1 − 3ෆ ෆ1)2 + (ෆ)2 d = ͙(−1)2 +ෆ ෆ (−4)2 d = ͙1 + 16 ෆ d = ͙17 ෆ To use the Pythagorean theorem (which is what the distance formula is derived from), draw the segment on a coordinate plane and create a right triangle where AC is the hypotenuse ෆෆ (−1,3) (−2,−1) The legs of the right triangle are and Use the Pythagorean theorem to find the length of the hypotenuse a2 + b2 = c2 12 + 42 = c2 + 16 = c2 17 = c2 ͙17 = c ෆ 189 – ACT MATH TEST PRACTICE – 30 Choice f is correct Since the line is parallel to 3y − 9x = 24, they have the same slope Put the equation into y = mx + b form to easily see the slope: 3y − 9x = 24 3y = 9x + 24 y = 3x + The equation above indicates that the slope is The line you are looking for also has a slope of You are looking for the line y = 3x + You must put the answer choices in y = mx + b form to compare them to this equation Equation f is correct f y = 3x + g y = 3x + h y = 2x + i y = 2x + 14 j y = −ᎏ1ᎏx − ᎏ9ᎏ 31 Choice d is correct You are looking for an x and a y value The x value must be since one of the equations is x = To find y, substitute for x in the second equation 3(9) + y = 27 + y = y = −23 The ordered pair is (9, −23) This point is a solution to both equations 32 Choice i is correct ͙37 is close to ͙36 which is ͙125 is close to ͙121 which is 11; (6)(11) = 66 ෆ ෆ, ෆ ෆ, 33 Choice d is correct The equation is quadratic, so it can be solved by either setting the equation equal to zero and factoring or using the quadratic formula In this case, factoring is the easiest option First, set the equation equal to zero: x2 − 6x − 27 = (x − 9)(x + 3) = x−9=0 x=9 x+3=0 x = −3 The solution set is {−3, 9} 34 g Find the probability of each event and multiply the answers to find the probability of both events occurring The probability of getting tails is ᎏ1ᎏ and the probability of rolling a is ᎏ1ᎏ; (ᎏ1ᎏ)(ᎏ1ᎏ) = ᎏ1ᎏ 12 6 The probability of getting tails and rolling a is ᎏ1ᎏ 12 35 Choice d is correct Multiply the decimal equivalent of ᎏ1ᎏ% by 90 The decimal equivalent of ᎏ1ᎏ% is 2 0.005 (0.005)(90) = 0.45 36 Choice j is correct ͙41 lies between ͙36 which is and ͙49 which is It lies between and ෆ ෆ, ෆ, 190 – ACT MATH TEST PRACTICE – 37 Choice e is correct Think about what you would if he used bags for each pizza 12 ÷ = pizzas Follow the same pattern and divide the 12 bags by ᎏ3ᎏ 12 ÷ ᎏ3ᎏ = 12 ᎏᎏ 48 × ᎏ4ᎏ = ᎏ3ᎏ = 16 Mike can make 16 pizzas 38 Choice g is correct Convert the given rate of meters per second to kilometers per hour 100΋ m ᎏᎏ ΋ 9.79s km ΋ 60 360,0 36.8 km ᎏ ᎏ ᎏ × ᎏᎏ × ᎏ0s × ᎏ΋ = ᎏ00 km = ᎏhᎏ 1000΋ m 1΋ 1h 9.790h Round the answer to 37 km/h 39 Choice c is correct Originally, Larry had 20 socks in the drawer Since he pulled a red one out already, there are only 19 left in the drawer and of them are red # ed P(red) = ᎏrtᎏ = ᎏ5ᎏ 19 to al 40 Choice i is correct Multiply the and to get 30 Then, multiply the powers of 10 10−4 × 108 = 104 Since the bases are the same, you can just add the exponents So far you have 30 × 104, but this is not an answer choice Change your answer to scientific notation by moving the decimal in 30 to 3.0 Since the decimal has been moved one place to the left, you must increase the power of 10 by one Therefore, the answer is × 105 opp it 41 b sin = ᎏosnuese First, find the hypotenuse by using the Pythagorean theorem, or noticing that it is a hypotᎏ e 3-4-5 triangle (multiplied by 2) Therefore, the missing side is × 2= 10 sin B = ᎏ6ᎏ = ᎏ3ᎏ 10 opp te 42 Choice h is correct Use the formula tan = ᎏocseint tan x = ᎏ4ᎏ adjaᎏ 43 Choice d is correct There are pairs of sides The two sides that measure × each have an area of 48 The two sides that measure × 10 each have an area of 60 The two sides that measure × 10 each have an area of 80 Since there are two of each side, multiply the area of each by 2, then add the areas 48 × = 96 60 × = 120 80 × = 160 96 + 120 + 160 = 376 44 Choice j is correct The radius of the large circle is (add the inner radius plus the extra from the