ACT Math

Provider · ACT, Inc. Format · 45 questions · 50 min Chapters · 8 Explorers · 3 interactive

Test format & strategy

🗻 ACT Math — Official Reporting Categories (enhanced ACT: 45 questions, 50 min)

Reporting Category	% of Test	Key Topics
Preparing for Higher Mathematics	80%	Broken into 5 sub-areas below
· Number & Quantity	10–12%	Integers, ratios, percents, complex numbers, vectors, matrices
· Algebra	17–20%	Linear/quadratic equations, systems, inequalities, absolute value
· Functions	17–20%	Definition, notation, graphs, transformations, composition, inverse
· Geometry	17–20%	Area, volume, angle relationships, trig, coordinate geometry
· Statistics & Probability	12–15%	Mean/median/mode, data interpretation, probability, expected value
Integrating Essential Skills	20%	Multi-step problems spanning rates, percentages, proportions, measurement, area/volume

The enhanced ACT Math test (from September 2025) is 45 multiple-choice questions in 50 minutes — about 67 seconds each — and a calculator is allowed throughout. 41 questions are scored; 4 are unscored field-test items you can't identify, so answer every one. Questions run roughly easy to hard, drawing on everything from pre-algebra to trigonometry.

What to expect

Four answer choices per question (the enhanced ACT dropped the fifth). There is no formula sheet, so the key formulas must be memorized. The content spans number & quantity, algebra, functions, geometry, and statistics & probability.

Questions

45 · multiple choice

Time

50 minutes

Choices

4 per question

Calculator

allowed throughout

▲ Three tactics that score

Don't linger — flag a hard item and move on so easy points aren't lost. Backsolve from the answer choices, and plug in convenient numbers for variables when a question is abstract.

⏱ 3-Tier Pacing Strategy (45 Questions / 50 Minutes)

Questions 1–15 (Easy): Spend ~40 seconds per question (10 minutes total). Bank time early while building confidence.
Questions 16–30 (Medium): Spend ~65 seconds per question (16 minutes total). Watch for multi-step algebra and word problems.
Questions 31–45 (Hard): Spend ~85 seconds per question (21 minutes total). These feature advanced geometry, trigonometry, and novel combinations.
Final 3 Minutes: Check flagged items and ensure EVERY single question has a bubbled answer!

▲ Zero Guessing Penalty & 4-Choice Advantage

No Negative Marking: There is absolutely no penalty for wrong answers on the ACT. An unanswered question is guaranteed 0 points; an educated guess gives a 25% to 50% chance of a point.
4-Choice Format: Enhanced ACT has 4 options (A, B, C, D) instead of 5, boosting blind guessing accuracy from 20% to 25%. Eliminating just one obvious wrong answer raises your odds to 33.3%, and eliminating two gives a 50% coin-flip.
Letter of the Day: If running out of time on the last 3–4 questions, pick one consistent letter (e.g., C) for all remaining items rather than alternating randomly.

ACT Math reporting category distribution. Preparing for Higher Mathematics comprises 80% (Algebra 19%, Geometry 19%, Functions 19%, Stats/Prob 13%, Number & Quantity 10%), while Integrating Essential Skills makes up the remaining 20%.

On the enhanced ACT Math test (45 questions in 50 minutes), a student allocates 10 minutes to the first 15 questions and 16 minutes to questions 16 through 30. How many seconds per question remain for the final 15 questions, assuming 3 minutes are reserved at the end to check answers?

\text{Time spent so far} = 10 + 16 = 26\text{ minutes}

. Leaving 3 minutes for review leaves

50 - 26 - 3 = 21\text{ minutes}

for the final 15 questions.

\text{Time per question} = \frac{21 \times 60\text{ seconds}}{15\text{ questions}} = \frac{1260}{15} = 84\text{ seconds per question}

📝 Chapter Quiz

Questions Level

Numbers, Percents & Ratios

🗻 Number & Quantity 7–10% · Integers & primes · Fractions, decimals, percents · Ratios & proportions · Real number properties

Pre-algebra fundamentals: fractions and decimals, percentages, ratios and proportions, factors and multiples.

▲ Reference — Proportional reasoning

Percent of a number: $\tfrac{p}{100}\times N$ . A ratio $a:b$ splits a total into $a+b$ equal parts. Set up a proportion $\tfrac{a}{b}=\tfrac{c}{d}$ and cross-multiply.

