ACT Math Flashcards

. Proportions

  • What I need to know: A proportion sets two equal ratios to solve for missing scale dimensions, shadow heights, or arc lengths.

  • Formula/rule: ab=cd→a×d=b×c\frac{a}{b} = \frac{c}{d} \rightarrow a \times d = b \times c

  • How to recognize the problem: Comparing two scaling scenarios, shadow/height problems, or arc lengths over total circle circumference (θ360∘=arc2πr\frac{\theta}{360^\circ} = \frac{\text{arc}}{2\pi r}).

  • Steps to solve:

    1. Set up equal fractions matching units (partwhole=partwhole\frac{\text{part}}{\text{whole}} = \frac{\text{part}}{\text{whole}}).

    2. Simplify fractions first if possible.

    3. Cross-multiply across equal sign.

    4. Isolate missing variable.

  • Tiny example: Person height 6 ft6\,\text{ft}, shadow 4 ft4\,\text{ft}; tree shadow 20 ft20\,\text{ft}: 420=6h→4h=120→h=30 ft\frac{4}{20} = \frac{6}{h} \rightarrow 4h = 120 \rightarrow h = 30\,\text{ft}.

  • Common ACT trap/mistake: Mixing up unit positions between numerators and denominators.

2. Averages and Means

  • What I need to know: An average distributes a total sum equally across all items.

  • Formula/rule: Average=Sum of termsTotal number of terms\text{Average} = \frac{\text{Sum of terms}}{\text{Total number of terms}}

  • How to recognize the problem: ACT gives a target average and partial sum, asking for the final missing score or value.

  • Steps to solve:

    1. Set up average equation including unknown xx in numerator.

    2. Multiply target average by total number of terms (including xx).

    3. Subtract given sum from product to solve for xx.

  • Tiny example: Target average 2000 calories/day2000\,\text{calories/day} over 7 days7\,\text{days}, consumed 11400 calories11400\,\text{calories} in 6 days6\,\text{days}: 11400+x7=2000→11400+x=14000→x=2600 calories\frac{11400 + x}{7} = 2000 \rightarrow 11400 + x = 14000 \rightarrow x = 2600\,\text{calories}.

  • Common ACT trap/mistake: Dividing partial sum by wrong count or failing to count missing term in denominator.

3. Basic Probability

  • What I need to know: Probability measures successful outcomes against all possible outcomes.

  • Formula/rule: P(Event)=Number of successful outcomesTotal number of possible outcomesP(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}

  • How to recognize the problem: Asking for probability of selecting an item, landing on a spinner sector, or hitting a winning code.

  • Steps to solve:

    1. Calculate total possible outcomes.

    2. Count favorable/winning outcomes.

    3. Divide winning outcomes by total outcomes and simplify.

  • Tiny example: 44-digit code (digits 0−90-9 allowed, repeatable): Total outcomes =10×10×10×10=10000= 10 \times 10 \times 10 \times 10 = 10000; 11 winning code →P=110000\rightarrow P = \frac{1}{10000}.

  • Common ACT trap/mistake: Miscounting digit options (e.g., counting digits 0−90-9 as 99 options instead of 1010).

4. Fundamental Counting Principle

  • What I need to know: Total combined outcomes for sequential choices are calculated by multiplying option counts.

  • Formula/rule: Ntotal=n1×n2×n3×…N_{\text{total}} = n_1 \times n_2 \times n_3 \times \dots

  • How to recognize the problem: Questions asking for total meal combinations, outfits, or branching paths.

  • Steps to solve:

    1. Count available options at each decision stage.

    2. Multiply stage counts together.

  • Tiny example: 2 mains×3 drinks×3 fruits=18 breakfast combinations2\,\text{mains} \times 3\,\text{drinks} \times 3\,\text{fruits} = 18\,\text{breakfast combinations}.

  • Common ACT trap/mistake: Adding options instead of multiplying (2+3+3=82 + 3 + 3 = 8 instead of 1818).

5. Functions and Custom Operators

  • What I need to know: Custom math symbols follow an explicit defined rule; substitute input values into defined symbol slots.

