Free printable and interactive math flashcards, organized by topic and grade level. Use them to drill the facts that need to be automatic, from times tables through to formulas you need on test day.

Math flashcard reference: 662 terms, formulas and test facts

Every unique card from the 72 decks above, written out in one list and grouped by topic. Terms and formulas that appear in more than one deck are shown once. Use it to look up a formula, revise without clicking through cards, or print the whole set.

Test Facts

ACT Math: calculator

A calculator is allowed on the entire Math section (check the approved-calculator list).

ACT Math: length & question count

60 multiple-choice questions in 60 minutes, about one minute per question.

ACT Math: no formula sheet

The ACT does NOT give you a formula sheet. You must memorize area, volume, quadratic, distance, trig, and all other formulas.

ACT Math: scoring

Scored on a scale of 1 to 36. It is one of four sections (English, Math, Reading, Science).

ATI TEAS 7 Math: calculator

A four-function on-screen calculator is provided for the Mathematics section, so you do not bring your own.

ATI TEAS 7 Math: content areas

Two areas: Numbers & Algebra and Measurement & Data. Numbers & Algebra carries the larger share of questions.

ATI TEAS 7 Math: length & questions

The Mathematics section has 38 questions (34 scored, 4 unscored) in about 57 minutes.

ATI TEAS 7 Math: what it covers

Whole numbers, fractions, decimals, percents, ratios and proportions, algebra, measurement and unit conversion, geometry, and data interpretation.

Calculator and format

Bring your own approved graphing calculator (most Texas Instruments graphing models are allowed); check the official approved-calculator list.

Can I retake ALEKS?

Usually yes, most schools allow several attempts (often after a 24-hour wait and some prep-module time).

Digital SAT Math: format

The SAT is a digital adaptive test. Math is two modules; the second adapts in difficulty to your first-module performance.

FSA calculator

A calculator is allowed on designated sections in grades 7 and up, and a reference sheet is provided.

FSA format

Items include multiple-choice, multi-select, and fill-in responses.

FSA math

The FSA (Florida Standards Assessment) math test is given by grade and aligned to Florida’s standards.

GED Math: calculator

An on-screen TI-30XS scientific calculator is allowed on Part 2 (questions 6 to 46). Part 1 (5 questions) is calculator-free.

GED Math: content mix

About 55% algebra (equations, expressions, functions) and 45% quantitative (number sense, ratios, geometry, data).

GED Math: length & passing score

115 minutes, 46 questions. Passing score is 145 (scale 100 to 200); 165 = College Ready.

GED Math: the formula sheet

The on-screen sheet gives area, perimeter, volume, Pythagorean, slope, distance, midpoint, and simple interest, but NOT the quadratic formula or percent change.

Georgia Milestones calculator

A calculator is allowed on designated sections, depending on grade.

Georgia Milestones format

Items include selected-response, constructed-response, and extended-response questions.

Georgia Milestones math

The Georgia Milestones math assessment is given by grade and aligned to Georgia’s standards.

Grade 3 math focus

Grade 3 math centers on multiplication and division, fractions, and area.

Grade 3 topics

Place value, multiplication & division within 100, fractions, area & perimeter, and telling time.

Grade 4 math focus

Grade 4 math focuses on multi-digit multiplication, fractions, and factors and multiples.

Grade 4 topics

Multi-digit arithmetic, equivalent fractions, decimals, factors & multiples, and angles.

Grade 5 math focus

Grade 5 math focuses on operations with fractions and decimals, and volume.

Grade 5 topics

Fraction & decimal operations, place value, volume, and the coordinate plane.

Grade 6 math focus

Grade 6 math focuses on ratios and rates, dividing fractions, and negative numbers.

Grade 6 topics

Ratios & rates, fraction division, integers, expressions & equations, and basic statistics.

Grade 7 math focus

Grade 7 math focuses on proportional relationships and operations with rational numbers.

Grade 7 topics

Proportions, percents, rational-number operations, expressions & equations, geometry, and probability.

Grade 8 math focus

Grade 8 math focuses on linear equations, functions, and the Pythagorean theorem.

Grade 8 topics

Linear equations & functions, systems, exponents, the Pythagorean theorem, and bivariate data.

HiSET Math: calculator

A calculator is allowed: an on-screen TI-30XS on the computer test, or bring your own scientific on paper.

HiSET Math: length & format

The Math subtest is about 50 multiple-choice questions in 90 minutes, roughly 1.8 minutes per question.

HiSET Math: scoring

Each subtest is scored on a 1 to 20 scale. You generally need about 8 to pass each individual subtest.

HiSET Math: the formula sheet

A formula reference page IS provided (area, volume, slope, distance, Pythagorean), but it does NOT include everything, e.g. The quadratic formula is not on it.

How is AFOQT math structured?

Arithmetic Reasoning: 25 questions in 29 minutes. Mathematics Knowledge: 25 questions in 22 minutes.

How is ASVAB math structured?

On the computer (CAT) ASVAB, Arithmetic Reasoning and Mathematics Knowledge each have about 15 questions; the paper version has 30 (AR) and 25 (MK).

How is GRE Quant scored?

On a 130 to 170 scale in 1-point increments.

How is HSPT math structured?

Mathematics: 64 questions in 45 minutes. Quantitative Skills: 52 questions in 30 minutes.

How is TABE math structured?

TABE 11/12 has a Mathematics test of roughly 40 questions, split into Computation and Applied Math.

How is the ACCUPLACER scored?

Each test is scored on a 200 to 300 scale; colleges set their own placement cutoffs.

How is the AFOQT scored?

As percentile composite scores (e.g., Quantitative and Pilot/Navigator composites), not a raw percent.

How is the GRE Quantitative Reasoning section structured?

Two sections: 12 questions in 21 minutes and 15 questions in 26 minutes (27 questions, ~47 minutes). The second section adapts to your first-section performance.

How is the HSPT scored?

As scaled scores and percentile ranks used by high schools for admissions and placement.

How is the Praxis Core Math test structured?

56 questions in 90 minutes (selected-response and numeric-entry).

How is the PSAT 10 Math section structured?

Digital and adaptive: 44 questions in two 22-question modules, 35 minutes each.

How is the PSAT 10 scored?

Math is scored 160 to 760 (total 320 to 1520).

How is the PSAT 8/9 Math section structured?

Digital and adaptive: 44 questions in two 22-question modules, 35 minutes each.

How is the PSAT 8/9 scored?

Math is scored 120 to 720 (total 240 to 1440).

How is the PSAT/NMSQT Math section structured?

Digital and adaptive: 44 questions in two 22-question modules, 35 minutes each.

How is the SHSAT Math section structured?

57 math questions (5 grid-in, 52 multiple-choice). You get 180 minutes for the whole test, about 90 minutes is suggested for math.

How is the SIFT Math Skills Test structured?

About 30 to 40 questions in 40 minutes. It is adaptive, harder questions follow correct answers.

How is the SIFT scored?

On a 20 to 80 scale; a minimum of 40 is required, and aviation programs prefer higher.

How long is it?

A 2 hour and 30 minute session of mostly multiple-choice questions.

How long is the ALEKS placement assessment?

Up to 30 questions, usually 60 to 90 minutes (most institutions give a 2 to 3 hour window). It is adaptive, so question count varies.

How many math questions are on the THEA?

About 50 multiple-choice questions. The THEA’s three sections share a 5-hour window and are not individually timed.

How many questions and how long?

About 40 selected-response questions. It is taken within the combined Core Subjects EC-6 session (about 5 hours total for all subtests).

How many questions are on CBEST Math?

50 multiple-choice questions. The whole CBEST battery allows up to 4 hours across all sections.