Real number line showing integers and rational fractions. On the ACT, distance between two coordinates

a

and

b

is given by the absolute difference

|a - b|

Percent of

\tfrac{p}{100}\times N

Percent change

\tfrac{\text{new}-\text{old}}{\text{old}}

Ratio parts

total

\div (a+b)

Proportion

cross-multiply

⚠️ Classic ACT Traps: Percents, Ratios & Number Properties

Percent Change Base Value: Always divide the difference by the original starting value, never the new value: $\text{Percent Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$ .
Successive Percentage Changes Do Not Add: A 20% discount followed by a 10% discount is NOT a 30% discount. The total multiplier is $0.80 \times 0.90 = 0.72$ , meaning an effective 28% total discount.
Ratio Parts vs. Whole: If boys to girls is $3:5$ , boys represent $\frac{3}{3+5} = \frac{3}{8}$ of the total students, NOT $\frac{3}{5}$ . Always add ratio terms to find total parts.
Number Property Facts: $2$ is the only even prime number; $1$ is neither prime nor composite; $0$ is an even integer (neither positive nor negative).

Worked example 1.A

What is $15\%$ of $200$ ? $0.15\times200=30.$

Worked example 1.B

Split $30$ in the ratio $2:3$ . Find the smaller part.

Five parts, each $6$ ; smaller is 2 parts:

12.

A jacket costs $120 after a 25% discount. What was the original price?

Let original =

x

0.75x = 120 \Rightarrow x = 160

. Original price: $160.

A merchant buys a gadget for $50, marks it up by 40%, and later sells it at a 20% discount off the marked price. What was the final selling price and the merchant's net profit?

Marked price:

50 \times 1.40 = \$70

. Sale price after 20% discount:

70 \times 0.80 = \$56

. Net profit:

56 - 50 = \$6

📝 Chapter Quiz

Questions Level

Elementary & Intermediate Algebra

🗻 Algebra 12–15% · Linear equations & inequalities · Systems of equations · Quadratics · Absolute value equations

Linear equations, quadratics, exponents, radicals and absolute value — the algebraic core of the test.

▲ Reference — Algebra essentials

Quadratic formula $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$ ; difference of squares $a^{2}-b^{2}=(a-b)(a+b)$ . Exponents: $a^{m}a^{n}=a^{m+n}$ . Absolute value $|x|=c$ gives $x=\pm c$ .

Graph of

y = x^2 - 4x + 3

. The vertex coordinates are given by

x = -b/(2a) = 2

and

y = f(2) = -1

. The

x

-intercepts

(1,0)

and

(3,0)

represent the real solutions to

x^2 - 4x + 3 = 0

Quadratic formula

\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}

Difference of squares

(a-b)(a+b)

Exponents

a^{m}a^{n}=a^{m+n}

Absolute value

x=\pm c

⚠️ Classic ACT Traps: Algebra & Quadratics

No Formula Sheet Provided: Unlike the SAT, the ACT provides zero reference formulas. You must memorize the Quadratic Formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ and the vertex formula $x = -\frac{b}{2a}$ .
Flipping Inequality Signs: When multiplying or dividing an inequality by a negative number, always reverse the inequality symbol: $-3x < 12 \Rightarrow x > -4$ .
Vieta's Sum of Roots Shortcut: For any quadratic $ax^2 + bx + c = 0$ , the sum of the solutions is $-b/a$ and the product of the solutions is $c/a$ . When the ACT asks for 'the sum of all solutions', never solve with the quadratic formula — simply evaluate $-b/a$ !
Discriminant Rules: The quantity $\Delta = b^2 - 4ac$ reveals solution count: $\Delta > 0$ (2 real roots), $\Delta = 0$ (1 real root), $\Delta < 0$ (2 complex/imaginary roots).

Worked example 2.A

Solve $3(x-2)=12$ .

$x-2=4$ :

x=6.

Worked example 2.B

Simplify $\sqrt{50}$ . $\sqrt{25\cdot 2}=5\sqrt{2}.$

For what values of

x

|2x - 3| < 7

-7 < 2x-3 < 7 \Rightarrow -4 < 2x < 10 \Rightarrow -2 < x < 5

What is the sum of the solutions of the quadratic equation

3x^2 - 12x - 5 = 0

By Vieta's formulas, for any quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

. Here

a = 3

and

b = -12

, so the sum of the solutions is

-\frac{-12}{3} = 4

📝 Chapter Quiz

Questions Level

Functions & Graphs

🗻 Functions & Graphs 12–15% · Function notation · Domain & range · Transformations · Composition & inverse

Function notation, evaluating and composing functions, and reading the vertex of a parabola.

▲ Reference — Functions

$f(a)$ substitutes $a$ for $x$ ; a composite $f(g(x))$ works inside-out. In vertex form $y=a(x-h)^{2}+k$ , the vertex is $(h,k)$ and the axis is $x=h$ .

Graph of rational function

f(x) = \frac{1}{x - 2} + 1

. The vertical asymptote occurs where the denominator equals zero (

x = 2

, restricted from the domain). The horizontal asymptote

y = 1

is the value approached as

x \to \pm\infty

Slide the vertex $(h,k)$ of $y=(x-h)^{2}+k$ . The amber dot is the vertex; the dashed line is the axis of symmetry.