  • Formula/rule: Identify defined rule equation and substitute given inputs into matching slots.

  • How to recognize the problem: Unfamiliar geometric symbols (e.g., □\square, ⋆\star) defined with formula.

  • Steps to solve:

    1. Match left and right input values to defined formula variables.

    2. Substitute inputs into formula.

    3. Evaluate expression using standard order of operations.

  • Tiny example: Defined operation l□m=l2−m2l \square m = l^2 - m^2; evaluate 2□7→22−72=4−49=−452 \square 7 \rightarrow 2^2 - 7^2 = 4 - 49 = -45.

  • Common ACT trap/mistake: Reversing order of inputs or getting intimidated by new symbols.

6. Equation of a Circle

  • What I need to know: Standard circle equation reveals center coordinates and radius directly.

  • Formula/rule: (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

  • How to recognize the problem: Equation with squared binomials summing to constant, asking for center or radius.

  • Steps to solve:

    1. Change signs inside binomials to get center (h,k)(h, k).

    2. Take square root of right-hand constant to get radius r=r2r = \sqrt{r^2}.

  • Tiny example: (x−7)2+(y+3)2=64→Center (7,−3), Radius r=64=8(x - 7)^2 + (y + 3)^2 = 64 \rightarrow \text{Center } (7, -3), \text{ Radius } r = \sqrt{64} = 8

  • Common ACT trap/mistake: Forgetting to reverse signs for (h,k)(h, k) or using r2r^2 as radius instead of r$.\n\n### 7. Percentages and Percent Changes\n- **What I need to know**: Multi-step percent changes apply sequentially and cannot be added together.\n- **Formula/rule**: \text{Final} = \text{Initial} \times (1 \pm p_1) \times (1 \pm p_2)\n- **How to recognize the problem**: Consecutive daily percentage increases/decreases or direct price adjustments.\n- **Steps to solve**:\n 1. Convert percent changes into decimal multipliers (100\% \pm p\%).\n 2. Multiply multipliers together.\n 3. Multiply result by baseline value.\n- **Tiny example**: Baseline 30\,\text{laps},,10\%increasetoday,increase today,20\%increasetomorrow:increase tomorrow:30 \times 1.10 \times 1.20 = 30 \times 1.32 = 39.6\,\text{laps}.\n- **Common ACT trap/mistake**: Adding percentages directly (10\% + 20\% = 30\%insteadofcompoundinstead of compound32\%).\n\n### 8. Factoring Polynomials\n- **What I need to know**: Factor algebraic expressions rapidly using key pattern rules.\n- **Formula/rule**:\n - Difference of Squares: a^2 - b^2 = (a + b)(a - b)\n - Trinomial (a=1):):x^2 + bx + c = (x + p)(x + q)wherewherep \times q = candandp + q = b\n- **How to recognize the problem**: Two squared terms subtracted, or quadratic trinomials.\n- **Steps to solve**:\n 1. Identify expression pattern type.\n 2. Find square roots or factors of constant term summing to middle coefficient.\n 3. Write in factored binomial form.\n- **Tiny example**: 9x^2 - 100 = (3x + 10)(3x - 10);;x^2 - x - 12 = (x - 4)(x + 3).\n- **Common ACT trap/mistake**: Incorrect sign placement in binomial factors.\n\n### 9. Exponent Rules\n- **What I need to know**: Rules for simplifying exponential expressions with identical bases.\n- **Formula/rule**:\n - Product: x^a \times x^b = x^{a+b}\n - Quotient: \frac{x^a}{x^b} = x^{a-b}\n - Negative: x^{-n} = \frac{1}{x^n}\n - Zero: x^0 = 1\n - Power Distribution: \left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}\n- **How to recognize the problem**: Multi-term algebraic fractions with exponent variables.\n- **Steps to solve**:\n 1. Add numerator exponents of matching bases.\n 2. Subtract denominator exponents.\n 3. Set equal base exponents equal to solve for unknown variables.\n- **Tiny example**: \frac{x^3 \times x^{5m}}{x^m} = x^{3 + 4m};ifequalto; if equal tox^{2+5m} \rightarrow 3 + 4m = 2 + 5m \rightarrow m = 1\n- **Common ACT trap/mistake**: Multiplying exponents when multiplying like bases (x^a \times x^b \neq x^{a \times b}).\n\n### 10. Pythagorean Theorem and Pythagorean Triples\n- **What I need to know**: Right triangle side length formulas and standard integer triples.\n- **Formula/rule**: a^2 + b^2 = c^2;KeyTriples:; Key Triples:(3,4,5),,(5,12,13),,(8,15,17),,(7,24,25).\n- **How to recognize the problem**: Right triangles with two given sides asking for third side.\n- **Steps to solve**:\n 1. Find Greatest Common Factor (GCF) of known sides.\n 2. Scale down sides to spot base triple.\n 3. Multiply missing triple number by GCF.\n- **Tiny example**: Legs 15andand36::\text{GCF} = 3 \rightarrow (5, 12, 13) \times 3 \rightarrow c = 39.\n- **Common ACT trap/mistake**: Applying triples to non-right triangles or mistaking legs for hypotenuse.\n\n### 11. Graphing Linear Inequalities\n- **What I need to know**: Inequality symbol determines boundary line style and shaded region side.\n- **Formula/rule**: Strict (<, >)=dashedline;Non−strict() = dashed line; Non-strict (\le, \ge) = solid line.\n- **How to recognize the problem**: Linear inequality matching test graph options.\n- **Steps to solve**:\n 1. Find x−and- andy-intercepts to plot boundary line.\n 2. Draw dashed or solid boundary line.\n 3. Test point (0,0) in inequality to identify valid shaded half-plane.\n- **Tiny example**: 3x - 2y > 6 \rightarrow \text{intercepts } (2,0), (0,-3);test; test(0,0) \rightarrow 0 > 6(False)(False)\rightarrowshaderegionNOTcontainingshade region NOT containing(0,0).\n- **Common ACT trap/mistake**: Forgetting to flip inequality direction when dividing or multiplying by negative numbers.\n\n### 12. Linear Equation Forms, Parallel and Perpendicular Lines\n- **What I need to know**: Parallel lines share identical slopes; perpendicular line slopes are opposite reciprocals.\n- **Formula/rule**:\n - Slope-intercept: y = mx + b\n - Point-slope: y - y_1 = m(x - x_1)\n - Standard: Ax + By = C\n - Parallel: m_1 = m_2;Perpendicular:; Perpendicular:m_2 = -\frac{1}{m_1}\n- **How to recognize the problem**: Comparing two line equations to check alignment relationship.\n- **Steps to solve**:\n 1. Convert equations to y = mx + b form.\n 2. Extract slopes m_1andandm_2$.