ring) Therefore, the area of the large circle is 49π Subtract the area of the inner circle, which is 16π 49π − 16π = 33π 191 – ACT MATH TEST PRACTICE – 45 Choice a is correct Take the square root of the area to find the length of one side ͙100 = 10, so the ෆ length of the sides of the square is 10 centimeters When the diagonal is drawn it creates a right triangle with legs of 10 cm each and the diagonal is the hypotenuse Z 10 cm 10 cm Y W 10 cm 10 cm X Use the Pythagorean theorem to find the diagonal 102 + 102 = x2 100 + 100 = x2 200 = x2 ͙200 = x ෆ 10͙2 = x ෆ 46 Choice i is correct Use the Pythagorean theorem to find the hypotenuse a2 + b2 = c2 42 + 92 = c2 16 + 81 = c2 97 = c2 ͙97 = c ෆ 47 Choice c is correct The formula for area of a circle is A = πr2 Change the radius to an improper frac49 tion; 3ᎏ1ᎏ = ᎏ7ᎏ Use the formula to find the area using the improper fraction A = π(ᎏ7ᎏ)2 = ᎏ4ᎏπ The area is 2 49 ᎏᎏπ square inches 48 Choice f is correct Substitute 15 for y in the equation and solve for x y = 4x2 − 15 = 4x2 − 16 = 4x2 = x2 2=x 49 Choice e is correct Find the number of students that voted for Kristen (male and female) by multiplying 540 by the decimal equivalent of 60% 540 × 0.60 = 324 324 students voted for Kristen Find 75% of that number 324 × 0.75 = 243 243 females voted for Kristen 192 – ACT MATH TEST PRACTICE – si 50 Choice f is correct Use the identity tanθ = ᎏnθ coᎏ sθ ᎏᎏ = sinθ ᎏ Multiply both sides by ᎏ6ᎏ to isolate the sinθ 17 ᎏᎏ 17 × ᎏ5ᎏ = sinθ ᎏᎏ 17 = sinθ 51 Choice d is correct The original formula was V = lwh If each dimension is tripled, the length is 3l, the width is 3w, and the height is 3h When these values are substituted into the equation, the equation becomes V = (3l)(3w)(3h) or V = 27lwh Thus, the new rectangular solid has a volume 27 times the original volume 52 Choice j is correct The sum of the measures of the angles in a triangle is 180° In a right triangle, the right angle is 90°, so another 90° is split between the remaining two angles 7x + 8x = 90 15x = 90 x=6 The value of x is 6, but the question asks for the measure of the smaller angle, which is 7x Substituting in for x, yields (7)(6) = 42 The measure of the angle is 42° 53 Choice c is correct Use the Pythagorean theorem to find the missing leg a2 + b2 = c2 92 + x2 = 102 81 + x2 = 100 x2 = 19 x = ͙19 ෆ The length of the missing side is ͙19 ෆ 54 Choice j is correct Call the width w and the length 2w The perimeter is then P = w + w + 2w + 2w 72 = 6w 12 = w The width is 12 Since the length is twice the width, the length is 24 193 – ACT MATH TEST PRACTICE – 55 Choice e is correct Three of the answer choices can be immediately eliminated because the length of the height cannot be negative Answer choices b, c, and d all have negative lengths b ᎏ Call the height h and the base h + The area of a triangle is ᎏ2h Substitute in h and h + for the base h(h + 5) and height, and set the area equal to 80 since the are of the given triangle is 80; ᎏ2ᎏ = 80 h + 5h Solve for h The equation is quadratic, so the quadratic formula will be used; ᎏ2ᎏ = 80 h2 + 5h = 160 h2 + 5h − 160 = h Use the quadratic formula to solve −5 ± ͙(5)2 − ෆ160) ෆ4(1)(−ෆ ᎏᎏᎏ 2(1) ෆ40 −5 ± ͙25 + 6ෆ ᎏᎏ ෆ −5 ± ͙665 ᎏᎏ The negative value is eliminated from the answer because it does not make sense The answer is −5 + ͙665 ෆ ᎏᎏ 56 Choice i is correct The easiest way to find the length of one side is to draw the square on the coordinate plane and count the spaces There are spaces between (−2, 3) and (5, 3) Therefore, the length of a side is (−2,3) (−2,−4) (5,3) (5,−4) The distance formula can also be used First, you must decide which points are consecutive vertices of the square Let’s use (−2, 3) and (5, 3) The distance formula is then: d = ͙(5 − (−ෆ(3 − 3)2 ෆ2))2 + ෆෆ d = ͙72 + ෆ d = ͙49 ෆ d=7 194 – ACT MATH TEST PRACTICE – 57 Choice e is correct The slope of the given equation is The slope of a line perpendicular to the line is the opposite, reciprocal of This is −ᎏ1ᎏ Arrange each