How many questions are on the ACCUPLACER math tests?

Each Next-Generation test (QAS and AAF) has 20 questions. The ACCUPLACER is untimed and adaptive.

How was the CHSPE math section scored?

Scaled 250 to 450; 350 was the passing score.

Is a calculator allowed on ALEKS?

ALEKS provides an on-screen calculator for the specific questions where one is needed; otherwise none.

Is a calculator allowed on GRE Quant?

Yes, an on-screen basic calculator is provided. You cannot bring your own.

Is a calculator allowed on the AFOQT?

No. Calculators are not permitted, only scratch paper.

Is a calculator allowed on the ASVAB?

No. Calculators are not permitted on any section.

Is a calculator allowed on the CBEST?

No. Calculators are not permitted.

Is a calculator allowed on the HSPT?

No. Calculators are not permitted (unless an approved accommodation applies).

Is a calculator allowed on the SHSAT?

No. Calculators and calculation aids are not permitted.

Is a calculator allowed on the SIFT?

No. Calculators are not permitted; work problems on scratch paper.

Is a calculator allowed on the TABE?

Only on the Applied Math part, a four-function calculator at Level M, a scientific calculator at Levels D and A. No calculator on Computation.

Is a calculator allowed on the THEA?

Yes, a basic, four-function non-programmable calculator for the math section, and a list of formulas is provided.

Is a calculator allowed?

An on-screen calculator appears on some questions only (a four-function/square-root tool on QAS, a graphing tool on AAF).

Is the CHSPE still offered?

No, California discontinued the CHSPE on June 30, 2023. The California Proficiency Program (CPP) replaced it using HiSET subtests. These cards still cover the same math skills.

Is the SHSAT changing?

Beginning with the 2025 to 26 cycle, NYC is moving the SHSAT to a digital, computer-adaptive format.

ISEE calculator & penalty

No calculator is allowed, and there’s no penalty for wrong answers, so never leave a question blank.

ISEE Lower Level

The Lower Level ISEE is for students applying to grades 5 to 6.

ISEE math sections

Two math sections, Quantitative Reasoning and Mathematics Achievement, cover numbers, algebra, geometry, and data.

ISEE Middle Level

The Middle Level ISEE is for students applying to grades 7 to 8.

ISEE no penalty

No calculator is allowed, and there’s no wrong-answer penalty, so never leave a blank.

ISEE Upper Level

The Upper Level ISEE is for students applying to grades 9 to 12.

PARCC calculator

Calculator use is allowed on designated sections, depending on grade.

PARCC content

Major grade-level content plus mathematical reasoning and modeling/application tasks.

PARCC math

PARCC math assesses the Common Core standards with multiple-choice, multi-select, and constructed-response items.

PERT calculator

No calculator is allowed on the PERT.

PERT content

Equations, inequalities, exponents, polynomials, and coordinate geometry.

PERT math format

Florida’s PERT math section has 30 questions and is untimed and adaptive.

PSSA calculator

A calculator is allowed on designated sections in grades 6 and up, and a formula sheet is provided in grades 7 to 8.

PSSA format

Items include multiple-choice and open-ended questions.

PSSA math

The PSSA (Pennsylvania System of School Assessment) math test is given in grades 3 to 8.

SAT Math: length & question count

About 44 math questions across two modules, 70 minutes total (about 35 minutes per module).

SAT Math: score & calculator

The Math section is scored 200 to 800. A calculator (built-in Desmos) is allowed throughout the entire math section.

SAT Math: the reference sheet

The built-in sheet gives circle, rectangle, triangle, volume formulas and special right triangles, but NOT the quadratic formula, slope, distance, or any trig.

SBAC calculator

An embedded calculator is available on some items in grades 6 and up.

SBAC format

It includes selected-response, constructed-response, and performance-task items.

Smarter Balanced math

The SBAC (Smarter Balanced) math test is computer-adaptive and aligned to the Common Core.

SSAT Elementary Level

The Elementary Level SSAT is for students applying to grades 4 to 5.

SSAT Elementary math

One quantitative section covers numbers, basic operations, geometry, and patterns.

SSAT Elementary scoring

The Elementary Level SSAT has no wrong-answer penalty.

SSAT math sections

The SSAT has two quantitative (math) sections covering number concepts, algebra, geometry, and data.

SSAT Middle Level

The Middle Level SSAT is for students applying to grades 6 to 8.

SSAT scoring

The SSAT has a guessing penalty, each wrong answer costs ¼ point, so skip questions you can’t narrow down.

SSAT Upper Level

The Upper Level SSAT is for students applying to grades 8 to 11.

STAAR calculator

A calculator is allowed on STAAR math in grades 5 and up, and a reference sheet is provided.

STAAR format

Questions are multiple-choice plus griddable (fill-in) items.

STAAR math

The STAAR (Texas) math tests are given by grade level and aligned to the TEKS standards.

TASC content

Numbers & quantity, algebra, functions, geometry, and statistics & probability.

TASC math sections

The TASC Math test has two sections, one allows a calculator and one does not, like the GED.

TASC purpose

The TASC is a high school equivalency test, an alternative to the GED and HiSET.

TSIA2 calculator

An on-screen calculator is available on the TSIA2.

TSIA2 content

Algebra, functions, geometry & measurement, and data analysis, statistics & probability.

TSIA2 math

Texas’s TSIA2 math is computer-adaptive and used for college placement.

Was a calculator allowed on the CHSPE?

No, calculators were not permitted on the CHSPE math section.

What are the subtests?

Subtest I: Algebra and Number Theory. Subtest II: Geometry, Probability and Statistics. Subtest III: Calculus and the History of Mathematics.

What are the TABE levels?

L (Literacy), E (Easy), M (Medium), D (Difficult), and A (Advanced), matched to your skill level.

What did CHSPE math cover?

Number sense and operations, patterns and algebra, statistics and probability, and geometry and measurement, 50 questions.

What do AR and MK cover?

AR is arithmetic word problems; MK covers algebra, geometry, exponents/roots, and number properties.

What does AFOQT math cover?

Arithmetic Reasoning is word problems; Mathematics Knowledge covers algebra, geometry, exponents, and number properties.

What does ALEKS cover?

A broad range from basic math through precalculus, depending on how you answer.

What does CBEST Math cover?

Three areas: Estimation/Measurement/Statistical Principles, Computation/Problem Solving, and Numerical/Graphical Relationships.

What does GRE Quant cover?

Arithmetic, algebra, geometry, and data analysis (statistics and probability). There is no trigonometry or calculus.

What does HSPT math cover?

Arithmetic, basic algebra, and geometry, plus number-series and reasoning items in Quantitative Skills.

What does it cover?

Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry & Trigonometry.

What does SHSAT math cover?

Arithmetic and number theory, algebra, geometry and coordinate geometry, probability, and statistics.

What does SIFT math cover?

Order of operations, algebra, geometry, and basic trigonometry, plus applied word problems.

What does TABE math cover?

Number sense and operations, algebra, geometry and measurement, and data analysis and statistics.

What does THEA math cover?

Fundamental mathematics, algebra, geometry, and problem solving, including ratios and proportions, measurement, and data interpretation.

What is a passing CBEST Math score?

A scaled score of 41 passes; you can pass with 37 if your three-section total is at least 123.

What is a passing score?

Most states set the Core Math passing score at 150 (scale 100 to 200).

What is a passing THEA math score?

A scaled score of 230 is the standard passing score.

What is the CSET Mathematics?

The single-subject CSET Mathematics, required to teach secondary math in California. It has three subtests.

What is the CSET Multiple Subjects Math?

The mathematics portion of CSET Multiple Subjects Subtest II, required for California multiple-subject (elementary) teachers.

