ACT Practice Test 3 Comprehensive Study Guide

ACT Overview and General Testing Guidelines

• Test Structure and Timing

  • The standard multiple-choice test battery consists of three core sections: English, Mathematics, and Reading. An optional Science section and an optional Writing section are also available.

  • English: 50 total questions (40 scored items), 35 minutes.

  • Mathematics: 45 total questions (41 scored items), 50 minutes.

  • Reading: 36 total questions (27 scored items), 40 minutes.

  • Science (Optional): 40 total questions (34 scored items), 40 minutes.

  • Writing (Optional): 1 essay prompt, 40 minutes.

  • Total time for the three multiple-choice sections (without Science or Writing) is 2 hours 20 minutes. Taking the test with Science requires 3 hours.

  • A 10-to-15-minute break is scheduled between Test 2 (Mathematics) and Test 3 (Reading).

English Section Guide and Core Concepts

Passage Analysis and Rhetorical Principles

• Passage I: "A Small Rescue"

  • Context: Forestry workers in Helena, Montana, discover two dehydrated two-week-old Western screech owlets (named Lila and Twyla) in a blanketed cardboard box and transfer them to Big Sky Bird Rescue.

  • Rehabilitation Process: The owlets were nourished with crickets and mice, tripling in weight before being moved to an outdoor enclosure to learn flight, hunting, and essential wild defensive behaviors (beak clacking and hissing) to avoid imprinting on humans.

  • Punctuation & Syntax: Semicolons separate independent clauses that are not joined by a coordinating conjunction. Clauses joined by dependent prepositional phrases require comma separation rather than semi-colons.

• Passage II: "The Musical Made Modern"

  • Context: Rodgers and Hammerstein's 1943 Broadway musical Oklahoma! revolutionized American musical theater.

  • Historical Contrast: Traditional musical theater consisted of plotless "revues," loosely connected love songs, and non-character chorus lines. Oklahoma! integrated every song directly into an overarching story arc, requiring every actor to stay in character.

  • Concision & Redundancy: Phrasings such as "from the outset" and "from the very beginning" are redundant when used alongside "opened… in 1943." Unnecessary modifiers should be omitted.

• Passage III: "Schmandt-Besserat and Ancient Accounting"

  • Context: Archaeologists excavating Uruk, Iraq (3100 BCE), discovered proto-cuneiform clay tablets and geometric clay objects (cones, discs, cylinders).

  • Archaeological Breakthrough: In the 1970s, Denise Schmandt-Besserat identified that these geometric objects ("tokens") functioned as a one-to-one counting system for specific commodities (e.g., 5 jars of oil represented by 5 ovoid tokens).

  • Evolution of Writing: Initial records were made by pressing tokens directly into wet clay; later, scribes used styluses to etch two-dimensional token shapes, originating formal counting, accounting, and cuneiform writing systems.

• Passage IV: "The Iron Man of Charleston"

  • Context: Philip Simmons of Charleston, South Carolina, began a blacksmith apprenticeship in 1925 at age 13.

  • Adaptation to Industrial Decline: As automobiles replaced horses and mass production replaced hand-forged machine parts, Simmons transitioned from functional blacksmithing to decorative ironwork.

  • Cultural Synthesis: Simmons combined classical European ironwork motifs present in Charleston with traditional West African ironworking techniques taught by his African American mentor, integrating organic shapes like live animals and asymmetrical plant leaves into hand-bent gates and railings.

• Passage V: "Zadar's Sea Organ"

  • Context: Located on the Riva promenade in Zadar, Croatia, the Sea Organ was designed by architect Nikola Bašić.

  • Acoustic Engineering: Underneath 200 feet of concrete coastal steps, 35 organ pipes of varying lengths utilize wave force to push air through vertical top slots, producing continuous, spectral, and slightly atonal acoustic chords.

• Passage VI: "Make Our Blocks Super"

  • Urban Planning Strategy: Pioneered by urbanist Salvador Rueda in Barcelona, Spain, a "superblock" (superilla) isolates a 3-by-3 block perimeter, banning general automobile traffic from interior streets.

  • Benefits: Reclaims urban street space for pedestrians, cyclists, parks, and playgrounds, significantly mitigating noise pollution and social isolation without requiring major structural road reconstruction.

Mathematics Concepts and Worked Problems

• Coordinate Geometry & Grid Navigation

  • Grid Problem: A uniform square grid is formed by streets where avenues run vertically and boulevards run horizontally.