answer choice in the y = mx + b format to quickly find the slope of each choice a y = 3x + ᎏ5ᎏ b y = 3x − c y = −ᎏ2ᎏx + ᎏ1ᎏ d y = −2x + e y = −ᎏ1ᎏx + ᎏ5ᎏ 3 58 Choice f is correct Arrange the equations of the two lines in y = mx + b format They both become y = ᎏ2ᎏx + Therefore, they are the same line 59 Choice b is correct To find the midpoint, take the average of the x values and the average of the y values −4 + −2 + (ᎏ2ᎏ, ᎏ2ᎏ) (−ᎏ1ᎏ, ᎏ6ᎏ) 2 (−0.5,3) The midpoint is (−0.5,3) 60 Choice i is correct Find a common denominator and add the fractions The common denominator is 15x ᎏᎏ 3x x−1 + ᎏ5ᎏ 20 ᎏᎏ 15x 3x(x − 1) + ᎏᎏ 15 20 + 3x(x − 1) ᎏᎏ 15x 20 + 3x2 − 3x ᎏᎏ 15x 3x2 − 3x + 20 ᎏᎏ 15x 61 Choice b is correct Since the exponent is negative, take the reciprocal of the fraction, then apply the exponent of (ᎏ1ᎏ)−3 2x2 2x (ᎏ1ᎏ)3 (2)3(x2)3 8x6 195 – ACT MATH TEST PRACTICE – 62 Choice f is correct Use the substitution method to solve for x and y The second equation can easily be solved for x in terms of y 2y − x = 2y = x Substitute this value for x in the first equation and solve for y 4x = 3y + 15 4(2y) = 3y + 15 8y = 3y + 15 5y = 15 y=3 Next, substitute the value for y in the second equation to find x 2(3) − x = 6−x=0 6=x 63 Choice d is correct The negative part of the exponent tells you to take the reciprocal of the number −3 36ᎏ2ᎏ (ᎏ1ᎏ)ᎏ2ᎏ 36 The denominator of the fractional exponent is the root and the numerator is the power Therefore, take the square root and raise that answer to the third power (Ίᎏ1ᎏ)3 36 ๶ (ᎏ1ᎏ)3 ᎏᎏ 216 64 Choice h is correct The cube root of a negative number is negative So, the answer must be negative (−3)3 = −27 and (−4)3 = − 64; −50 falls between −27 and −64 The value of x must be between −3 and −4 65 Choice e is correct Every triangle has a total of 180° 112° are used in the top angle, leaving 68° to be shared equally between the bottom two angles 68 ÷ = 34° 196 – ACT MATH TEST PRACTICE – 66 Choice f is correct Recall that all triangles have 180° Next, using the two angle measures given, find the two bottom angles of the triangle The bottom left angle is supplementary (adds to 180°) with 120°, therefore, it is 60° The bottom right angle is a corresponding angle to the 21° angle and, therefore, is 21° 21° x° 21° 120° 60° The three angles in the triangle must add to 180°, so x is 99° 67 Choice d is correct Keeping in mind that a tangent line will only intersect the circle in one place, draw the graph of the circle on the coordinate plane to see that the radius must be 4 (4,−2) 68 Choice g is correct The equation of a circle is in the form (x − h)2 + (y − k)2 = r2 where r is the radius Since the given equation is already in this form, you can find the radius quickly r2 = 36; therefore, r = Use the formula A = πr2 to find the area of the circle; A = π(6)2 or 36π 69 Choice b is correct Find the measures of angles DBC and BDC by using the supplements given (remember that supplementary angles add to 180°) ∠DBC = 60° and ∠BDC = 70° The three angles of a triangle must add to 180° Therefore, ∠BCD = 50° A B 120° 60° D E 70° 110° 50° C 197 – ACT MATH TEST PRACTICE – 70 Choice g is correct A triangle that has sides in the ratio 1:1:͙2 has angle measures of 45°, 45°, and ෆ 90° The side that measures ͙2 is opposite the 90° angle and is the hypotenuse See the diagram ෆ below 45° √¯¯¯ 45° The smallest angle is one of the 45° angles To find the sine of 45°, you need to know the side opposite 45°(1) and the hypotenuse (͙2) ෆ sin 45 = ᎏ ෆ Rationalize the denominator by multiplying the numerator and denominator by ͙2 ᎏ × ͙2 ෆ ᎏ ͙2 ෆ = ͙2 ෆ ᎏ The sine of 45° is ͙2 ෆ ᎏ 71 Choice e is correct −1 ≤ cos x ≤ 1; therefore, −9 ≤ 9cos x ≤ The minimum value is −9 ෆ 72 Choice f is correct A 30-60-90 triangle has side lengths in the ratio 1:͙3:2 If the smallest side is 7, the largest side is twice 7, or 14 The hypotenuse is 14 73 Choice c is correct Refer to the drawing below to see the dimensions of the pool and the walkway Notice