⊙ Explorer · Vertex forminteractive

h (horizontal) = 0

k (vertical) = 0

vertex—

axis of symmetry—

Moving $h$ slides the curve left/right; $k$ slides it up/down.

Evaluate

f(a)

Composite

f(g(x))

Vertex form

a(x-h)^{2}+k

Rate of change

\dfrac{f(b)-f(a)}{b-a}

▲ ACT Traps & Pitfalls — Functions & Graphs

Order of Composition: $(f \circ g)(x) = f(g(x))$ means evaluate $g(x)$ FIRST, then substitute the result into $f$ . Evaluating outside-in is one of the most common ACT distractor answer choices.
Domain Restrictions: Watch for two key restrictions: denominators cannot equal zero (e.g., $\frac{1}{x-3} \implies x \neq 3$ ), and expressions under an even radical cannot be negative (e.g., $\sqrt{2x-8} \implies 2x-8 \ge 0 \implies x \ge 4$ ).
Vertical Line Test vs. Horizontal Line Test: A vertical line tests whether a graph is a function (each input has at most one output). A horizontal line tests whether a function is one-to-one (has an invertible function $f^{-1}(x)$ ).
Function Transformations: $f(x - c)$ shifts RIGHT by $c$ , while $f(x + c)$ shifts LEFT by $c$ (horizontal operations are counter-intuitive). Vertical shifts $f(x) + c$ go UP, and $f(x) - c$ go DOWN.

Worked example 3.A

If $f(x)=x^{2}-1$ , find $f(3)$ . $9-1=8.$

Worked example 3.B

If $f(x)=2x+1$ and $g(x)=x^{2}$ , find $f(g(2))$ .

$g(2)=4$ , then $f(4)=9$ :

9.

f(x) = x^2 - 1

and

g(x) = 2x + 3

, find

f(g(1))

g(1) = 5

f(5) = 25 - 1 = 24

What is the domain of the function

f(x) = \dfrac{\sqrt{x + 5}}{x - 3}

For the radical in the numerator to be defined, the radicand must be non-negative:

x + 5 \ge 0 \Rightarrow x \ge -5

. For the rational expression to be defined, the denominator cannot equal zero:

x - 3 \neq 0 \Rightarrow x \neq 3

. Combining both restrictions gives

x \ge -5

and

x \neq 3

, or in interval notation

[-5, 3) \cup (3, \infty)

f(x) = 3x - 7

, what is the formula for the inverse function

f^{-1}(x)

Set

y = 3x - 7

. To find the inverse, interchange

x

and

y

x = 3y - 7

. Solve for

y

x + 7 = 3y \Rightarrow y = \dfrac{x + 7}{3}

. Therefore,

f^{-1}(x) = \dfrac{x + 7}{3}

📝 Chapter Quiz

Questions Level

Coordinate Geometry

🗻 Coordinate Geometry 12–15% · Slope & equations of lines · Distance & midpoint · Circles · Parabolas

Lines in the plane: slope, intercepts, distance and midpoint, and the equations of lines.

▲ Reference — The coordinate plane

Slope $m=\dfrac{y_2-y_1}{x_2-x_1}$ ; line $y=mx+b$ . Distance $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$ ; midpoint $\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)$ .

Adjust slope $m$ and intercept $b$ of $y=mx+b$ ; the amber dots are the intercepts.

⊙ Explorer · Slope & interceptsinteractive

Slope m = 1

Intercept b = 0

slope—

y-intercept—

x-intercept—

Parallel lines share a slope; perpendicular slopes multiply to $-1$ .

Perpendicular lines

L_1

and

L_2

intersecting at right angles at

(2, 3)

. The product of their slopes satisfies

m_1 \cdot m_2 = 2 \cdot (-0.5) = -1

(negative reciprocals).

Slope

\dfrac{y_2-y_1}{x_2-x_1}

Distance

\sqrt{\Delta x^{2}+\Delta y^{2}}

Midpoint

\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right)

Line

y=mx+b

⚠️ Classic ACT Traps: Coordinate Geometry

Perpendicular Slopes: Perpendicular lines have slopes that are negative reciprocals: $m_2 = -\frac{1}{m_1}$ . A line perpendicular to $y = \frac{2}{3}x - 5$ has slope $-\frac{3}{2}$ , NOT $\frac{3}{2}$ and NOT $-\frac{2}{3}$ .
Circle Equation Center Signs: The equation $(x - h)^2 + (y - k)^2 = r^2$ subtracts coordinates. A circle with equation $(x + 3)^2 + (y - 4)^2 = 49$ has center $(-3, 4)$ and radius $r = 7$ , NOT $49$ !
Finding an Missing Endpoint from Midpoint: If $M(x_m, y_m)$ is the midpoint of $\overline{AB}$ and you know $A(x_a, y_a)$ , do NOT average $A$ and $M$ ! Use the step method: $x_b = 2x_m - x_a$ and $y_b = 2y_m - y_a$ .
Horizontal and Vertical Lines: Horizontal lines have slope $m = 0$ and equation $y = c$ ; vertical lines have undefined slope and equation $x = c$ .

