ACT Form 1874FPRE Practice Test Comprehensive Study Guide

Examinee Statements, Certification, and Signature

• Examinee Statements Agreement: By registering for, launching, starting, or submitting answer documents for an ACT test, the examinee agrees to comply with the "Terms and Conditions: Testing Rules and Policies for the ACT Test" ("Terms").

• Score Cancellation and Arbitrations: ACT reserves the right to cancel scores if there is reason to believe they are invalid. The Terms limit available damages and require arbitration for dispute resolution, meaning the examinee waives the right to a judge or jury.

• Ownership and Confidentiality: ACT owns all test questions and responses. Examinees are prohibited from sharing them via any communication form before, during, or after the test. Taking the test for another person is illegal and subject to legal penalties.

• Data Privacy and Transfer: Examinees consent to the collection and processing of personally identifying information (PII) according to the ACT Privacy Policy (www.act.org/privacy.html). ACT is permitted to transfer PII to the United States or third-party providers, where it is subject to U.S. laws, including access by law enforcement or national security authorities.

• Certification Process: The examinee must copy a specific italicized certification statement, then sign, date, and print their name. The certification confirms agreement with the statements and verifies the examinee's identity.

• Form Identification: This document follows ACT Form $1874FPRE$ ($2019$-$2020$).

General Test Directions and Administration

• Test Contents: The booklet contains separate tests for English, mathematics, reading, and science, designed to measure skills related to high school coursework and college success.

• Answering Mechanics: Students must use a soft lead pencil (not ink or mechanical) to fill in ovals on a separate answer document. Marks must be heavy and black. If an answer is changed, the original mark must be erased thoroughly.

• Scoring Policy: Scores are based solely on the number of questions answered correctly within the time limit. There is no penalty for guessing, so answering every question is advantageous.

• Time Management Rules:

  • Testing staff dictates when to start and stop work.

  • If a student finishes a test early, they may reconsider questions ONLY within that specific test.

  • Looking back at a previous test or moving ahead to a future test results in disqualification.

  • Pencils must be laid down immediately when time is called; altering ovals after time is called results in disqualification.

• Material Integrity: Test booklets must not be folded or torn. The material is confidential copyrighted property; unauthorized reproduction or transfer is subject to civil and criminal penalties.

Mathematics Test Overview and Rules

• Format: $60$ minutes to answer $60$ questions.

• Calculator Usage: Calculators are permitted on the mathematics test only. While they can be used for any problem, some problems may be solved more efficiently without one.

• Standard Assumptions:

  1. Illustrative figures are NOT necessarily drawn to scale.

  2. Geometric figures lie in a 2-dimensional plane.

  3. The term "line" refers to a straight line.

  4. The term "average" refers to the arithmetic mean.

Mathematics Practice Problems: Discrete Questions

• Casserole Modification: Marcus's recipe uses $3$ eggs for $6$ servings. If he uses $5$ eggs and increases all other ingredients proportionally, the new recipe will produce $10$ servings.

• History Club Representative: A $35$-member club must choose a representative at random who is NOT one of the $3$ officers. For a non-officer member like Hiroko, the probability of being chosen is $\frac{1}{32}$.

• Exponential Equation: For the equation $2^{2x+7} = 2^{15}$ to be true, the value of $x$ must be $4$.

• Function Evaluation: For the function $f(x) = 5x^2 - 7(4x+3)$, the value of $f(3)$ is computed as $5(3)^2 - 7(4(3)+3) = 45 - 7(15) = 45 - 105 = -60$.

• Probability (Wallet): A lost wallet contains $5$ five-dollar bills, $7$ ten-dollar bills, and $8$ twenty-dollar bills (Total = $20$ bills). The probability of drawing a twenty-dollar bill at random is $\frac{8}{20} = \frac{2}{5}$.

• Book Club Cost Equilibrium: ABC Book Club charges $40$ monthly plus $2$ per book. Easy Book Club charges $35$ monthly plus $3$ per book. They are equal cost ($50$) when $5$ books are read in a month.