that the walkway is 10 feet longer and 10 feet wider than the pool (NOT feet) because feet is added on EACH side of the pool To find the area of the walkway, find the area of the large rectangle (walkway and pool combined), and subtract the area of the pool 34 ft ft ft 24 ft 22 ft 12 ft ft ft Area of the walkway and pool = 34 × 22 = 748 square feet Area of the pool = 12 × 24 = 288 square feet Area of walkway = 748 − 288 = 460 square feet 198 – ACT MATH TEST PRACTICE – 74 Choice f is correct Since YW is an altitude in an equilateral triangle, it bisects the opposite side ෆW ෆෆ Xෆ and WZ are both inches See the diagram below ෆෆ Y n 14 i n i 14 X in W in Z An altitude also makes a right angle and, therefore, the Pythagorean theorem can be used to find the length of the altitude Refer to triangle WXY The hypotenuse is 14 inches and one leg is inches a2 + b2 = c2 72 + b2 = 142 49 + b2 = 196 b2 = 147 b = ͙147 ෆ b = 7͙3 ෆ The length of the hypotenuse is 7͙3 ෆ 75 Choice c is correct The equation is quadratic Set it equal to zero and factor 2x2 − 2x − 12 = 2(x2 − x − 6) = 2(x − 3)(x + 2) = Set each factor equal to zero and solve (2 can be ignored because ≠ 0) x−3=0 x=3 x+2=0 x = −2 The sum of the solutions is + −2 = 76 Choice j is correct Use the identity sin2 A + cos2 A = sin2 A + (ᎏ9ᎏ)2 = 10 81 sin2 A + ᎏ0ᎏ = 1 19 sin2 A = ᎏ0ᎏ 19 sin A = Ίᎏ0ᎏ = ๶ ෆ ͙19 ᎏ 10 77 Choice d is correct The triangle given is a 45-45-90 triangle so the sides are in the ratio 1:1:͙2 ෆ Use a proportion to find x ͙2 ෆ ᎏ = x ᎏ x = ͙10 ෆ 199 – ACT MATH TEST PRACTICE – 2 x y 78 Choice f is correct An ellipse is defined by an equation such as ᎏᎏ + ᎏᎏ = Therefore, answer choices f a2 b2 and i are possibilities Choice f is the correct choice because the square root of the number under the x is where the ellipse crosses the x-axis Another way to check is to substitute the given ordered pairs into the equations to see which one works 79 Choice b is correct Notice that the y-intercept is and the slope is −3 Thus, the equation must be y = − 3x + Answer choices b and c are possibilities The shading will determine which one Substitute (0, 0) in for x and y Since the shading is over the point (0, 0), (0, 0) must be a solution to the inequality ≤ − 3(0) + 0≤2 TRUE Therefore, choice b is the correct answer 80 Choice j is correct The only constraint on this function is that the denominator must not be zero To find which values will yield a denominator of zero, set the denominator equal to zero and solve x2 + 3x − = (x + 4)(x − 1) = Set each factor equal to zero and solve x+4=0 x = −4 x−1=0 x=1 These are the values of x that not work All other real numbers work The domain is all real numbers, such that x ≠ −4 and x ≠ This is written as {x | x ≠ − and x ≠ 1} 200 – ACT MATH TEST PRACTICE – Glossar y of Math Terms This glossary is a tool to prepare you for the ACT Math Test You will not be asked any vocabulary questions on the ACT Math Test, so there is no need to memorize any of these terms or definitions However, reading through this list will familiarize you with general math words and concepts, as well as terms you may encounter in the practice questions These terms come from all the areas of math found on the ACT, but it is not guaranteed that any of the terms below will be included on an official ACT Math Test Base—A number used as a repeated factor in an exponential expression In 85, is the base number Base 10—see Decimal numbers Binary System—One of the simplest numbering systems The base of the binary system is 2, which means that only the digits and can appear in a binary representation of any number Circumference—The distance around the outside of a circle Composite number—Any integer that can be divided evenly by a number other than itself and All numbers are either prime or composite Counting numbers—Include all whole numbers, with the exception of Decimal—A number in the base 10 number system Each place value in a decimal number is worth ten times the