*   *Midpoint Calculation:* To find the midpoint between Main Street (x = 0) & Oak Boulevard and 4th Avenue (x = 4) & Elm Boulevard along a straight line:

        $\text{Midpoint } x\text{-coordinate} = \frac{0 + 4}{2} = 2$         The location lies along 2nd Avenue.

*   *Translation of Vertices:* A rectangle with vertices $A(1,1)$, $B(5,1)$, $C(5,3)$, and $D(1,3)$ is shifted right 2 units and down 5 units.

        $\text{New } B = (5 + 2, 1 - 5) = (7, -4)$

• Algebraic Analysis & Equations

  • Mean Value Computation: Given four numbers with a mean of 45, where the first three numbers are 26, 39, and 58:         $\text{Total Sum} = 4 \times 45 = 180$         $\text{Fourth Number} = 180 - (26 + 39 + 58) = 180 - 123 = 57$

  • Multivariable Evaluation: For $a = 7$, $b = -4$, $c = 5$:         $abc - ac^2 - ab = (7)( -4)(5) - (7)(5)^2 - (7)(-4)$         $= -140 - 175 + 28 = -287$

  • Absolute Value Equations: Solving $|4x - 3| = 7$:         $4x - 3 = 7 \implies 4x = 10 \implies x = 2.5$         $4x - 3 = -7 \implies 4x = -4 \implies x = -1$

  • Combining Like Terms: Simplifying $3xy^2 - 4x^2y + 12x^2y - 7xy^2$:         $(-4x^2y + 12x^2y) + (3xy^2 - 7xy^2) = 8x^2y - 4xy^2$

  • Binomial Expansion: Expanding $(x - 5)^4$ using Pascal's Triangle coefficients (1, 4, 6, 4, 1):         $(x - 5)^4 = x^4 - 4(x^3)(5) + 6(x^2)(25) - 4(x)(125) + 625$         $= x^4 - 20x^3 + 150x^2 - 500x + 625$

• Functions, Transformations & Graphs

  • Triangle Angle Sum: An interior triangle has angles $(6x)^{\circ}$, $(2x)^{\circ}$, and $x^{\circ}$:         $6x + 2x + x = 180 \implies 9x = 180 \implies x = 20$

  • Absolute Value Graph Intercepts:

*   The x-intercepts where $y = 0$ are located at **$(-3,0)$ and $(7,0)$**.

*   *Linear Modeling from Tabular Data:*

*   Evaluating $y = kx$ for integer $k$:

        $\frac{36}{2.99} \approx 12.04, \quad \frac{116}{9.70} \approx 11.96, \quad \frac{348}{29.00} = 12.00$         Using integer $k = 12$, the predicted y-value for $x = 66.00$ is:         $y = 12 \times 66.00 = 792$

*   *Horizontal Shift:* The function $r(x) = p(x + 6)$ represents a horizontal shift of $y = p(x)$ by **6 units in the negative x-direction**.

• Complex Numbers & Quadratic Motion

  • Complex Number Operations: With $i = \sqrt{-1}$:         $\sqrt{9} + \sqrt{-144} = 3 + 12i$

  • Projectile Motion Max Height: An object is launched vertically at $44.1\,\text{m/s}$ from a $931\text{-meter}$ platform according to $s = -4.9t^2 + 44.1t + 931$:         $\text{Time of peak height } t = -\frac{b}{2a} = -\frac{44.1}{2(-4.9)} = 4.5\,\text{seconds}$         $s(4.5) = -4.9(4.5)^2 + 44.1(4.5) + 931 = -99.225 + 198.45 + 931 = 1030.2\,\text{meters}$

• Sequences, Logarithms & Probability

  • Arithmetic Sequence: Sequence: $-3, -7, -11, -15, -19,\dots$         $a_1 = -3, \quad d = -4$         $a_{21} = a_1 + 20d = -3 + 20(-4) = -83$

  • Logarithmic Form Conversion: Given $y = \log_{10}(2x)$, solve for $x$:         $10^y = 2x \implies x = \left(\frac{1}{2}\right)10^y$

  • Two-Way Table Conditional Probability:

*   Percentage of customers who like white cake:

        $\text{Percentage} = \frac{101}{229} \times 100\% \approx 44\%$

*   Given $P(\text{Yes} \mid 7\text{th Grade}) = \frac{49}{139}$, with $x = 49$:

        $x + y + 71 = 139 \implies 49 + y + 71 = 139 \implies y = 19$

• Trigonometry & Geometry

  • Ray Angle Bisector:

*   Ray $PR$ bisects $\angle QPS$. If $m\angle QPS = 64^{\circ}$ and $m\angle RPS = (2x + 10)^{\circ}$:

        $2x + 10 = \frac{64}{2} = 32 \implies 2x = 22 \implies x = 11$

*   *Tangent Function Identity:* Given $\sin(\theta) = \frac{1}{3}$ and $\cos(\theta) = \frac{2\sqrt{2}}{3}$:

        $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{1/3}{\frac{2\sqrt{2}}{3}} = \frac{1}{2\sqrt{2}}$

*   *Line Intersection:*

*   Line $k$ is horizontal at $y = -6$. Line $j$ passes through $(-10,0)$ and $(0,8)$ with slope $m = \frac{8}{10} = \frac{4}{5}$.