• Parallelogram Geometry: In parallelogram $ABCD$, $\angle ABC = 40^\circ$ and diagonal $AC$ forms $\angle ACD = 57^\circ$. Because alternating interior angles are equal and adjacent angles in a parallelogram are supplementary, the measure of $\angle CAD$ is $83^\circ$.

• Rational Algebraic Expressions: When $x = \frac{1}{2}$, the expression $\frac{8x^3 - 3}{x}$ evaluates to $-4$.

• Midpoint Formula: The midpoint of a line segment with endpoints $(3,8)$ and $(1,-4)$ is $(2,2)$.

• Slope Calculation: The slope of a line passing through the points $(-2,1)$ and $(2,-5)$ is $\frac{-5-1}{2 - (-2)} = -\frac{6}{4} = -\frac{3}{2}$.

• Speeding Fines: In Cherokee County, the fine is $17$ for each $1\text{ mph}$ over the limit. Kirk was fined $221$ on a $30\text{ mph}$ road. This indicates he was traveling $13\text{ mph}$ over the limit, totaling $43\text{ mph}$.

• Solution Sums: For $8x = 12$ and $2y + 10 = 22$, the sum of the solutions ($x = 1.5, y = 6$) is $7.5$. Note: Options provided are whole numbers; based on page-specific text, the correct sum is $7.5$.

• Averages and Medians: For $5$ distinct scores with a sum of $420$, the average is $84$. Since the average equals the median, the median is $84$. The sum of the other $4$ scores is $420 - 84 = 336$.

• Absolute Value: The expression $||-8+4| - |3-9||$ simplifies to $|4 - 6| = |-2| = 2$.

• Exponents and Radicals: The expression $x^{3/4}$ is equivalent to $\sqrt[4]{x^3}$.

• Line Equation Slope: The slope of the line $4x = 7y + 5$ is $\frac{4}{7}$.

• Integer Parity: The sum of integers $m$ and $n$ will always be odd if and only if one is odd and the other is even.

• Right Triangle Midpoint: In a right triangle with legs $24$ and $32$, the hypotenuse is $40$. The midpoint of the hypotenuse is $20\, \text{inches}$ from the vertex.

• Paint Area Calculation: For a room with two $10\text{-foot}$ walls and two $15\text{-foot}$ walls (all $8\text{ feet}$ high), total wall area is $400\, \text{sq ft}$. Subtracting a window and door area of $60\, \text{sq ft}$ yields a paintable surface of $340\, \text{sq ft}$.

• Rectangular Perimeter: A rectangle with a perimeter of $40\, \text{inches}$ where the length is $5\, \text{inches}$ longer than the width ($L = W + 5$) has a width of $7.5\, \text{inches}$.

• Percentage and Proportion: $8\%$ of $60$ is $4.8$. $4.8$ is $\frac{1}{5}$ of the number $24$.

• Basketball Season Pass: Individual tickets cost $14$. A season pass is $175$. Armin must attend at least $13$ games ($13 \times 14 = 182$) for the pass to be cheaper.

• Scientific Notation: $\frac{4.8 \times 10^{-7}}{1.6 \times 10^{-11}} = 3.0 \times 10^{4}$.

• Circle Coordinates: A circle with center $C(-1,2)$ passing through $A(2,6)$ has a diameter $AB$. Point $B$ coordinates are $(-4,-2)$.

• Factoring: The expression $x - 4$ is a factor of $x^3 - 64$, as $(x-4)(x^2+4x+16) = x^3 - 64$.

Scenario: Springdale Park (Problems 32-34)

• Scale and Layout: Intern Mikea is designing a trapezoidal park. In her drawing, $1\, \text{inch}$ represents $1.5\, \text{feet}$. The drawing dimensions are: North side = $28\, \text{inches}$, West side = $16\, \text{inches}$, South side = $40\, \text{inches}$, with a height of $16\, \text{inches}$.

• Area: The area of the scale drawing is $\frac{1}{2} \times (28 + 40) \times 16 = 544\, \text{square inches}$.

• Perimeter: The calculated perimeter based on the drawing dimensions converted to feet (using scaling factor $1.5$) totals $156\, \text{feet}$.

• Side Relationships: The length of the south side ($40\, \text{inches}$) is approximately $142.8\% \approx 142\%$ of the north side ($28\, \text{inches}$).