What is the FTCE Elementary Math (604)?

Subtest 4 (Mathematics) of the FTCE Elementary Education K-6 (060) exam.

What is the FTCE GK Math (828)?

The mathematics subtest of the FTCE General Knowledge Test, required of new Florida teachers.

What is the FTCE Math 6-12 (026)?

The secondary mathematics content test required to teach grades 6-12 math in Florida.

What is the FTCE Middle Grades Math 5-9 (025)?

The mathematics content test required to teach middle-grades math (5-9) in Florida.

What is the GACE Elementary Education Math?

The mathematics content within GACE Elementary Education Test II (002), required for Georgia elementary teachers.

What is the GACE Mathematics test?

The secondary mathematics content test for Georgia high-school math teachers (Tests I and II, fields 022/023, combined 522).

What is the GACE Program Admission Math (211)?

Test II (Mathematics) of the GACE Program Admission Assessment, the basic-skills entry test for Georgia educator programs.

What is the ILTS Mathematics (208)?

The secondary mathematics content test for Illinois high-school math teachers.

What is the ILTS TAP Math (400)?

The mathematics subtest of the Illinois Test of Academic Proficiency, a basic-skills test for Illinois educators.

What is the MoGEA Math?

The mathematics subtest of the Missouri General Education Assessment, a basic-skills test for Missouri teacher candidates.

What is the MTEL General Curriculum Math (178)?

The mathematics subtest of the MTEL General Curriculum, required for many Massachusetts elementary licenses.

What is the MTEL Mathematics test?

The secondary mathematics content test for Massachusetts high-school math teachers (the Mathematics field, now numbered 63).

What is the MTEL Middle School Math test?

The middle-school mathematics content test for Massachusetts teachers (the Middle School Mathematics field, now numbered 65).

What is the MTTC Elementary Education Math?

The mathematics subarea of the MTTC Elementary Education (103) test, required for Michigan elementary teachers.

What is the MTTC Mathematics (Secondary) (022)?

The secondary mathematics content test for Michigan high-school math teachers.

What is the NES Essential Academic Skills Math?

The mathematics subtest (003) of the NES Essential Academic Skills, used to show basic math skills for teacher candidates in several states.

What is the NYSTCE Mathematics CST (004)?

The Content Specialty Test in Mathematics, required for New York secondary math teachers.

What is the NYSTCE Multi-Subject Math (222)?

Part Two (Mathematics) of the NYSTCE Multi-Subject: Teachers of Childhood (Grade 1-6), required for New York elementary teachers.

What is the OAE Elementary Education Math?

The mathematics content of OAE Elementary Education Subtest II (019), required for Ohio elementary teachers.

What is the OAE Mathematics (027)?

The secondary mathematics content test for Ohio high-school math teachers.

What is the ParaPro Math?

The mathematics section of the ETS ParaPro Assessment (1755), used to qualify paraprofessionals in many states.

What is the Praxis Elementary Math (5003)?

The mathematics subtest of the Praxis Elementary Education: Multiple Subjects test, used in many states.

What is the Praxis Math (5165)?

The Praxis Mathematics: Content Knowledge test, the secondary math content test used in many states.

What is the TExES EC-6 Math (902)?

The math subtest of the TExES Core Subjects EC-6 (391), required to teach early childhood through grade 6 in Texas.

What is the TExES Math 7-12 (235)?

The secondary mathematics content test required to teach grades 7-12 math in Texas. It goes through calculus.

What is the WEST-B Math?

The mathematics subtest of the Washington Educator Skills Test-Basic (WEST-B), a basic-skills test for Washington educators.

What score do you need?

A score of 250 passes; Georgia may also accept a lower single-test score if the three-test combined total is at least 750.

What’s the difference between QAS and AAF?

QAS is for non-STEM placement (arithmetic, algebra, statistics); AAF is for STEM placement (advanced algebra, functions, trig).

Why does ASVAB math matter?

AR and MK both feed your AFQT score, which determines eligibility to enlist.

Why does the PSAT/NMSQT matter?

Math is scored 160 to 760; a high Selection Index can qualify juniors for the National Merit Scholarship Program.

Algebra

Add or scale matrices

Add matrices entry by entry (same size). To scale, multiply every entry by the number.

Combine like terms?

Add/subtract terms with the same variable and exponent: 3x + 5x = 8x; 3x and 3x² are not alike.

Determinant of a 2×2 matrix

For [[a, b], [c, d]], the determinant is ad − bc.

Difference of squares?

a² − b² = (a + b)(a − b)

Equation balance

An equation stays true when the same valid operation is performed on both sides.

Equivalent expression

Equivalent expressions have the same value for every allowed input, even when they look different.

Evaluate an expression

Substitute the given value for the variable, then simplify using order of operations.

Expression vs. Equation

An expression has no equals sign (2x + 3); an equation sets two expressions equal (2x + 3 = 11).

Factor x² + bx + c

Find two numbers that ADD to b and MULTIPLY to c. X² + 5x + 6 = (x + 2)(x + 3).

Factor x² + bx + c (Reverse FOIL)

Find two numbers that ADD to b and MULTIPLY to c. X² + 5x + 6 = (x + 2)(x + 3).

FOIL (multiply binomials)

(x + a)(x + b) = x² + (a + b)x + ab. First, Outer, Inner, Last.

FOIL (multiply two binomials)?

(a+b)(c+d) = ac + ad + bc + bdFirst, Outer, Inner, Last.

Function

A rule that assigns each input x exactly one output y, written y = f(x). F(4) = 47 means the input 4 gives the output 47.

Function input and output

A function assigns exactly one output to each allowed input. Evaluate by replacing the input variable carefully.

Inequality sign flip

When you multiply or divide both sides by a negative, flip the inequality. −2x > 4 → x < −2.

Inequality symbols

< less than, > greater than, ≤ less than or equal to, ≥ greater than or equal to, ≠ not equal to.

Linear equation

ax + b = c. Undo addition/subtraction first, then multiplication/division. 2x + 3 = 11 → x = 4.

Matrix dimensions

An m × n matrix has m rows and n columns. To multiply A·B, A’s columns must equal B’s rows.

Perfect-square trinomials

a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)².

Perfect-square trinomials?

(a ± b)² = a² ± 2ab + b²

Polynomial degree

The largest exponent in the polynomial. 2x³ − 5x + 1 has degree 3. The leading term sets end behavior.

Slope as a rate of change

Slope compares vertical change with horizontal change and includes direction through its sign.

Solve a linear equation

Undo addition/subtraction first, then multiplication/division. 2x + 3 = 11 → 2x = 8 → x = 4.

Solve a one-step equation

Undo the operation with its inverse. 3x = 12 → divide by 3 → x = 4.

Solve a one-variable equation?

Isolate the variable using inverse operations, doing the same thing to both sides.

Solve a quadratic by factoring

Factor to (x + a)(x + b) = 0, then set each factor to 0. X² + 4x + 3 = 0 → x = −3 or x = −1.

Solve a system by elimination

Add or subtract the equations to cancel one variable, then solve for the other and back-substitute.

Solve a system by substitution

Solve one equation for a variable, substitute it into the other, then solve for both. If x + y = 3 and 4x − y = 2, then y = 3 − x gives x = 1, y = 2.

Solve a system of equations

Use substitution (solve one for a variable, plug in) or elimination (add equations to cancel a variable).

Solve a two-step equation

Undo addition/subtraction first, then multiplication/division. 2x + 3 = 11 → 2x = 8 → x = 4.

Solving an inequality, key rule?

Solve like an equation, but flip the inequality sign when multiplying or dividing by a negative.