        $y = \frac{4}{5}x + 8 \implies -6 = \frac{4}{5}x + 8 \implies -14 = \frac{4}{5}x \implies x = -\frac{35}{2}$         The intersection point is $\left(-\frac{35}{2}, -6\right)$.

*   *FSA Conditional Probability:* Probability a patient has no FSA given they are in Company A:

        $P(\text{No FSA} \mid \text{Company A}) = \frac{20}{28 + 20} = \frac{20}{48} = \frac{5}{12}$

*   *Volumetric Ratio:* Cleaning solution concentration ratio $3:40$. Concentrated solution required for $5\,\text{gallons}$ ($80\,\text{cups}$) of water:

        $\text{Concentrate} = 80 \times \frac{3}{40} = 6\,\text{cups}$

*   *Circle Area from Inscribed Right Triangle:*

*   Triangle $WCK$ is right-angled at center $C$ with hypotenuse $WK = 5\sqrt{2}\,\text{inches}$. Since legs equal radius $r$:

        $r^2 + r^2 = (5\sqrt{2})^2 = 50 \implies 2r^2 = 50 \implies r^2 = 25$         $\text{Area} = \pi r^2 = 25\pi \approx 78.5\,\text{square inches}$

*   *3D Geometry Cross-Section:*

*   A right pyramid has a square base $PQRS$ of side length $5\,\text{cm}$ and height $12\,\text{cm}$. Triangle $MNT$ passes through the midpoints $M$ and $N$ and peak $T$, producing a cross-sectional area of:

        $\text{Area} = \frac{1}{2} \times 5 \times 12 = 30\,\text{cm}^2$

*   *Shaded Region Area:*

*   Main circle radius $OA = R = 12\,\text{cm}$. Total Area = $\pi (12)^2 = 144\pi\,\text{cm}^2$.
*   Unshaded inner circle diameter $OA = 12\,\text{cm}$ ($r = 6\,\text{cm}$): Area = $\pi (6)^2 = 36\pi\,\text{cm}^2$.
*   Unshaded triangle $OBC$ legs = $12\,\text{cm}$: Area = $\frac{1}{2}(12)(12) = 72\,\text{cm}^2$.
*   $\text{Shaded Area} = 144\pi - (36\pi + 72) = 108\pi - 72 \approx 108(3.14159) - 72 \approx 267\,\text{cm}^2$

*   *Matrix Algebra:* Solving $a\begin{pmatrix} 5 & b \\ 1 & bc \end{pmatrix} = \begin{pmatrix} 10 & 2 \\ c & d \end{pmatrix}$:

        $5a = 10 \implies a = 2$         $ab = 2 \implies 2b = 2 \implies b = 1$         $a(1) = c \implies c = 2$         $d = a(bc) = 2(1 \times 2) = 4$

*   *Distance Unit Conversions:* A marathon is $26.2\,\text{miles}$ ($1.0\,\text{mile} \approx 1.6\,\text{km}$). Difference from a $10,000\text{-meter}$ ($10\,\text{km}$) race:

        $\text{Marathon in km} = 26.2 \times 1.6 = 41.92\,\text{km}$         $\text{Difference} = 41.92 - 10 = 31.92 \approx 31.9\,\text{km}$

*   *Trigonometric Inverse Value:* Evaluating $\cos\left(\arcsin\left(\frac{8}{17}\right)\right)$:

        $\text{If } \sin(\theta) = \frac{8}{17}, \quad \text{adjacent} = \sqrt{17^2 - 8^2} = \sqrt{225} = 15$         $\cos(\theta) = \frac{15}{17}$

*   *Expected Value of a Discrete Random Variable:*

        $E(X) = \sum x \cdot P(X) = (8)(0.1) + (9)(0.3) + (10)(0.4) + (11)(0.2)$         $= 0.8 + 2.7 + 4.0 + 2.2 = 9.7$

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