Worked example 4.A

Find the distance between $(0,0)$ and $(3,4)$ . $\sqrt{9+16}=5.$

Worked example 4.B

Find the midpoint of $(2,4)$ and $(6,8)$ . $(4,\,6).$

What is the equation of the circle with centre

(-2, 4)

and radius

5

(x+2)^2 + (y-4)^2 = 25

Point

M(3, -2)

is the midpoint of line segment

\overline{AB}

. If point

A

has coordinates

(-1, 4)

, what are the coordinates of point

B

Using the midpoint formula for each coordinate:

x_M = \frac{x_A + x_B}{2} \Rightarrow 3 = \frac{-1 + x_B}{2} \Rightarrow 6 = -1 + x_B \Rightarrow x_B = 7

y_M = \frac{y_A + y_B}{2} \Rightarrow -2 = \frac{4 + y_B}{2} \Rightarrow -4 = 4 + y_B \Rightarrow y_B = -8

.
Coordinates of

B

: (7, -8).

📝 Chapter Quiz

Questions Level

Plane Geometry

🗻 Plane Geometry 12–15% · Angles & parallel lines · Triangles · Polygons & circles · Area & volume

Angles, triangles and polygons, circles, and area, perimeter and volume.

▲ Reference — Shapes

Triangle angles sum to $180^{\circ}$ ; a polygon of $n$ sides sums to $(n-2)\cdot180^{\circ}$ . Pythagoras $a^{2}+b^{2}=c^{2}$ . Circle area $\pi r^{2}$ , circumference $2\pi r$ .

Parallel lines cut by a transversal. Alternate interior angles and corresponding angles are equal (

120^\circ

), while consecutive interior angles are supplementary (

120^\circ + 60^\circ = 180^\circ

Triangle area

\tfrac12 bh

Pythagoras

a^{2}+b^{2}=c^{2}

Polygon angles

(n-2)\cdot180^{\circ}

Circle

\pi r^{2},\ 2\pi r

⚠️ Classic ACT Traps: Plane Geometry

Figures Are Not Necessarily to Scale: Never trust eye measurements on the ACT. An angle that looks like 90° is only 90° if marked with a square or proven with given equations.
The Triangle Inequality Theorem: The sum of any two sides must strictly exceed the third: $a + b > c$ . If two sides are $7$ and $11$ , the third side $x$ must satisfy $11 - 7 < x < 11 + 7 \Rightarrow 4 < x < 18$ .
Special Right Triangles (No Formula Sheet!): Memorize $30^\circ-60^\circ-90^\circ$ sides $x : x\sqrt{3} : 2x$ , and $45^\circ-45^\circ-90^\circ$ sides $x : x : x\sqrt{2}$ . Common Pythagorean triples include $3-4-5$ , $5-12-13$ , and $8-15-17$ .
Regular Polygon Angles: The sum of interior angles is $(n-2)\cdot180^\circ$ . Each interior angle of a regular $n$ -gon is $\frac{(n-2)\cdot180^\circ}{n}$ , and each exterior angle is $\frac{360^\circ}{n}$ .

Worked example 5.A

A right triangle has legs $9$ and $12$ . Find the hypotenuse. $\sqrt{81+144}=15.$

Worked example 5.B

Find the interior-angle sum of a pentagon. $(5-2)\cdot180^{\circ}=540^{\circ}.$

A cylinder has radius 3 cm and height 8 cm. Find its total surface area.

TSA

= 2\pi r^2 + 2\pi r h = 2\pi(9) + 2\pi(3)(8) = 18\pi + 48\pi = 66\pi \approx 207.3

^2

Two sides of a non-degenerate triangle measure 7 inches and 11 inches. What is the range of possible perimeters for this triangle?

Let

x

be the third side. By the Triangle Inequality Theorem,

11 - 7 < x < 11 + 7 \Rightarrow 4 < x < 18

. The perimeter is

P = 7 + 11 + x = 18 + x

. Substituting the bounds for

x

18 + 4 < P < 18 + 18 \Rightarrow 22 < P < 36

inches.

📝 Chapter Quiz

Questions Level

Trigonometry

🗻 Trigonometry 7–10% · SOH-CAH-TOA · Unit circle · Trig identities · Graphs of trig functions

Right-triangle ratios, the values of common angles, and reading the sine curve.