place value of the digit to its right Denominator—The bottom number in a fraction The denominator of ᎏ1ᎏ is 2 Diameter—A chord which passes through the center of the circle and has endpoints on the circle Difference—The result of subtracting one number from another Divisible by—Capable of being evenly divided by a given number, without a remainder Dividend—The number in a division problem that is being divided In 32 ÷ = 8, 32 is the dividend Even number—A counting number that is divisible by Expanded notation—A method of writing numbers as the sum of their units (hundreds, tens, ones, etc.) The expanded notation for 378 is 300 + 70 + Exponent—A number that indicates an operation of repeated multiplication For instance, 34 indicates that the number should be multiplied by itself times Factor—One of two or more numbers or variables that are being multiplied together Fractal—A geometric figure that is self-similar; that is, any smaller piece of the figure will have roughly the same shape as the whole Improper fraction—A fraction whose numerator is the same size as or larger than its denominator Improper fractions are equal to or greater than Integer—All of the whole numbers and negatives too Examples are −3, −2, −1, 0, 1, 2, and Note that integers not include fractions, or decimals Multiple of—A multiple of a number has that number as one of its factors 35 is a multiple of 7; it is also a multiple of Negative number—A real number whose value is less than zero Numerator—The top number in a fraction The numerator of ᎏ1ᎏ is 201 – ACT MATH TEST PRACTICE – Odd number—A counting number that is not divisible by Percent—A ratio or fraction whose denominator is assumed to be 100, expressed using the percent sign; 98% 98 is equal to ᎏ0ᎏ Perimeter—The distance around the outside of a polygon Polygon—A closed two-dimensional shape made up of several line segments that are joined together Positive number—A real number whose value is greater than zero Prime number—A real number that is divisible by only positive factors: and itself Product—The result when two numbers are multiplied together Proper fraction—A fraction whose denominator is larger than its numerator Proper fractions are equal to less than Proportion—A relationship between two equivalent sets of fractions in the form ᎏaᎏ = ᎏcᎏ d b Quotient—The result when one number is divided into another Radical—The symbol used to signify a root operation Radius—Any line segment from the center of the circle to a point on the circle The radius of a circle is equal to half its diameter Ratio—The relationship between two things, expressed as a proportion Real numbers—Include fractions and decimals in addition to integers Reciprocal—One of two numbers which, when multiplied together, give a product of For instance, since 3 ᎏᎏ × ᎏᎏ is equal to 1, ᎏᎏ is the reciprocal of ᎏᎏ 3 Remainder—The amount left over after a division problem using whole numbers Divisible numbers always have a remainder of zero Root (square root)—One of two (or more) equal factors of a number The square root of 36 is 6, because × = 36 The cube root of 27 is because × × = 27 Simplify terms—To combine like terms and reduce an equation to its most basic form Variable—A letter, often x, used to represent an unknown number value in a problem Whole numbers—0, 1, 2, 3, and so on They not include negatives, fractions, or decimals 202 ... b 0.095 c 0.0595 d 0.024 e 0.092 165 – ACT MATH TEST PRACTICE – Which of the following is NOT the graph of a function? f g h i j 166 – ACT MATH TEST PRACTICE – 4.6 × 105 = ? a 4.60000 b 0.000046... QUATIONS BY FACTORING Before factoring a quadratic equation to solve for the variable, you must set the equation equal to zero x2 − 7x = 30 x2 − 7x − 30 = 151 – ACT MATH TEST PRACTICE – Next, factor... − 32 Solutions −5 + × −5 + 16 11 143 – ACT MATH TEST PRACTICE – + (6 + × 4) − 32 + (6 + 8) − 32 + 14 − 23 − 14 F RACTIONS Addition of Fractions To add fractions, they must have a common denominator