    1. Compare sign and inverted fraction properties.

  • Tiny example: y=23x+4y = \frac{2}{3}x + 4 vs 3x+2y=6→y=−32x+3→m1=23,m2=−323x + 2y = 6 \rightarrow y = -\frac{3}{2}x + 3 \rightarrow m_1 = \frac{2}{3}, m_2 = -\frac{3}{2} (Perpendicular).

  • Common ACT trap/mistake: Forgetting to negate sign when taking reciprocal for perpendicular lines.

13. Slope Formula

  • What I need to know: Slope measures rate of change and line steepness between points.

  • Formula/rule: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

  • How to recognize the problem: Finding steepness or rate of change given two coordinate points.

  • Steps to solve:

    1. Subtract yy-coordinates for numerator.

    2. Subtract xx-coordinates in same order for denominator.

    3. Simplify resulting fraction.

  • Tiny example: Points (6,2)(6,2) and (4,8)→m=8−24−6=6−2=−3(4,8) \rightarrow m = \frac{8 - 2}{4 - 6} = \frac{6}{-2} = -3.

  • Common ACT trap/mistake: Subtracting coordinates in reverse order in denominator relative to numerator.

14. Midpoint Formula

  • What I need to know: Midpoint gives exact center coordinates between two endpoints.

  • Formula/rule: M=(x1+x22,y1+y22)M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

  • How to recognize the problem: Finding coordinate location halfway between two map points.