▲ Reference — SOH-CAH-TOA

$\sin\theta=\dfrac{\text{opp}}{\text{hyp}}$ , $\cos\theta=\dfrac{\text{adj}}{\text{hyp}}$ , $\tan\theta=\dfrac{\text{opp}}{\text{adj}}$ . Common values: $\sin30^{\circ}=\tfrac12$ , $\cos60^{\circ}=\tfrac12$ , $\tan45^{\circ}=1$ .

Unit circle with terminal angle

\theta = 150^\circ

(

\frac{5\pi}{6}

rad). The terminal point has coordinates

(\cos\theta, \sin\theta) = (-\frac{\sqrt{3}}{2}, \frac{1}{2})

. In Quadrant II, sine is positive while cosine and tangent are negative.

Slide the angle $\theta$ along the sine curve. The readouts give $\sin\theta$ and $\cos\theta$ .

⊙ Explorer · The sine curveinteractive

Angle θ = 30

sin θ—

cos θ—

Sine peaks at $90^{\circ}$ , is zero at $0^{\circ}$ and $180^{\circ}$ , and bottoms at $270^{\circ}$ .

Sine

\tfrac{\text{opp}}{\text{hyp}}

Cosine

\tfrac{\text{adj}}{\text{hyp}}

Tangent

\tfrac{\text{opp}}{\text{adj}}

Identity

\sin^{2}\theta+\cos^{2}\theta=1

▲ ACT Traps & Pitfalls — Trigonometry

Radian vs. Degree Mode: ACT problems frequently alternate between degrees and radians. Always verify your calculator status line before evaluating trig functions, especially when finding angles or amplitudes.
Pythagorean Identity Variations: $\sin^2\theta + \cos^2\theta = 1$ is the most tested trig identity on the ACT. Look for algebraic rearrangements such as $1 - \sin^2\theta = \cos^2\theta$ and $\frac{\sin^2\theta}{1 - \cos\theta} = 1 + \cos\theta$ .
ASTC Quadrant Signs: Remember 'All Students Take Calculus': Quadrant I (All positive), Quadrant II (Sine positive), Quadrant III (Tangent positive), Quadrant IV (Cosine positive). A common error is missing the negative sign when given $\theta$ in Quadrants II or III.
Period of a Sinusoid: For $y = A \sin(Bx)$ or $y = A \cos(Bx)$ , the period is $\frac{2\pi}{B}$ (or $\frac{360^\circ}{B}$ ), NOT $2\pi B$ . Amplitude is $|A|$ , and the vertical midline is $y = D$ .

Worked example 6.A

A right triangle has opposite $3$ and hypotenuse $5$ . Find $\sin\theta$ . $\sin\theta=\tfrac{3}{5}.$

Worked example 6.B

Evaluate $\tan45^{\circ}$ . $1.$

In right triangle

ABC

with

C = 90°

a = 5

c = 13

. Find

\sin A

\cos A

\tan A

b = \sqrt{169-25} = 12

\sin A = 5/13

\cos A = 12/13

\tan A = 5/12

\cos\theta = -\frac{4}{5}

and

\theta

lies in Quadrant II, what is the value of

\tan\theta

In Quadrant II,

x = -4

and

r = 5

. Using

x^2 + y^2 = r^2

, we have

(-4)^2 + y^2 = 5^2 \Rightarrow 16 + y^2 = 25 \Rightarrow y = 3

(since sine is positive in Quadrant II). Therefore,

\tan\theta = \frac{y}{x} = -\frac{3}{4}

What is the period of the trigonometric function

f(x) = 4\sin(3x) - 2

For any sinusoidal function of the form

f(x) = A\sin(Bx - C) + D

, the period is given by

\frac{2\pi}{B}

. Here

B = 3

, so the period is

\frac{2\pi}{3}

📝 Chapter Quiz

Questions Level

Statistics & Probability

🗻 Statistics & Probability 8–12% · Mean, median, mode · Data spread · Probability · Expected value

Averages and spread, basic counting, and the probability of simple and combined events.

▲ Reference — Data & chance

Mean is total over count; the median is the middle value. Probability $=\dfrac{\text{favourable}}{\text{total}}$ ; independent events multiply: $P(A\text{ and }B)=P(A)\cdot P(B)$ .

Five-number summary box-and-whisker plot: Minimum = 14, First Quartile

Q_1 = 22

, Median

= 28

, Third Quartile

Q_3 = 35

, Maximum = 44. The Interquartile Range is

\text{IQR} = Q_3 - Q_1 = 35 - 22 = 13

, containing the middle 50% of the distribution.