Variable

A letter that stands for an unknown or unspecified number, such as x.

Vertex of a parabola

For y = ax² + bx + c, the vertex is at x = −b/(2a); plug in to get y. Vertex form is y = a(x − h)² + k.

Zero product property

If (x + a)(x + b) = 0, then x + a = 0 or x + b = 0. (x + 2)(x + 4) = 0 gives x = −2 or x = −4.

Algebra & Functions

Graph transformations

f(x) + k shifts up k; f(x − h) shifts right h; −f(x) reflects over the x-axis; a·f(x) stretches vertically by a.

Polynomial end behavior

Set by the leading term. Even degree: both ends go the same way. Odd degree: ends go opposite ways. Positive leading coefficient: right end rises.

Solving inequalities

Solve like an equation, but reverse the inequality sign when multiplying or dividing both sides by a negative number.

Analytic Geometry

Dot product

a · b = a₁b₁ + a₂b₂ = |a||b|cos θ. The vectors are perpendicular when a · b = 0.

Ellipse and hyperbola

Ellipse: (x−h)²/a² + (y−k)²/b² = 1. Hyperbola: (x−h)²/a² − (y−k)²/b² = 1, asymptote slopes ±b/a.

Parabola

y = a(x − h)² + k has vertex (h, k). Opens up if a > 0, down if a < 0.

Slope and line forms

Slope m = (y₂ − y₁)/(x₂ − x₁). Slope-intercept: y = mx + b. Point-slope: y − y₁ = m(x − x₁). Standard: Ax + By = C.

Vector magnitude

For v = ⟨x, y⟩, |v| = √(x² + y²). Add vectors componentwise.

Calculus: Derivatives

Chain rule

d/dx f(g(x)) = f′(g(x))·g′(x). Differentiate the outside, then multiply by the derivative of the inside.

Continuity

f is continuous at a if f(a) is defined, the limit exists, and lim_x→a f(x) = f(a), no breaks, holes, or jumps.

Critical points and extrema

Critical points occur where f′(x) = 0 or is undefined. F′ changing + to − marks a local maximum; − to + a local minimum.

Definition of the derivative

f′(x) = lim_h→0 [f(x + h) − f(x)] / h. It gives the slope of the tangent line and the instantaneous rate of change.

Derivatives of eˣ and ln x

d/dx eˣ = eˣ; d/dx ln x = 1/x; d/dx aˣ = aˣ·ln a.

Derivatives of trig functions

d/dx sin x = cos x; d/dx cos x = −sin x; d/dx tan x = sec²x.

Limit

lim_x→a f(x) = L means f(x) gets arbitrarily close to L as x approaches a. The limit can exist even if f(a) is undefined.

Mean value theorem

If f is continuous on [a, b] and differentiable on (a, b), some c has f′(c) = (f(b) − f(a))/(b − a).

Power rule

d/dx (xⁿ) = n·x^(n−1). Example: d/dx (x³) = 3x².

Second derivative and concavity

f″ > 0: concave up (cup); f″ < 0: concave down (cap). An inflection point is where concavity changes.

Calculus: Integrals

Area between curves

A = ∫_a^b [top(x) − bottom(x)] dx over the interval where they overlap.

Average value of a function

The average value of f on [a, b] is (1/(b − a)) ∫_a^b f(x) dx.

Common antiderivatives

∫ (1/x) dx = ln|x| + C; ∫ eˣ dx = eˣ + C; ∫ cos x dx = sin x + C; ∫ sin x dx = −cos x + C.

Definite integral

∫_a^b f(x) dx is the signed area between the curve and the x-axis from a to b.

Fundamental theorem of calculus

∫_a^b f(x) dx = F(b) − F(a), where F is an antiderivative of f. Differentiation and integration are inverse processes.

Power rule for integration

∫ xⁿ dx = x^(n+1)/(n + 1) + C, for n ≠ −1. Always add the constant C for indefinite integrals.

u-substitution

Reverse of the chain rule: let u = inside function, du = u′ dx, rewrite the integral in u, integrate, then substitute back.

Volume of revolution (disks)

Rotating about the x-axis: V = π ∫_a^b [R(x)]² dx, where R(x) is the radius at x.

Complex Numbers

Add or subtract complex numbers

(a+bi) +/- (c+di)=(a +/- c)+(b +/- d)i

What it means: Combine the real parts and the imaginary parts separately.

How to use it

1. Group the real parts a and c.
2. Group the imaginary parts b and d.
3. Write the result as real + imaginary i.

Complex conjugate and division

1a+bi=a-bia^2+b^2

What it means: Multiply numerator and denominator by the conjugate a−bi to clear i from the bottom.

How to use it

1. Find the conjugate of the denominator.
2. Multiply top and bottom by it.
3. Simplify using (a+bi)(a−bi)=a²+b².

Conjugate pair

For a polynomial with real coefficients, nonreal complex zeros occur in conjugate pairs.

Imaginary unit and powers of i

i=sqrt(-1), i^2=-1, i^3=-i, i^4=1

What it means: Powers of i cycle every four. Reduce the exponent mod 4 to simplify.

How to use it

1. Divide the exponent by 4 and keep the remainder.
2. Use i⁰=1, i¹=i, i²=−1, i³=−i.
3. Replace i² with −1 whenever it appears.

Modulus of a complex number

|a+bi|=sqrt(a^2+b^2)

What it means: The modulus is the distance from the origin to a+bi on the complex plane.

How to use it

1. Treat a and b as legs of a right triangle.
2. Square each, add, take the square root.
3. The result is the length of the vector.

Multiply complex numbers

(a+bi)(c+di)=(ac-bd)+(ad+bc)i

What it means: FOIL, then replace i² with −1 and combine like terms.

How to use it

1. Distribute every term (FOIL).
2. Replace i² with −1.
3. Combine real and imaginary parts.

Conic Sections

Circle

(x-h)^2+(y-k)^2=r^2

What it means: Center (h, k), radius r. Complete the square to reach this form.

How to use it

1. Identify the center (h, k).
2. Take the square root of the right side for r.
3. Plot the center, then points r units away.

Ellipse

(x-h)^2a^2+(y-k)^2b^2=1

What it means: Center (h, k); a and b are the horizontal and vertical radii.

How to use it

1. Find the center (h, k).
2. Take √ of each denominator for the radii.
3. The larger denominator marks the major axis.

Hyperbola

(x-h)^2a^2-(y-k)^2b^2=1

What it means: Opens left/right when x² is positive; asymptote slopes are ±b/a.

How to use it

1. Decide which variable is positive (opening direction).
2. Vertices are a units from the center.
3. Draw asymptotes with slopes ±b/a.

Parabola (conic form)

y=a(x-h)^2+k or x=a(y-k)^2+h

What it means: A parabola is a conic; vertical or horizontal depending on which variable is squared.

How to use it

1. See which variable is squared.
2. Read the vertex (h, k).
3. Use a for direction and width.

Coordinate Geometry

Distance between two points?

d = √[(x₂−x₁)² + (y₂−y₁)²]

Equation of a circle

(x − h)² + (y − k)² = r², with center (h, k) and radius r.

Equation of a circle (center (h,k), radius r)?

(x − h)² + (y − k)² = r²

Graph of a parabola

y = ax² + bx + c is a parabola: opens up if a > 0, down if a < 0. Its axis of symmetry is x = −b/(2a).

Midpoint of a segment?

(x₁+x₂2, y₁+y₂2)

Parallel vs. Perpendicular lines

Parallel lines have equal slopes. Perpendicular slopes are negative reciprocals: m₁ · m₂ = −1.

Slope of a line

m = (y₂ − y₁) / (x₂ − x₁) = rise / run.

