ACT Math

Provider · ACT, Inc. Format · 60 questions · 60 min Chapters · 8 Explorers · 3 interactive

Test format & strategy

🗻 ACT Math — Official Reporting Categories (60 questions, 60 min)

Reporting Category	% of Test	Key Topics
Preparing for Higher Mathematics	57–60%	Broken into 5 sub-areas below
· Number & Quantity	7–10%	Integers, ratios, percents, complex numbers, vectors, matrices
· Algebra	12–15%	Linear/quadratic equations, systems, inequalities, absolute value
· Functions	12–15%	Definition, notation, graphs, transformations, composition, inverse
· Geometry	12–15%	Area, volume, angle relationships, trig, coordinate geometry
· Statistics & Probability	8–12%	Mean/median/mode, data interpretation, probability, expected value
Integrating Essential Skills	40–43%	Multi-step problems spanning rates, percentages, proportions, measurement, area/volume

The ACT Math test is 60 multiple-choice questions in 60 minutes — about a minute each — and a calculator is allowed throughout. Questions run roughly easy to hard, drawing on everything from pre-algebra to trigonometry.

What to expect

Five answer choices per question (not four). There is no formula sheet, so the key formulas must be memorized. The content spans number & quantity, algebra, functions, geometry, and statistics & probability.

Questions

60 · multiple choice

Time

60 minutes

Choices

5 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.

📝 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.

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

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.

📝 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 $.

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 $

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 $x$ is $|2x - 3| < 7$?

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

📝 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 $.

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} $

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 $:

\[ 9. \]

If $f(x) = x^2 - 1$ and $g(x) = 2x + 3$, find $f(g(1))$.

$g(1) = 5$. $f(5) = 25 - 1 = 24$.

📝 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 $.

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 $

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 $5$?

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

📝 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 $.

Triangle area

$ \tfrac12 bh $

Pythagoras

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

Polygon angles

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

Circle

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

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

📝 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 $.

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 $

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 $\sin A$, $\cos A$, $\tan A$.

$b = \sqrt{169-25} = 12$. $\sin A = 5/13$, $\cos A = 12/13$, $\tan A = 5/12$.

📝 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) $.

Mean

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

Median

middle value

Probability

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

Independent

$ P(A)\cdot P(B) $

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 $= 15-4 = 11$.

📝 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 $.

Imaginary

$ i^{2}=-1 $

Matrix

add entrywise

Vector

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

Logarithm

$ \log_b x $

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 $a+bi$.

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

📝 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

\[ \tfrac12\cdot\tfrac12=\tfrac14. \]

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Based on the ACT Mathematics test: 60 questions, 60 minutes, 5 answer choices, calculator permitted. Covers Pre-Algebra through Trigonometry.

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ACT Math

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