Mean

\dfrac{\text{sum}}{\text{count}}

Median

middle value

Probability

\dfrac{\text{fav}}{\text{total}}

Independent

P(A)\cdot P(B)

▲ ACT Traps & Pitfalls — Statistics & Probability

Outlier Sensitivity (Mean vs. Median): The mean is heavily pulled by extreme outliers, whereas the median is resistant. If an unusually high score is added to a dataset, the mean increases significantly, but the median often changes little or not at all.
Expected Value (Net vs. Gross): On ACT raffle/carnival game problems, expected value is $E(X) = \sum [x \cdot P(x)]$ . Watch out whether the question asks for net winnings (subtracting the ticket cost) or gross payout!
Mutually Exclusive vs. Independent: For mutually exclusive events, $P(A \text{ or } B) = P(A) + P(B)$ . For non-mutually exclusive events, you must subtract the overlap: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . For independent events, $P(A \text{ and } B) = P(A) \times P(B)$ .
Combinations vs. Permutations: Order matters $\implies$ Permutation $_nP_r = \frac{n!}{(n-r)!}$ (e.g. officer positions: President, VP, Secretary). Order does NOT matter $\implies$ Combination $_nC_r = \frac{n!}{r!(n-r)!}$ (e.g. committee, team, selecting toppings).

Worked example 7.A

Find the mean of $4,\,8,\,10,\,18$ . $\frac{40}{4}=10.$

Worked example 7.B

A fair coin is tossed twice. Find $P(\text{two heads})$ . $\tfrac12\cdot\tfrac12=\tfrac14.$

A data set is: 4, 7, 7, 9, 12, 15. Find the mean, median, and range.

Mean

= 54/6 = 9

. Median

= (7+9)/2 = 8

. Range

= 15-4 = 11

A carnival game costs $5 to play. There is a

\frac{1}{10}

probability of winning $30, a

\frac{2}{10}

probability of winning $10, and a

\frac{7}{10}

probability of winning nothing. What is the expected net profit or loss per play?

The expected gross payout is

E(\text{gross}) = 30\left(\frac{1}{10}\right) + 10\left(\frac{2}{10}\right) + 0\left(\frac{7}{10}\right) = 3 + 2 + 0 = \$5

. Since the ticket costs $5, the expected net profit is

5 - 5 = \$0

(a fair game).

A committee of 3 students is to be chosen from a group of 8 candidates. How many different committees can be formed?

Because the order of committee members does not matter, this is a combination:

_8C_3 = \dfrac{8!}{3!(8-3)!} = \dfrac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56

different committees.

📝 Chapter Quiz

Questions Level

Matrices, Vectors & Complex Numbers

🗻 Advanced Topics (Matrices/Vectors/Complex) ~5% · Matrix operations · Vector components · Complex arithmetic · Polar form

The hardest ACT items touch matrices, vectors, logarithms and the imaginary unit $i$ .

▲ Reference — Advanced topics

$i^{2}=-1$ ; add matrices entrywise; a vector $(a,b)$ has magnitude $\sqrt{a^{2}+b^{2}}$ ; $\log_b x=y$ means $b^{y}=x$ .

Complex number

z = 3 + 4i

on the complex plane (real axis horizontal, imaginary axis vertical). The modulus is the distance from the origin

|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5

Imaginary

i^{2}=-1

Matrix

add entrywise

Vector

\sqrt{a^{2}+b^{2}}

Logarithm

\log_b x

▲ ACT Traps & Pitfalls — Advanced Topics

Matrix Dimensions for Multiplication: To multiply $A_{m \times k} \times B_{k \times n}$ , the inner dimensions MUST match (number of columns of $A$ = number of rows of $B$ ). The product has dimensions $m \times n$ . Matrix multiplication is NOT commutative: $AB \neq BA$ in general.
Matrix Determinant & Inverse: For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ , $\det(A) = ad - bc$ . If $\det(A) = 0$ , the matrix has NO inverse. To form $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ , swap the main diagonal and negate the off-diagonal entries!
Powers of $i$ Cycle of 4: $i^1 = i$ , $i^2 = -1$ , $i^3 = -i$ , $i^4 = 1$ . To evaluate $i^n$ , divide $n$ by 4 and look at the remainder: remainder 0 $\implies 1$ , remainder 1 $\implies i$ , remainder 2 $\implies -1$ , remainder 3 $\implies -i$ .
Vector Magnitude & Addition: Vector addition is component-wise: $\langle u_1, u_2 \rangle + \langle v_1, v_2 \rangle = \langle u_1+v_1, u_2+v_2 \rangle$ . Magnitude is $|\vec{v}| = \sqrt{v_1^2 + v_2^2}$ . Do not simply add magnitudes together!