  • Steps to solve:

    1. Assign coordinate positions to given locations.

    2. Average xx-coordinates and yy-coordinates.

  • Tiny example: Home (0,0)(0,0) and airport (−7,12)→M=(0−72,0+122)=(−3.5,6)(-7,12) \rightarrow M = \left(\frac{0 - 7}{2}, \frac{0 + 12}{2}\right) = (-3.5, 6).

  • Common ACT trap/mistake: Subtracting coordinates instead of averaging/adding them.

15. Logarithms and Exponential Conversions

  • What I need to know: Logarithms are inverse functions of exponential expressions.

  • Formula/rule: log⁡b(x)=n  ⟺  bn=x\log_b(x) = n \iff b^n = x

  • How to recognize the problem: Equations involving log⁡b(x)\log_b(x) asking for unknown base or variable.

  • Steps to solve:

    1. Apply base bb exponentially to both sides to remove logarithm.

    2. Solve resulting exponential equation.

  • Tiny example: log⁡x(25)=2→x2=25→x=5\log_x(25) = 2 \rightarrow x^2 = 25 \rightarrow x = 5.

  • Common ACT trap/mistake: Forgetting logarithm bases must always be positive values.

16. Distance Formula

  • What I need to know: Calculates straight-line distance between two grid points.

  • Formula/rule: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

  • How to recognize the problem: Finding straight-line distance between directional coordinates.

  • Steps to solve:

    1. Calculate horizontal difference Δx\Delta x and vertical difference Δy\Delta y.

    2. Square both differences and add them.

    3. Take square root of sum.

  • Tiny example: Points (2,4)(2,4) and (−5,−20)→Δx=7,Δy=24→d=72+242=625=25 miles(-5,-20) \rightarrow \Delta x = 7, \Delta y = 24 \rightarrow d = \sqrt{7^2 + 24^2} = \sqrt{625} = 25\,\text{miles}.

  • Common ACT trap/mistake: Forgetting to square negative differences properly before adding.

17. Basic Trigonometry and SOH CAH TOA

  • What I need to know: Ratios connecting right triangle angles to side lengths.

  • Formula/rule:

    • sin⁡(θ)=OppHyp\sin(\theta) = \frac{\text{Opp}}{\text{Hyp}}, cos⁡(θ)=AdjHyp\cos(\theta) = \frac{\text{Adj}}{\text{Hyp}}, tan⁡(θ)=OppAdj\tan(\theta) = \frac{\text{Opp}}{\text{Adj}}

    • Reciprocals: csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}, sec⁡(θ)=1cos⁡(θ)\sec(\theta) = \frac{1}{\cos(\theta)}, cot⁡(θ)=1tan⁡(θ)\cot(\theta) = \frac{1}{\tan(\theta)}

  • How to recognize the problem: Right triangles asking for missing side using trigonometric expressions.

  • Steps to solve:

    1. Identify given angle and available/target sides.

    2. Set up matching SOH CAH TOA ratio.

    3. Rearrange formula to solve for target variable.

  • Tiny example: Angle 15∘15^\circ, adjacent leg 88, hypotenuse x→cos⁡(15∘)=8x→x=8cos⁡(15∘)=8sec⁡(15∘)x \rightarrow \cos(15^\circ) = \frac{8}{x} \rightarrow x = \frac{8}{\cos(15^\circ)} = 8\sec(15^\circ).

  • Common ACT trap/mistake: Setting up incorrect side ratio orientation or using incorrect calculator mode (Radians vs Degrees).

18. Special Right Triangles

  • What I need to know: Fixed side ratio rules for 30∘−60∘−90∘30^\circ-60^\circ-90^\circ and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangles.

  • Formula/rule:

    • 30∘−60∘−90∘30^\circ-60^\circ-90^\circ sides: x:x3:2xx : x\sqrt{3} : 2x (Short leg : Long leg : Hypotenuse)

    • 45∘−45∘−90∘45^\circ-45^\circ-90^\circ sides: x:x:x2x : x : x\sqrt{2} (Leg : Leg : Hypotenuse)

  • How to recognize the problem: Right triangles with explicit 30∘,60∘30^\circ, 60^\circ, or 45∘45^\circ angles.

