ACT Math Concepts
Complete Review Guide
A complete ACT Math review built from 100 key concepts across number properties, fractions, algebra, geometry, coordinate geometry, and trigonometry.
This guide organizes the key ACT Math concepts into a review page you can actually study from. The source material covers different ideas, beginning with number properties and divisibility and ending with circles, solids, and trigonometric functions. Instead of adding general outside material, this page stays anchored to the concepts, rules, and examples presented in the document.
| ACT Math Area | Core Ideas Included | Examples From the Document | Study Priority |
|---|---|---|---|
| Number Properties | Undefined values, integers, rational numbers, PEMDAS, absolute value, divisibility | Division by zero, signed numbers, consecutive integers | High |
| Fractions & Percents | Reducing fractions, operations, reciprocals, decimals, percent formulas | Mixed numbers, repeating decimals, percent increase | High |
| Ratios & Averages | Ratios, proportions, rate problems, averages, counting, probability | Average speed, missing number, counting outfits | High |
| Algebra | Expressions, factoring, solving equations, quadratics, systems, inequalities | FOIL, factoring difference of squares, quadratic formula | High |
| Coordinate Geometry | Distance, slope, intercepts, circles, parabolas, ellipses | Distance formula, slope-intercept form, circle equation | Medium |
| Geometry & Trig | Triangles, polygons, circles, solids, SOHCAHTOA, trig graphs | 30-60-90, 45-45-90, arc length, sphere volume, sin²x+cos²x=1 | High |
Number Properties and Divisibility
The opening concepts cover undefined expressions, real versus imaginary numbers, integers, rational and irrational numbers, signed number operations, order of operations, absolute value, and consecutive integers. One of the clearest takeaways is that on the ACT, undefined almost always means division by zero. The guide also emphasizes that important irrational numbers include √2, √3, and π.
Undefined
If an expression has division by zero, it is undefined. The document points to this as the main ACT meaning of undefined.
Signed numbers
For multiplication and division with positives and negatives, multiply the number parts and use a negative sign only if there are an odd number of negatives.
PEMDAS
Parentheses, exponents, multiplication and division left to right, then addition and subtraction left to right.
For the expression 9 − 2 × (5 − 3)² + 6 ÷ 3, the guide says to begin with the parentheses, then the exponent, then multiplication and division, and finally addition and subtraction.
(5 − 3) = 22² = 49 − 2 × 4 + 6 ÷ 39 − 8 + 2 = 3Final value: 3
Divisibility rules the document highlights
- A number is divisible by 2 if its last digit is even.
- A number is divisible by 4 if its last two digits form a multiple of 4.
- A number is divisible by 3 if the sum of its digits is divisible by 3.
- A number is divisible by 9 if the sum of its digits is divisible by 9.
- A number is divisible by 5 if the last digit is 5 or 0.
- A number is divisible by 10 if the last digit is 0.
The document uses prime factorization to support relative primes, least common multiple, and greatest common factor. For example, 36 = 2 × 2 × 3 × 3, and 48 = 2 × 2 × 2 × 2 × 3, so the GCF is 2 × 2 × 3 = 12.
Fractions, Decimals, and Percents
The source walks through reducing fractions, adding and subtracting fractions with common denominators, multiplying and dividing fractions, converting between mixed numbers and improper fractions, and using reciprocals. It also explains how to compare fractions either by giving them a common denominator or by converting them to decimals.
Add and subtract fractions
Find a common denominator first, then combine the numerators.
Multiply fractions
Multiply numerator by numerator and denominator by denominator.
Divide fractions
Invert the second fraction and multiply.
The guide gives the repeating decimal form of 2/27 as a repeating cluster 037. It then explains how to find a particular digit by using the length of the repeating cluster.
Because the repeating cluster has 3 digits, every 3rd digit repeats the pattern. To find the 50th digit, look for the multiple of 3 just below 50, which is 48.
The 48th digit is 7, then the pattern restarts with 0 at the 49th digit, so the 50th digit is 3.
The percent framework from the document
To increase by a percent, add that percent to 100 percent, convert to a decimal, and multiply. The document’s example increases 40 by 25 percent by using 1.25 × 40 = 50.
The guide recommends starting with 100. If a value goes up 10 percent and then up 20 percent, the document shows 100 → 110 → 132, which is a combined 32 percent increase.
Ratios, Rates, Averages, and Probability
The document explains how to set up a ratio by putting the number associated with of on top and the quantity associated with to on the bottom, then reducing. It also covers part-to-part and part-to-whole ratios, solving proportions by cross multiplication, using units to solve rate problems, and finding average rate the right way.
The document says average rate is not simply the average of two rates. You must use total A over total B. For average speed, that means total distance over total time.
The guide finds the average speed for 120 miles at 40 mph and 120 miles at 60 mph.
Total distance = 120 + 120 = 240Time for first leg = 3 hoursTime for second leg = 2 hoursTotal time = 5 hoursAverage speed = 240 / 5 = 48 mphAverage speed: 48 miles per hour
Averages and counting
Average formula
Add the numbers and divide by the number of terms.