Worked example 8.A

Simplify $i^{2}+i^{4}$ . $-1+1=0.$

● Complex Numbers & Polar Form

A complex number $z = a+bi$ has modulus $|z| = \sqrt{a^2+b^2}$ and argument $\theta = \arctan(b/a)$ . Polar form: $z = r(\cos\theta + i\sin\theta)$ . Multiplying in polar: multiply moduli, add arguments.

Worked example — Matrix determinant & inverse

Find $A^{-1}$ for $A = \begin{pmatrix}3&1\\5&2\end{pmatrix}$ .

$\det A = 6-5 = 1$ . $A^{-1} = \dfrac{1}{1}\begin{pmatrix}2&-1\\-5&3\end{pmatrix} = \begin{pmatrix}2&-1\\-5&3\end{pmatrix}$ .

Simplify

(2+3i)(1-i)

and write in the form

a+bi

(2+3i)(1-i) = 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5+i

A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}

and

B = \begin{pmatrix} 5 \\ -2 \end{pmatrix}

, what is the matrix product

AB

The product of a

2 \times 2

matrix and a

2 \times 1

matrix is a

2 \times 1

matrix. Compute row by column: the first entry is

2(5) + (-1)(-2) = 10 + 2 = 12

; the second entry is

3(5) + 4(-2) = 15 - 8 = 7

. Thus

AB = \begin{pmatrix} 12 \\ 7 \end{pmatrix}

What is the value of the complex expression

i^{47} - i^{18}

Divide the exponents by 4:

47 = 4(11) + 3

, so

i^{47} = i^3 = -i

18 = 4(4) + 2

, so

i^{18} = i^2 = -1

. Therefore,

i^{47} - i^{18} = -i - (-1) = 1 - i

📝 Chapter Quiz

Questions Level

Practice set

Fifty ACT-style problems spanning numbers, algebra, functions, geometry, trigonometry and statistics, tagged by difficulty. Click any problem for a full worked solution.

Evaluate $15\%$ of $200$ .

Basic

0.15\times200=30.

Simplify $\dfrac{3}{4}+\dfrac{1}{8}$ .

Basic

$\tfrac{6}{8}+\tfrac{1}{8}$ :

\tfrac{7}{8}.

Split $30$ in the ratio $2:3$ . Find the smaller part.

Intermediate

Five parts, each $6$ :

2\times6=12.

$12$ is what percent of $48$ ?

Intermediate

\frac{12}{48}=25\%.

Find the least common multiple of $6$ and $8$ .

Intermediate

\text{LCM}=24.

A recipe is scaled from 2 to 5 servings. How many cups replace $4$ cups?

Intermediate

Multiply by $\tfrac{5}{2}$ :

4\times\tfrac{5}{2}=10.

A \$80 item is marked down $20\%$ . Find the sale price.

Advanced

80\times0.80=\$64.

A car travels $120$ miles in $3$ hours. Find the average speed.

Intermediate

\frac{120}{3}=40\ \text{mph}.

Solve $2x+7=19$ .

Basic

$2x=12$ :

x=6.

Solve $3(x-2)=12$ .

Intermediate

$x-2=4$ :

x=6.

Factor $x^{2}-9$ .

Intermediate

(x-3)(x+3).

Solve $x^{2}-7x+10=0$ .

Intermediate

$(x-2)(x-5)=0$ :

x=2,\ 5.

Simplify $2x+3x-5$ .

Intermediate

5x-5.

If $y=2x+1$ , find $x$ when $y=9$ .

Intermediate

$2x=8$ :

x=4.

Solve $|x-3|=5$ .

Advanced

$x-3=\pm 5$ :

x=8,\ -2.

Simplify $x^{2}\cdot x^{4}$ .

Intermediate

x^{6}.

Solve $5x-3=2x+9$ .

Intermediate

$3x=12$ :

x=4.

Simplify $\sqrt{50}$ .

Advanced

\sqrt{25\cdot2}=5\sqrt{2}.

Evaluate $2^{3}+3^{2}$ .

Intermediate

8+9=17.

Solve the system $x+y=7$ , $2x-y=2$ .

Advanced

Add: $3x=9\Rightarrow x=3$ , then $y=4$ :

x=3,\ y=4.

If $f(x)=x^{2}-1$ , find $f(3)$ .

Intermediate

9-1=8.

State the vertex of $y=(x-2)^{2}+3$ .

Intermediate

(2,\,3).

If $f(x)=2x+1$ and $g(x)=x^{2}$ , find $f(g(2))$ .

Advanced

$g(2)=4$ , then $f(4)=9$ :

9.

For what value of $x$ is $\dfrac{1}{x-2}$ undefined?

Intermediate

The denominator is zero:

x=2.

Find the y-intercept of $y=3x-6$ .

Intermediate

(0,\,-6).

For $f(x)=x^{2}$ , find the average rate of change on $[1,3]$ .