  • Steps to solve:

    1. Identify base short leg xx or leg length.

    2. Apply standard multiplier rule to find missing hypotenuse or long leg.

    3. Rationalize radical denominators if dividing.

  • Tiny example: 30∘−60∘−90∘30^\circ-60^\circ-90^\circ long leg 12→short leg=123=43→hypotenuse y=2(43)=8312 \rightarrow \text{short leg} = \frac{12}{\sqrt{3}} = 4\sqrt{3} \rightarrow \text{hypotenuse } y = 2(4\sqrt{3}) = 8\sqrt{3}.

  • Common ACT trap/mistake: Applying 2\sqrt{2} multiplier to a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle.

19. Perimeter, Area, and Volume Formulas

  • What I need to know: Core formulas for 2D shapes, 3D solids, and stepped orthogonal polygons.

  • Formula/rule:

    • Circle: Circumference C=2πrC = 2\pi r, Area A=πr2A = \pi r^2

    • Trapezoid Area: A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h

    • Triangle Area: A=12bhA = \frac{1}{2}bh

    • Cylinder: Volume V=πr2hV = \pi r^2 h, Surface Area SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi rh

    • Cone Volume: V=13πr2hV = \frac{1}{3}\pi r^2 h

    • Pyramid Volume: V=13AbasehV = \frac{1}{3}A_{\text{base}}h

    • Orthogonal Step Perimeter: P=2×bounding width+2×bounding heightP = 2 \times \text{bounding width} + 2 \times \text{bounding height}

  • How to recognize the problem: Geometry questions asking for perimeter, area, volume, or stepped shape borders.

  • Steps to solve:

    1. Select core geometric formula.

    2. For orthogonal stepped shapes, project inner edges outward to form bounding rectangle.

    3. Calculate perimeter using bounding rectangle dimensions.

  • Tiny example: Stepped polygon with overall width WW and height H \rightarrow \text{Perimeter} = 2W + 2H$.\n- **Common ACT trap/mistake**: Individually calculating small internal step edges instead of using bounding rectangle rule.\n\n### 20. Parallel Lines, Transversals, and Angle Relationships\n- **What I need to know**: Intersecting lines create congruent angle pairs and supplementary linear pairs.\n- **Formula/rule**:\n - Congruent (= , equal): Vertical, Alternate Interior, Alternate Exterior, Corresponding angles.\n - Supplementary (sum to 180^\circ): Linear pairs, Consecutive Interior angles.\n- **How to recognize the problem**: Parallel lines cut by transversals asking for missing angle values.\n- **Steps to solve**:\n 1. Transfer equal angles using alternate interior or corresponding properties.\n 2. Apply linear pair supplementary rule (180^\circ - \text{angle}) to find adjacent angles.\n 3. Step through figure to target angle.\n- **Tiny example**: Interior angle 120^\circ \rightarrow \text{linear pair angle} = 180^\circ - 120^\circ = 60^\circ$.

  • Common ACT trap/mistake: Assuming corresponding angles are congruent when lines are not parallel.

21. Solving Inequalities and Sign Flipping Rules

  • What I need to know: Isolate variables like standard equations, flipping sign direction when multiplying/dividing by negatives.

  • Formula/rule: Flip direction (<→>,≤→≥< \rightarrow >, \le \rightarrow \ge) ONLY when multiplying or dividing both sides by a negative number.

  • How to recognize the problem: Algebraic inequality expressions involving negative coefficients.

  • Steps to solve:

    1. Collect variable terms on one side.

    2. Isolate variable; flip sign ONLY if dividing/multiplying by negative.

    3. Rewrite variable on left side maintaining arrow direction.

  • Tiny example: −3x−8<3x→−8<6x→−43<x→x>−43-3x - 8 < 3x \rightarrow -8 < 6x \rightarrow -\frac{4}{3} < x \rightarrow x > -\frac{4}{3}.

  • Common ACT trap/mistake: Flipping inequality sign when adding or subtracting negative numbers.