Special slopes

A horizontal line has slope 0; a vertical line has undefined slope.

Data

Center versus spread

A measure of center describes a typical location; a measure of spread describes how far values vary around it.

Correlation is not causation

A pattern between two variables does not by itself prove that changing one variable causes the other to change.

Outlier influence

An extreme value can pull the mean strongly while often changing the median less. Choose a summary that fits the distribution.

Data & Probability

Binomial probability

P(k)=nkp^k(1-p)^ n-k

What it means: Probability of exactly k successes in n independent trials with success rate p.

How to use it

1. Identify n, k, and the success probability p.
2. Compute the combination nCk.
3. Multiply by p^k and (1−p)^n−k.

Combinations and permutations

_nC_r=n!r!(n-r)!, _nP_r=n!(n-r)!

What it means: Use combinations when order does not matter, permutations when it does.

How to use it

1. Decide whether order matters.
2. Order matters → permutation; not → combination.
3. Plug n and r into the formula.

Complement probability

P(not A)=1-P(A)

What it means: The event and its complement add to 1.

How to use it

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Correlation direction

Positive correlation rises left to right. Negative correlation falls left to right. No correlation has no clear trend.

Study move

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Counting principle

If one choice has m options and another has n, there are m × n ways together.

Find a missing value from the mean

Total = mean × number of values. Subtract the known values to find the missing one.

Independent events

P(A and B)=P(A) x P(B)

What it means: Use this when events A and B are independent, meaning one event does not affect the other.

How to use it

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Line of best fit

y=mx+b

What it means: Use a trend line to predict y from x.

How to use it

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Mean

mean=(sum of values) / (number of values)

What it means: Add all values, then divide by how many values there are.

How to use it

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Mean (average)

Mean = (sum of the values) ÷ (number of values).

Median

The middle value of a sorted list. With an even count, average the two middle values.

Median, mode, and range

Median = middle value after the data are ordered. If there are two middle values, average them. Mode = most frequent value; a data set may have one mode, more than one mode, or no mode. Range = maximum minus minimum.

Study move

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Mode

The value that appears most often in the data set.

Normal distribution (empirical rule)

68% (+/- 1), 95% (+/- 2), 99.7% (+/- 3)

What it means: In a bell curve, data clusters within 1, 2, and 3 standard deviations of the mean.

How to use it

1. Locate the mean μ at the center.
2. Count standard deviations from the mean.
3. Apply 68 to 95 to 99.7 for the interval.

Probability

P(event)=(favorable outcomes) / (total outcomes)

What it means: Use this formula when all outcomes are equally likely. Probability is between 0 and 1, inclusive.

How to use it

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

Probability of A and B

For independent events: P(A and B) = P(A) × P(B).

Range

Range = largest value − smallest value.

Residual

residual=actual y-predicted y

What it means: A positive residual means the data point is above the prediction line.

How to use it

1. Identify whether you are summarizing data, reading a graph, or counting outcomes.
2. For data, order and label values; for probability, label favorable and total outcomes.
3. Interpret the answer in words and check that it is reasonable.

z-score

z=x-

What it means: Measures how many standard deviations a value is from the mean.

How to use it

1. Subtract the mean from the value.
2. Divide by the standard deviation.
3. Positive z is above the mean; negative below.

Data & Statistics

Independent vs. Dependent variable

The independent variable is the input you change (often the x-axis). The dependent variable responds and is measured (often the y-axis).

Permutations & combinations

Order matters: ₙPᵣ = n!/(n − r)!. Order doesn’t matter: ₙCᵣ = n!/(r!(n − r)!).

Probability of an event

P = (number of desired outcomes) ÷ (number of total outcomes). Always between 0 and 1.

Reading a table or graph

Check the title, axis labels, and units first. Find the row and column (or bar) you need, then read the value where they meet.

Scatterplot & line of best fit

A line of best fit models the trend in scatterplot data; its slope estimates the rate of change.

Standard deviation

Measures how spread out data are around the mean. Larger = more spread; smaller = tightly clustered.

Sum from the average

Sum = average × number of terms. Use it to find a missing value once you know the mean.

Two-way table

A table of counts by two categories. Add the right row and column cells to find a conditional probability.

Exponential & Logarithmic

Change of base

log_b(x) = ln(x) / ln(b) = log(x) / log(b).

Compound interest

A=P(1+rn)^nt, A=Pe^rt

What it means: Discrete compounding uses n periods per year; continuous compounding uses e.

How to use it

1. Identify P, r, n, and t.
2. Plug into the matching formula.
3. Use e^rt when interest is continuous.

Definition of a logarithm

b^x=y log_b y=x

What it means: A logarithm answers: what exponent turns the base into y?

How to use it

1. Rewrite the log as an exponential equation.
2. Match base, exponent, and result.
3. Solve for the unknown.

Exponent laws

xᵃ · xᵇ = xᵃ⁺ᵇ; xᵃ ÷ xᵇ = xᵃ⁻ᵇ; (xᵃ)ᵇ = xᵃᵇ; x⁻ᵃ = 1/xᵃ; x^(1/n) = ⁿ√x; x⁰ = 1.

Exponential growth and decay

y=a(1 +/- r)^t

What it means: Use + for growth and − for decay; a is the starting amount, r the rate.

How to use it

1. Identify the starting value a.
2. Use (1+r) to grow or (1−r) to decay.
3. Raise to the time t and evaluate.

Logarithm properties

log_b(xy)=log_b x+log_b y, log_bxy=log_b x-log_b y

What it means: Products become sums, quotients become differences, powers come out front.

How to use it

1. Expand products into sums of logs.
2. Turn quotients into differences.
3. Bring exponents to the front as coefficients.

Power rule and change of base

log_b(x^n)=nlog_b x, log_b x=ln xln b

What it means: Move exponents in front and rewrite any log with natural logs for a calculator.

How to use it

1. Bring an exponent down as a coefficient.
2. To evaluate, divide ln x by ln b.
3. Keep the base positive and not equal to 1.

Solve an exponential equation

b^x=c x=log_b c=ln cln b

What it means: Take a logarithm of both sides to bring the variable down from the exponent.

How to use it

1. Isolate the exponential term.
2. Take the log of both sides.
3. Solve the resulting linear equation.

The number e

e ≈ 2.718, the base of natural logarithms. Ln x = log_e x, and d/dx eˣ = eˣ.

Exponentials and Logs

Exponential growth factor

For a growth rate r per period, multiply by 1+r each period; for decay, multiply by 1-r.

Logarithm as an exponent

The statement log base b of x equals y means exactly that b raised to y equals x.

Logarithm domain

A real logarithm requires a positive argument. Bases must be positive and cannot equal 1.

