College Mathematics

About This Exam

CLEP College Mathematics covers the material of a one-semester general-education math course for non-STEM majors. It is one of the broadest — and, for most test-takers, one of the most passable — CLEP math exams, because it favors reasoning and everyday application over symbolic manipulation. An on-screen scientific calculator is available during the exam. Many schools award a full 6 credits for a passing score of 50.

Content Breakdown

• Counting, Probability & Statistics (~25%): permutations and combinations, probability rules, measures of center and spread
• Algebra and Functions (~20%): linear and quadratic equations, inequalities, function evaluation, lines, exponents
• Financial Mathematics (~20%): percents, simple and compound interest, loans, depreciation, inflation
• Logic and Sets (~15%): set operations, Venn diagrams, conditionals, negation, truth values
• Numbers (~10%): real number system, primes, GCD/LCM, scientific notation, sequences
• Geometry (~10%): perimeter, area, volume, angles, Pythagorean theorem, similar figures

Exam Tips

• Probability/statistics and financial math together are nearly half the exam — study them first and hardest.
• Memorize the small formula set in this guide; most questions are one correct formula away from solved.
• Answer everything — there is no penalty for guessing, and eliminating one or two options tilts the odds your way.
• Watch for the classic traps drilled in the practice set below: perimeter vs. area, GCD vs. LCM, permutation vs. combination, converse vs. contrapositive.

Sets and Set Operations

A set is a collection of distinct objects, called elements. Sets are written with braces: A = {1, 2, 3}. The exam tests four core operations: union (A ∪ B — elements in either set), intersection (A ∩ B — elements in both), complement (A′ — elements of the universal set not in A), and set difference (A − B — elements of A not in B).

Key facts to memorize

• Subsets: a set with n elements has exactly 2ⁿ subsets (each element is in or out). The empty set ∅ is a subset of every set.
• Inclusion–exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|. This appears constantly in survey-style word problems — adding the two group sizes double-counts the overlap.
• Venn diagrams: practice translating word problems into two- and three-circle diagrams; fill in the innermost overlap first and work outward.

Logic: Statements and Connectives

A statement is a sentence that is either true or false. Compound statements are built with connectives: and (∧, conjunction — true only when both parts are true), or (∨, disjunction — true when at least one part is true), and not (¬, negation).

Conditionals — the exam's favorite topic

The conditional p → q ("if p, then q") is false in exactly one case: when p is true and q is false. From any conditional you can form three related statements:

• Converse: q → p (swap) — not equivalent to the original
• Inverse: ¬p → ¬q (negate both) — not equivalent to the original
• Contrapositive: ¬q → ¬p (swap and negate) — always equivalent to the original

Negating statements

The negation of "all A are B" is "at least one A is not B" — never "no A are B." De Morgan's laws govern compound negations: ¬(p ∨ q) = ¬p ∧ ¬q and ¬(p ∧ q) = ¬p ∨ ¬q. Notice the connective flips each time.

The Real Number System

Know the nested families: natural numbers (1, 2, 3, …) ⊂ whole numbers (add 0) ⊂ integers (add negatives) ⊂ rational numbers (any ratio of integers — including all terminating and repeating decimals) ⊂ real numbers. The reals also contain the irrational numbers — non-repeating, non-terminating decimals such as √2, π, and e. A classic trap: 22/7 is rational (it's a fraction); it merely approximates the irrational number π.

Primes, Factors, and Multiples

A prime has exactly two divisors, 1 and itself (2 is the only even prime; 1 is not prime). Every integer greater than 1 factors uniquely into primes — the Fundamental Theorem of Arithmetic.

• GCD (greatest common divisor): the largest number dividing both. LCM (least common multiple): the smallest number both divide into.
• Shortcut: GCD × LCM = a × b for any two positive integers a and b.
• Divisibility tests: by 3 or 9 — digit sum divisible by 3 or 9; by 4 — last two digits divisible by 4; by 6 — divisible by both 2 and 3.

Scientific Notation and Number Sense

Scientific notation writes a number as m × 10^e with 1 ≤ m < 10. Moving the decimal left raises the exponent; moving it right lowers it. Also expect questions on sequences (arithmetic: constant difference; geometric: constant ratio), absolute value, and converting repeating decimals to fractions (0.333… = 1/3 via the "multiply by 10 and subtract" technique).

Equations and Inequalities

Solve linear equations by isolating the variable — undo addition/subtraction first, then multiplication/division, performing each operation on both sides. Inequalities follow the same rules with one critical exception: multiplying or dividing by a negative number reverses the inequality sign.