Evenly spaced numbers
Average the smallest and largest numbers.
Missing number
Use the average to find the total sum, then subtract the known values.
Probability
Probability = favorable outcomes divided by total possible outcomes.
If one event can happen in m ways and another can happen in n ways, then the two events together can happen in m × n ways. The document’s example uses 5 shirts and 7 pairs of pants to make 35 outfits.
Powers, Roots, and Radicals
The document reviews multiplying and dividing powers with the same base, raising powers to powers, simplifying square roots, and adding, subtracting, multiplying, and dividing radicals. It emphasizes that when multiplying powers with the same base, you add exponents, and when dividing them, you subtract exponents.
You can add or subtract radicals only when the part under the radical is the same. The guide shows 2√3 + 3√3 = 5√3.
Algebraic Expressions and Factoring
The source reviews evaluating expressions, adding and subtracting monomials and polynomials, multiplying monomials, multiplying binomials with FOIL, and multiplying larger polynomials by distributing every term. Then it shifts into factoring common divisors, factoring the difference of squares, recognizing perfect square trinomials, and factoring quadratics by thinking about FOIL in reverse.
To multiply (x + 3)(x + 4), the guide uses FOIL.
First: x × x = x²Outer: x × 4 = 4xInner: 3 × x = 3xLast: 3 × 4 = 12x² + 4x + 3x + 12 = x² + 7x + 12Final expression: x² + 7x + 12
a² − b² = (a − b)(a + b)a² + 2ab + b² = (a + b)²a² − 2ab + b² = (a − b)²
Equations, Quadratics, Systems, and Inequalities
To solve a linear equation, the guide says to do whatever is necessary to both sides to isolate the variable. It also includes solving one variable in terms of another, translating English into algebra, solving quadratics by factoring or the quadratic formula, solving systems of equations by eliminating or substituting, handling absolute value equations through cases, and solving inequalities while remembering to reverse the sign when multiplying or dividing by a negative.
If a quadratic does not factor easily, the guide uses the quadratic formula with coefficients from ax² + bx + c = 0.
To solve x² + 12 = 7x, the document first rewrites the equation in standard form.
x² − 7x + 12 = 0(x − 3)(x − 4) = 0x = 3 or x = 4Solutions: 3 and 4
To solve |x − 12| = 3, the guide treats it as two cases.
x − 12 = 3x − 12 = −3x = 15 or x = 9Solutions: 15 and 9
When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Coordinate Geometry
The guide includes two ways to find distance between points, using either the Pythagorean theorem or the distance formula. It then covers slope from two points, slope from an equation written in y = mx + b form, finding x- and y-intercepts, and recognizing equations for circles, parabolas, and ellipses.
The document defines slope as change in y over change in x. It also shows how to isolate y in an equation like 3x + 2y = 4 to reveal the slope as −3/2.
Triangles and Angle Rules
The document reviews interior angles of a triangle, exterior angles of a triangle, similar triangles, area of a triangle, the Pythagorean theorem, and the special right triangles that save time on the ACT.
Interior angles
The three interior angles of a triangle add to 180 degrees.
Exterior angle rule
An exterior angle equals the sum of the two remote interior angles.
Triangle area
Area = 1/2 × base × height.
Similar triangles
Corresponding angles are equal and corresponding sides are proportional.
- 3-4-5 triangles
- 5-12-13 triangles
- 30-60-90 triangles with side ratio 1 : √3 : 2
- 45-45-90 triangles with side ratio 1 : 1 : √2
Polygons and Circles
The polygon section includes rectangles, parallelograms, squares, trapezoids, the area formulas for those figures, and the sum of interior angles of a polygon as (n − 2) × 180. The circle section includes circumference, arc length, area, and sector area.
The document uses radius 5 and central angle 72 degrees.
(72 / 360)(2π × 5)(1 / 5)(10π)2πArc length: 2π
The sum of the interior angles of an n-sided polygon is (n − 2) × 180. The document uses an octagon to show a total of 1080 degrees.
Solids and Trigonometry
For solids, the guide includes surface area and volume of rectangular solids, volume of cubes, cylinders, cones, and spheres. For trigonometry, it reviews sine, cosine, tangent, the reciprocal functions cotangent, secant, and cosecant, trig values for angles greater than 90 degrees using the unit circle, the identity sin²x + cos²x = 1, and the basic idea of graphing trig functions using key angles.
Rectangular solid
Surface area = 2lw + 2wh + 2lh. Volume = lwh.
Cube
Volume = e³.
Cylinder and cone
Cylinder volume = πr²h. Cone volume = 1/3 πr²h.
Sphere
Volume = 4/3 πr³.
The source explicitly highlights sin²x + cos²x = 1 and uses it to simplify trig expressions.
For sin 210°, the document sketches a unit circle and notes that 210 degrees lands in Quadrant III. Using the 30-60-90 reference triangle, the point coordinates are (−√3/2, −1/2).
Since sine is the y-coordinate on the unit circle, sin 210° = −1/2.
Frequently Asked Questions
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