Exponents & Radicals

Fractional exponent

x^1/n=sqrt([)n]x

What it means: A denominator in the exponent becomes a root. For even roots in the real-number system, the radicand must be nonnegative.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Negative exponent

x^-n=(1) / (x^n), x != 0

What it means: A negative exponent moves the factor to the other side of the fraction. The base cannot be 0.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Power of a power

(x^a)^b=x^ab

What it means: Multiply the exponents.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Power of a product

(ab)^n=a^n b^n

What it means: Distribute the exponent to every factor inside the parentheses.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Power of a quotient

((a) / (b))^n=(a^n) / (b^n), b != 0

What it means: Distribute the exponent to numerator and denominator. The denominator cannot be 0.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Product rule

x^a x x^b=x^a+b

What it means: Same base: add exponents.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Quotient rule

(x^a) / (x^b)=x^a-b, x != 0

What it means: Same nonzero base: subtract exponents.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Rational exponent

x^m/n=sqrt([)n]x^m=(sqrt([)n]x)^m

What it means: The numerator is a power, and the denominator is a root. For even roots in the real-number system, use nonnegative radicands.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Scientific notation

a x 10^n, 1 <= |a|<10

What it means:For positive numbers, 1 <= a<10. For negative numbers, the coefficient is negative and 1 <= |a|<10.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Square root equation check

x^2=a, a >= 0 x= +/- sqrt(a)

What it means: If a > 0, there are two real solutions. If a = 0, there is one real solution. If a < 0, there is no real solution in Algebra 1’s real-number system.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Square root product rule

sqrt(ab)=sqrt(a)sqrt(b), a >= 0, b >= 0

What it means: Use this to pull perfect-square factors out of a radical when the factors are nonnegative.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Square root quotient rule

sqrt((a) / (b))=(sqrt(a)) / (sqrt(b)), a >= 0, b>0

What it means: Use when a is nonnegative and b is positive.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Zero exponent

x^0=1 (x != 0)

What it means: Any nonzero base to the zero power equals 1.

How to use it

1. Make sure the bases match before using exponent rules.
2. Apply one rule at a time and rewrite negative exponents as fractions.
3. Simplify radicals by pulling out perfect-square factors.

Exponents & Roots

Exponent

Tells how many identical factors to multiply. 2³ = 2 × 2 × 2 = 8. The base is 2.

Exponential growth / decay

y = a·bˣ. If b > 1 the quantity grows; if 0 < b < 1 it decays. A is the starting amount.

Log rules

log(MN) = log M + log N, log(M/N) = log M − log N, and log(Mᵏ) = k·log M.

Logarithm definition

log_b(x) = y means bʸ = x. Example: log₂(8) = 3 because 2³ = 8.

Negative exponent?

x⁻ⁿ = 1xⁿMove the base to flip the sign of the exponent.

Power of a power?

(x^a)^b = x^abMultiply the exponents.

Powers of a fraction between 0 and 1

If 0 < x < 1, then x² < x. Raising a proper fraction to a higher power makes it smaller.

Product of powers

xᵃ · xᵇ = xᵃ⁺ᵇ. Same base → add the exponents.

Product rule for exponents?

x^a · x^b = x^a+bSame base → add exponents.

Quotient of powers

xᵃ ÷ xᵇ = xᵃ⁻ᵇ. Same base → subtract the exponents.

Quotient rule for exponents?

x^ax^b = x^a−bSame base → subtract exponents.

Scientific notation?

Written as a × 10ⁿ, where 1 ≤ a < 10. E.g. 47,000 = 4.7 × 10⁴.

Simplify a radical

√(xy) = √x · √y. Pull out the largest perfect square: √72 = √36 · √2 = 6√2.

Simplify a square root?

√(ab) = √a · √bPull out perfect-square factors: √50 = √25·√2 = 5√2.

Square root

√x is the number r with r² = x. √36 = 6.

Zero & negative exponents

x⁰ = 1 for any nonzero x, and x⁻ᵃ = 1/xᵃ. A negative exponent means take the reciprocal.

Zero exponent & fractional exponent?

x⁰ = 1x^1/n = ⁿ√xAny nonzero base to the 0 power is 1.

Foundations

Study decision 1: Name the goal

State what the question asks for before choosing a formula or operation.

Study decision 10: Check arithmetic

Recompute key products, signs, and decimal placement independently.

Study decision 11: Check algebra

Substitute the result into the original equation or relationship.

Study decision 12: Check reasonableness

Compare the result with your estimate, units, context, and domain.

Study decision 13: Compare methods

When possible, verify with a second representation or inverse operation.

Study decision 14: Explain the relationship

Describe why the quantities are connected before relying on a memorized procedure.

Study decision 2: List what is known

Write the given values, relationships, and units so no condition is lost.

Study decision 3: Define the unknown

Use a symbol or clear phrase for the quantity you must find.

Study decision 4: Choose a representation

A diagram, table, equation, graph, or number line can make the relationship visible.

Study decision 5: Predict direction

Decide whether the answer should increase, decrease, be positive, or fall within a certain range.

Study decision 6: Estimate magnitude

Use compatible numbers or benchmarks to predict a reasonable size.

Study decision 7: Separate steps

Complete one valid transformation at a time so signs and operations remain easy to check.

Study decision 8: Preserve equality

When solving an equation, perform the same valid operation on both sides.

Study decision 9: Track restrictions

Record values excluded by denominators, radicals, logarithms, geometry, or context.

Fractions & Decimals

Add or subtract fractions?

ab ± cd = ad ± bcbdGet a common denominator, then add/subtract numerators.

Convert a fraction to a decimal?

Divide the numerator by the denominator. 34 = 3 ÷ 4 = 0.75.

Convert a fraction to a percent?

Convert to a decimal, then multiply by 100. 14 = 0.25 = 25%.

Divide fractions?

ab ÷ cd = ab × dcMultiply by the reciprocal (flip the second fraction).

Multiply fractions?

ab × cd = acbdMultiply across; simplify.

Fractions & Percents

Add or subtract fractions

Rewrite over a common denominator, add or subtract the numerators, then simplify.

Decimal

A fraction written in a special place-value form. Instead of 1/2 you can write 0.5.

Decimal → fraction

Write the digits over their place value, then simplify. 0.6 = 6/10 = 3/5.

Discount

Multiply the regular price by the discount rate to get the discount. Selling price = original price − discount.

Divide fractions

Keep the first fraction, flip the second, and multiply (multiply by the reciprocal).

Fraction → percent

Divide to get a decimal, then multiply by 100. 3/4 = 0.75 = 75%.

Mixed number

A whole number combined with a fraction, such as 2 2/3. It equals 2 + 2/3.

Mixed number → improper fraction

a b/c = (a·c + b)/c. Example: 2 3/4 = (2×4 + 3)/4 = 11/4.

Multiply fractions

Multiply the numerators together and the denominators together, then simplify. (a/b)(c/d) = ac/bd.

Percent of change

(New Value − Old Value) ÷ Old Value × 100%. Positive = increase, negative = decrease.

Percent: find the part

part = (percent ÷ 100) × whole. Example: 20% of 80 = 0.20 × 80 = 16.

Sales tax

Multiply the tax rate (as a decimal) by the taxable amount, then add it to the price. 8% tax on 50 dollars = 0.08 × 50 = 4 dollars.

Functions

Arithmetic sequence explicit formula

a_n=a_1+(n-1)d

What it means: d is the common difference. This formula assumes n = 1 gives the first term.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Arithmetic sequence recursive formula

a_n=a_n-1+d, with a given starting value such as a_1

What it means: Use the previous term plus the common difference. A recursive formula must include a starting term.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Average rate of change of f on [a, b]?

f(b) − f(a)b − aSame as slope between the two points.

Composition of functions

For (f composed with g)(x), evaluate the inside function g(x) first, then use that result as the input of f.

Direct variation

y=kx

What it means: k is the constant of variation, and the graph passes through the origin.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Domain and range

Domain = possible input values. Range = possible output values.

Study move

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Domain restriction for an inverse

A function such as a quadratic may need a restricted domain before its inverse can be a function.

Domain vs range?

Domain = all possible inputs (x). Range = all possible outputs (y).

Evaluate a function

Substitute the input for x and simplify. If f(x) = 2x + 3, then f(5) = 2(5) + 3 = 13.

Exponential decay

y=a(1-r)^x, 0<r<1

What it means: Use when the percent rate r is a decrease written as a decimal.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Exponential function

y=ab^x, a != 0, b>0, b != 1

What it means: a is the starting value and b is the growth or decay factor.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Exponential growth

y=a(1+r)^x

What it means: Use when the percent rate r is positive and written as a decimal.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Function notation

f(x) means the output of function f when the input is x. To find f(3), substitute 3 for x.