Scenario: Smith Family Cabin (Problems 35-37)

• Structure and Layout: A $3$-room cabin (2 BR, 1 LR) measures $30\text{'}$ by $24\text{'}$. The roof is an isosceles triangle ($ABC$) with a $30^\circ$ pitch.

• Walkway Area: Mr. Smith builds a $3\text{-foot}$-wide walkway around the outside. The area of the top surface is $396\, \text{square feet}$.

• Fixtures and Curtains: Ceiling fans ($3$ total) cost $52.00$ each. Curtains for small windows ($S$) cost $39.50$; curtains for the large window ($L$) cost $79.00$ ($2 \times 39.50$). The total price for fixtures and curtains is closest to $341$.

• Probability (Independent Events): There is a $20\%$ ($0.20$) chance of rain daily. The probability of rain on two consecutive independent days is $0.20 \times 0.20 = 0.04$.

Advanced Mathematical Concepts

• Irrational Numbers: The expression $\sqrt{2} + \sqrt{8}$ results in an irrational number ($3\sqrt{2}$), whereas $\sqrt{2} \times \sqrt{8} = \sqrt{16} = 4$ (rational).

• Trigonometry: For a line through the origin and $(10,4)$, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{10} = \frac{2}{5}$.

• Cumulative Frequency: Ms. Hernandez's data shows $12$ students scored between $65$-$70$ and $13$ students scored between $65$-$80$. Thus, exactly $1$ student scored in the interval $71$-$80$.

• Logarithmic Modeling (Decibels): Given $d = 10 \log(\frac{I}{K})$, if sound intensity $I = 1,000K$, then $d = 10 \log(1,000) = 10(3) = 30\, \text{decibels}$.

• Weighted Scoring: Mario's basketball stats: $1$-point shots ($80$ attempts, $75\%$ success = $60$ points); $2$-point shots ($60$ attempts, $90\%$ success = $108$ points); $3$-point shots ($60$ attempts, $25\%$ success = $45$ points). Total points = $60 + 108 + 45 = 213$.

• Transformations: The graph of $y = |x - 6|$ is a translation of $y = |x|$ to the right by $6$ units.

• Volume Displacements: A container starts with water at depth $4\, \text{cm}$ ($V = 8 \times 6 \times 4 = 192\, \text{cm}^3$). Submerging a toy soldier raises the depth to $6.6\, \text{cm}$ ($V = 8 \times 6 \times 6.6 = 316.8\, \text{cm}^3$). The soldier's volume is $316.8 - 192 = 124.8 \approx 125\, \text{cm}^3$.

• Packing Material: A cube ($18\, \text{in}$ side) holds a cylinder ($r=6, h=12$). Packing material volume = $18^3 - \pi(6^2)(12)$.

• Unit Conversion: A floor $15\, \text{ft} \times 21\, \text{ft}$ has an area of $315\, \text{sq ft}$. Since $1\, \text{sq yard} = 9\, \text{sq ft}$, the area is $35\, \text{sq yards}$.

• Statistical Analysis: For the list $41, 35, 30, X, Y, 15$ with median $25$ and mode $15$, variables are determined, showing the mean is approximately $26$.

• Arithmetic Sequences: In a sequence where the $3^{\text{rd}}$ term is $13$ and the $4^{\text{th}}$ term is $18$ ($d = 5$), the $50^{\text{th}}$ term is $13 + (47 \times 5) = 248$.

• Trigonometric Properties:

  • The graph of $y = \sin^2(x) + \cos^2(x)$ is a horizontal line at $y = 1$.

  • The period of $f(x) = \csc(4x)$ is $\frac{2\pi}{4} = \frac{\pi}{2}$.

• Complex Numbers: If $i^n = 1$, then the remainder when $n$ is divided by $4$ must be $0$.

• Matrices: The determinant of matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is defined as $ad - bc$.

• Angle Bisection: If ray $PK$ bisects $\angle LPM$ ($11x^\circ$), then $\angle LPK = \angle KPM = \frac{11x}{2}$. Given $\angle LPK = (4x + 18)^\circ$, solving for $x$ gives $x = 12$. The measure of $\angle KPM$ is $66^\circ$.