Advanced

\frac{9-1}{3-1}=4.

Find the slope of the line through $(0,0)$ and $(2,6)$ .

Basic

\frac{6}{2}=3.

Find the midpoint of $(2,4)$ and $(6,8)$ .

Intermediate

\left(\tfrac{2+6}{2},\tfrac{4+8}{2}\right)=(4,6).

Find the distance between $(0,0)$ and $(3,4)$ .

Intermediate

\sqrt{9+16}=5.

Find the slope of $y=2x+5$ .

Intermediate

m=2.

Find the x-intercept of $2x+3y=12$ .

Intermediate

Set $y=0$ : $2x=12$ :

x=6.

A line passes through $(1,2)$ with slope $3$ . Find $y$ when $x=4$ .

Advanced

$y-2=3(4-1)$ :

y=11.

Give the equation of the horizontal line through $(0,5)$ .

Intermediate

y=5.

Find the area of a rectangle $7\times4$ .

Basic

7\times4=28.

A triangle has angles $50^{\circ}$ and $60^{\circ}$ . Find the third.

Intermediate

180^{\circ}-110^{\circ}=70^{\circ}.

A right triangle has legs $9$ and $12$ . Find the hypotenuse.

Intermediate

\sqrt{81+144}=15.

Find the area of a triangle with base $10$ and height $8$ .

Intermediate

\tfrac12\times10\times8=40.

Find the area of a circle with radius $5$ .

Intermediate

\pi(5)^{2}=25\pi.

Find the circumference of a circle with $r=7$ . Take $\pi=\tfrac{22}{7}$ .

Intermediate

2\times\tfrac{22}{7}\times7=44.

Find the interior-angle sum of a pentagon.

Advanced

(5-2)\cdot180^{\circ}=540^{\circ}.

Find the volume of a box $2\times3\times4$ .

Intermediate

2\times3\times4=24.

A square has area $36$ . Find its perimeter.

Advanced

Side $=6$ :

4\times6=24.

A right triangle has opposite $3$ and hypotenuse $5$ . Find $\sin\theta$ .

Intermediate

\sin\theta=\tfrac{3}{5}.

Evaluate $\sin30^{\circ}$ .

Intermediate

\tfrac12.

Evaluate $\cos60^{\circ}$ .

Intermediate

\tfrac12.

Evaluate $\tan45^{\circ}$ .

Advanced

1.

A right triangle has adjacent $4$ and hypotenuse $5$ . Find $\cos\theta$ .

Intermediate

\cos\theta=\tfrac{4}{5}.

A right triangle has opposite $6$ and adjacent $8$ . Find $\tan\theta$ .

Advanced

\tan\theta=\tfrac{6}{8}=\tfrac{3}{4}.

Find the mean of $4,\,8,\,10,\,18$ .

Intermediate

\frac{40}{4}=10.

A fair coin is tossed twice. Find $P(\text{two heads})$ .

Intermediate

\tfrac12\cdot\tfrac12=\tfrac14.

✎

Test generator

Build a randomised test from this subject’s question bank. Pick the number of questions and a difficulty, generate, then reveal the worked answers when you’re ready — or print a clean copy to hand out.

Questions

Difficulty

Source

Full official exam paper

Based on the enhanced ACT Mathematics test: 45 questions, 50 minutes, 4 answer choices, calculator permitted. Covers Pre-Algebra through Trigonometry.

↓

Downloads

Take the ACT material offline. The PDF prints from your browser; the editable Word worksheet downloads instantly.

PDF · ready now

ACT notes & practice

All eight chapters plus every worked solution, formatted as a clean print document.

DOCX · editable

Practice worksheet

The 50-problem set with solutions as an editable Word document you can adapt for a quiz or drill.

Official · Past Papers

Past Papers & Official Tests

Officially released exams and mark schemes from the examining body.

ACT Free Practice Tests Official ACT full-length practice tests ACT Prep Guide PDF Official ACT PDF with full-length practice test

Worksheet Library

8 original ClipSAT worksheets with full answer keys, organized by unit — free to download and print.

Loading worksheet library…

📋 Mistake Log

📊 Your Progress

ACT Math

Test format & strategy

Numbers, Percents & Ratios

Elementary & Intermediate Algebra

Functions & Graphs

Coordinate Geometry

Plane Geometry

Trigonometry

Statistics & Probability

Matrices, Vectors & Complex Numbers

Practice set

Test generator

Downloads

ACT notes & practice

Practice worksheet

Past Papers & Official Tests

Worksheet Library

Privacy Policy

What's stored on your device

Visitor counter

Advertising & cookies

Google Forms integration

Third-party links

Contact

Terms of Service

Use of content

No guarantee of results

AI tutor

Contact

Mock Exam

⚙️ AI API Key