Study move

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Function notation f(x)

f(x) is the output for input x. To find f(3), substitute 3 for every x. If f(x)=2x+1, f(3)=7.

Function notation f(x)?

f(x) is the output for input x. F(3) means substitute 3 for x and evaluate.

Geometric sequence explicit formula

a_n=a_1r^n-1

What it means: r is the common ratio. This formula assumes n = 1 gives the first term.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Geometric sequence recursive formula

a_n=r x a_n-1, with a given starting value such as a_1

What it means: Multiply the previous term by the common ratio. A recursive formula must include a starting term.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Inverse variation

y=(k) / (x) or xy=k

What it means: When k > 0 and x > 0, as x increases, y decreases. The product xy = k stays constant.

How to use it

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

Inverse-function check

Functions f and g are inverses when both compositions return x on the relevant domains.

Linear vs. Exponential model

Linear changes by a constant amount each step (y = mx + b). Exponential changes by a constant factor (y = a·bˣ).

Linear vs. Quadratic function

Linear: y = mx + b, graphs as a straight line. Quadratic: y = ax² + bx + c, graphs as a parabola.

One-to-one function

A one-to-one function never assigns the same output to two different inputs; its graph passes the horizontal-line test.

Rate of change

How fast the output changes per unit of input. For a line it equals the slope, m.

Transformations of f(x)

f(x)+k shifts up k; f(x−h) shifts right h; −f(x) flips over the x-axis; a·f(x) stretches vertically.

Vertical line test

A graph is a function if every vertical line touches it at most once.

Study move

1. Identify the input and output before calculating.
2. Use a table or graph to see the pattern.
3. Check whether each input has exactly one output.

What is a function?

A rule assigning exactly one output to each input. Passes the vertical-line test.

Zeros / roots of a function

The x-values where f(x) = 0, where the graph crosses the x-axis. Found by factoring or the quadratic formula.

Functions & Transformations

Absolute value function

y=|x-h|+k

What it means: Graphs as a V with vertex (h, k); the slope is ±1 times any stretch factor.

How to use it

1. Plot the vertex (h, k).
2. Draw lines with slope +1 and −1.
3. Apply any stretch factor to the slopes.

Even and odd functions

even: f(-x)=f(x); odd: f(-x)=-f(x)

What it means: Even functions are symmetric about the y-axis; odd about the origin.

How to use it

1. Replace x with −x in the function.
2. If you get f(x), it is even.
3. If you get −f(x), it is odd.

Function composition

(f degrees g)(x)=f(g(x))

What it means: Apply g first, then feed the result into f. Order matters.

How to use it

1. Evaluate the inside function g(x).
2. Substitute that output into f.
3. Simplify the combined expression.

Inverse function

swap x and y, solve for y; f^-1 reflects over y=x

What it means: An inverse undoes a function; its graph is the mirror image across y=x.

How to use it

1. Replace f(x) with y, then swap x and y.
2. Solve the new equation for y.
3. Write y as f⁻¹(x).

Transformations of graphs

y=a f(b(x-h))+k

What it means: h shifts left/right, k up/down, a stretches/flips vertically, b horizontally.

How to use it

1. Find h and k for the translation.
2. Use a for vertical stretch or flip.
3. Apply b for horizontal stretch or flip.

Geometry

Angle relationships

Vertical angles are equal. Complementary add to 90°; supplementary add to 180°. Parallel lines cut by a transversal make equal alternate and corresponding angles.

Angles in a triangle & on a line?

Triangle interior angles sum to 180°. Angles on a straight line sum to 180°. Around a point: 360°.

Area & perimeter of a rectangle?

A = lwP = 2l + 2w

Area of a circle?

A = πr²

Area of a parallelogram & trapezoid?

Parallelogram: A = bhTrapezoid: A = 12(b₁ + b₂)h

Area of a triangle?

A = 12bhb = base, h = perpendicular height.

Area versus perimeter

Perimeter measures boundary length in linear units; area measures covered surface in square units.

Circumference of a circle?

C = 2πr = πd

Complementary vs supplementary angles?

Complementary add to 90°. Supplementary add to 180°.

Cone

Volume = ⅓πr²h. Surface area = πrs + πr² (s = slant height).

Cylinder

Volume = πr²h. Surface area = 2πrh + 2πr².

Draw and label

Sketch the figure, label known measurements and the unknown, and mark equal or perpendicular parts before selecting a formula.

Equilateral & isosceles triangles

Equilateral: 3 equal sides, all angles 60°. Isosceles: 2 equal sides and equal base angles.

Exterior angle of a triangle

An exterior angle equals the sum of the two remote (non-adjacent) interior angles.

Parallelogram area

A = b × h (base × height).

Polygon interior angles

The interior angles of an n-sided polygon add to (n − 2) × 180°.

Pyramid

Volume = ⅓ × (base area) × h.

Pythagorean theorem?

a² + b² = c²Right triangles only; c = hypotenuse.

Pythagorean triples

Whole-number right-triangle sides: 3-4-5, 5-12-13, 8-15-17, 7-24-25 (and their multiples).

Rectangle

Area = l × w. Perimeter = 2l + 2w. (Square when l = w.)

Rectangular (right) prism

Volume = l × w × h = Bh. Surface area = 2lw + 2lh + 2wh.

Rectangular prism

Volume = Bh = l × w × h. Surface area = 2lw + 2lh + 2wh = ph + 2B (p = base perimeter).

Rectangular prism (box)

Volume = l × w × h. Surface area = 2lw + 2lh + 2wh.

Scale-factor effect

If every length is multiplied by k, perimeter scales by k, area by k squared, and volume by k cubed.

Similar figures

Same shape, corresponding angles equal, and corresponding sides in proportion. A 3-4-5 triangle is similar to a 6-8-10 triangle.

Similar triangles

Same shape, proportional sides, equal angles. Set up a proportion to find a missing side.

Special right triangles

45°-45°-90°: sides x, x, x√2. 30°-60°-90°: sides x, x√3, 2x (opposite 30°, 60°, 90°).

Sphere

Volume = (4/3)πr³. Surface area = 4πr².

Square

Area = s². Perimeter = 4s.

Sum of interior angles of a polygon?

(n − 2) × 180°n = number of sides.

Triangle inequality

Each side of a triangle is shorter than the sum, and longer than the difference, of the other two.

Triangle: perimeter & angles

Perimeter = a + b + c. The interior angles always add to 180°.

Volume of a cone & sphere?

Cone: V = 13πr²hSphere: V = 43πr³

Volume of a cylinder?

V = πr²h

Volume of a pyramid?

V = 13(base area)(height)

Volume of a rectangular prism (box)?

V = lwh

Geometry & Coordinate Formulas

Circle area

A=pi r^2

What it means: Use radius, not diameter.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Circle circumference

C=2pi r=pi d

What it means: Circumference is distance around the circle.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Cylinder volume

V=pi r^2h

What it means: Area of the circular base times height.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Distance formula

d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

What it means: This is the Pythagorean theorem on the coordinate plane.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Midpoint formula

((x_1+x_2) / (2),(y_1+y_2) / (2))

What it means: Average the x-values and average the y-values.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Pythagorean theorem

a^2+b^2=c^2

What it means: Use only for right triangles. C is the hypotenuse.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Rectangle area

A=lw

What it means: Multiply length by width.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Rectangular prism volume

V=lwh

What it means: Multiply length, width, and height.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Trapezoid area

A=(1) / (2)(b_1+b_2)h

What it means: Add the two bases, multiply by height, then divide by 2.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Triangle area

A=(1) / (2) bh

What it means: Use base times height, then divide by 2.