Quadratics

Exam quadratics nearly always factor: find two numbers whose product is the constant term and whose sum is the x-coefficient, then set each factor to zero. Useful checks: the solutions of x² + bx + c = 0 sum to −b and multiply to c. The quadratic formula x = [−b ± √(b² − 4ac)] / 2a handles anything that won't factor.

Functions

A function assigns exactly one output to each input. The notation f(x) means "the output of f at input x" — so evaluating f(−3) means substituting −3 everywhere x appears (parentheses prevent sign errors: (−3)² = 9, not −9). The domain is the set of allowed inputs (exclude division by zero and even roots of negatives); the range is the set of possible outputs. On graphs, the vertical line test identifies functions.

Linear Functions and Exponents

Lines

Slope m = (y₂ − y₁)/(x₂ − x₁) measures steepness; slope-intercept form y = mx + b reads the slope and y-intercept directly. Parallel lines share a slope; perpendicular lines have slopes that multiply to −1.

Exponent rules

• x^a · x^b = x^a+b  and  x^a/x^b = x^a−b
• (x^a)^b = x^ab
• x⁰ = 1 (x ≠ 0)  and  x^−a = 1/x^a

Counting Techniques

The multiplication principle: if one task can be done m ways and a second n ways, together they can be done m × n ways. From it come the two staple formulas:

• Permutations — order matters (officers, rankings, arrangements): P(n, r) = n!/(n−r)!
• Combinations — order doesn't matter (committees, hands of cards): C(n, r) = n!/[r!(n−r)!]

Ask one question every time: does rearranging the selection produce a genuinely different outcome? If yes, permutation; if no, combination.

Probability

For equally likely outcomes, P(event) = favorable outcomes ÷ total outcomes — always a number from 0 to 1. Core rules:

• Complement: P(not A) = 1 − P(A)
• Independent events: P(A and B) = P(A) × P(B)
• Mutually exclusive events: P(A or B) = P(A) + P(B); in general subtract the overlap: P(A) + P(B) − P(A and B)

Memorize the 36-outcome grid for two dice — sums of 7 occur 6 ways (most likely), 2 and 12 only 1 way each.

Statistics

The three measures of center: mean (sum ÷ count), median (middle value of the ordered list; average the two middle values when the count is even), mode (most frequent value). The range (max − min) and standard deviation measure spread. Outliers drag the mean but barely move the median — that's why home prices are reported as medians. For normal distributions, the empirical rule puts 68% of data within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. Finally: correlation is not causation — a reliable exam concept question.

Percents, Markups, and Discounts

Percent problems reduce to one identity: part = percent × whole. A p% discount means paying (100 − p)% of the price; a p% markup or tax means paying (100 + p)%. Successive percent changes multiply — they never simply add: a 20% rise followed by a 20% fall leaves 1.20 × 0.80 = 0.96 of the original, a net 4% loss. Percent change = (new − old)/old × 100%.

Interest

Simple interest

I = P·r·t — interest accrues only on the original principal P. The accumulated value is A = P(1 + rt).

Compound interest

A = P(1 + r/n)^nt, where n is the number of compounding periods per year. Compounding earns interest on previously earned interest, so for the same nominal rate, more frequent compounding always yields more. This is also why APY (the effective annual yield) exceeds the nominal APR whenever compounding happens more than once a year.

Loans, Depreciation, and Real Value

• Amortized loans: each fixed payment covers the month's interest first; the remainder reduces principal. Early in the loan, the balance is large, so most of the payment is interest.
• Straight-line depreciation: annual loss of value = (cost − salvage value) ÷ useful life.
• Inflation: real return ≈ nominal rate − inflation rate. If inflation outpaces your interest rate, your balance grows in dollars while shrinking in purchasing power.

Perimeter, Area, and Volume

The formulas that account for nearly every measurement question:

• Rectangle: A = lw, P = 2(l + w)  ·  Triangle: A = ½bh  ·  Circle: A = πr², C = 2πr = πd
• Rectangular box: V = lwh  ·  Cylinder: V = πr²h

Watch units: area answers are square units, volume answers cubic units — the options often differ only by that suffix.

Angles and Triangles

• Triangle angles sum to 180°; quadrilaterals to 360°; any n-gon to (n − 2) × 180°.
• Complementary angles sum to 90°; supplementary to 180°; vertical angles are equal.
• Pythagorean theorem (right triangles only): a² + b² = c². Memorize the common triples 3-4-5, 5-12-13, 8-15-17 and their multiples.
• Similar figures have equal corresponding angles and proportional sides — find the scale factor from one known pair of sides and apply it to the rest. Areas scale by the square of the factor.