How to use it

1. Draw and label the shape or coordinate points first.
2. Choose the formula that matches the labeled parts.
3. Check units: length, square units, or cubic units.

Graphing

Graph a line from two points

m=(y_2-y_1) / (x_2-x_1)

What it means: Plot both points, draw the line through them, then compute slope to describe the line. The slope triangle on the graph shows the rise and run visually.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph a quadratic from factored form

y=a(x-r)(x-s), a != 0

What it means: Factored form gives the x-intercepts immediately: r and s. Plot the zeros, find the midpoint for the axis of symmetry, then sketch the parabola.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph a quadratic from standard form

y=x^2-2x-3

What it means: Find the axis of symmetry, vertex, y-intercept, and zeros if possible. The graph below shows how these features shape the parabola.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph a system of equations

cases y=x+1 y=-2x+4cases

What it means: Graph both lines on the same coordinate plane. The intersection point is the solution because it makes both equations true.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph an absolute value function

y=|x+1|

What it means: The graph is a V. Find the vertex by setting the inside expression equal to zero, then draw both arms with slope 1 and -1.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph distance between two points

d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)

What it means: Draw the horizontal and vertical legs between the two points. The distance formula is the Pythagorean theorem on that right triangle.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph exponential growth and decay

Growth: y=a(1+r)^x Decay: y=a(1-r)^x, 0<r<1

What it means: Use + for growth and – for decay. The rate r must be written as a decimal. A table of values helps students sketch the curve accurately.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph inverse variation

y=(k) / (x)

What it means: The graph has two curved branches. It gets close to the axes but does not cross them when k is nonzero.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph slope-intercept form

y=mx+b

What it means: Start at the y-intercept b. From that point, use the slope m as rise over run. The graph below shows y = x + 1, so it crosses the y-axis at 1 and rises 1 for every run of 1.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Graph standard form by intercepts

Ax+By=C

What it means: Set y = 0 to find the x-intercept. Set x = 0 to find the y-intercept. Plot both intercepts, then draw the line. If both intercepts are the same point, such as (0, 0), find one additional point before drawing the line.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Read a scatter plot and trend line

y=mx+b

What it means: The line of best fit shows the overall trend. The slope describes the predicted change in y for each 1-unit increase in x.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Use the discriminant to predict the graph

D=b^2-4ac

What it means: The discriminant tells how many x-intercepts the parabola has: if D > 0, it crosses the x-axis twice; if D = 0, it touches the x-axis once; if D < 0, it has no real x-intercepts.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Use the vertical line test

one inputone output

What it means: A relation is a function only if every vertical line hits the graph at most once. If a vertical line hits twice, one x-value has two outputs.

How to use it

1. Mark the key features first: intercept, slope, vertex, boundary, or asymptote.
2. Draw the graph from those features, not from random points only.
3. Check one point or table value so the graph matches the rule.

Linear Algebra & Matrices

Determinant of a 2×2

For [[a, b],[c, d]], det = ad − bc. The matrix is invertible exactly when det ≠ 0.

Determinant of a 3×3

Expand along a row using cofactors: alternate + and − signs, each times the 2×2 minor formed by deleting that entry’s row and column.

Identity matrix

I has 1’s on the main diagonal and 0’s elsewhere. AI = IA = A.

Inverse of a 2×2

[[a,b],[c,d]]⁻¹ = (1/(ad − bc)) · [[d, −b],[−c, a]], when ad − bc ≠ 0.

Matrix addition and scalar multiples

Add matrices of the same size entry by entry. Multiply by a scalar by multiplying every entry.

Solving systems with matrices

Write Ax = b. Solve by Gaussian elimination on the augmented matrix, by x = A⁻¹b, or by Cramer’s rule using determinants.

Vectors and linear combinations

A vector has magnitude and direction. A linear combination is c₁v₁ + c₂v₂ + …; the span is all such combinations.

Linear Equations

Average rate of change

(delta y) / (delta x)=(f(b)-f(a)) / (b-a), a != b

What it means: For a linear function, this is the slope. The inputs must be different so the denominator is not 0.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Find intercepts by substitution

x-intercept: set y=0, y-intercept: set x=0

What it means: This method works for more than just lines.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Horizontal line

y=k

What it means: Horizontal lines have slope 0.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Intercepts from standard form

x-intercept=((C) / (A),0), A != 0 y-intercept=(0,(C) / (B)), B != 0

What it means: For the x-intercept, set y = 0. For the y-intercept, set x = 0.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Linear function

f(x)=mx+b

What it means: The input is x, the output is f(x), and the graph is a line.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Parallel slopes

Nonvertical parallel lines have equal slopes: m_1=m_2. Vertical lines are also parallel to each other, but their slopes are undefined.

Study move

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Parallel vs perpendicular slopes?

Parallel lines: equal slopes. Perpendicular lines: slopes are negative reciprocals (m₁·m₂ = −1).

Perpendicular slopes

For nonvertical, nonhorizontal lines, perpendicular slopes are negative reciprocals: m_1m_2=-1. A horizontal line is perpendicular to a vertical line.

Study move

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Point-slope form

y-y_1=m(x-x_1)

What it means: Use this when you know one point and the slope.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Point-slope form?

y − y₁ = m(x − x₁)

Slope between two points

m=(y_2-y_1) / (x_2-x_1), x_2 != x_1

What it means:Slope is rise over run. Use the same point order on top and bottom. If x_2=x_1, the line is vertical and the slope is undefined.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Slope between two points?

m = y₂ − y₁x₂ − x₁Rise over run.

Slope from standard form

Ax+By=C, B != 0 m=-(A) / (B)

What it means: For Ax + By = C, solve for y to see the slope. If B = 0, the line is vertical and its slope is undefined.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Slope-intercept form

y=mx+b

What it means: m is the slope and b is the y-intercept.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Slope-intercept form?

y = mx + bm = slope, b = y-intercept.

Solve a system of equations?

Use substitution or elimination. The solution (x, y) is where the two lines intersect.

Solve a two-step linear equation

ax+b=c, a != 0 x=(c-b) / (a)

What it means: Undo addition or subtraction first, then undo multiplication or division. The coefficient a cannot be 0 because division by 0 is undefined.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Standard form of a line

Ax+By=C

What it means: A, B, and C are usually integers. This form makes intercepts easy.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Standard form of a line?

Ax + By = C

Vertical line

x=k

What it means: Vertical lines have undefined slope.

How to use it

1. Identify whether the problem asks you to solve an equation, find slope, or write or graph a line.
2. Keep equation operations balanced; for graphing, label slope and intercepts before drawing.
3. Check by substituting the result into the original equation or by testing a point on the line.

Measurement

How to convert units

Multiply by a conversion factor written as a fraction equal to 1, canceling the unit you don’t want. 5 ft × (12 in / 1 ft) = 60 in.

Metric ↔ US customary

1 in ≈ 2.54 cm; 1 mile ≈ 1.61 km; 1 kg ≈ 2.2 lb; 1 L ≈ 1.06 quarts.

Metric length conversions

1 km = 1,000 m; 1 m = 100 cm; 1 cm = 10 mm.

Metric mass & volume conversions

1 kg = 1,000 g; 1 g = 1,000 mg; 1 L = 1,000 mL.

Metric prefixes

kilo = 1,000; centi = 1/100; milli = 1/1,000. So 1 km = 1,000 m and 1 cm = 0.01 m.

Time conversions

1 min = 60 s; 1 hour = 60 min; 1 day = 24 hours; 1 week = 7 days; 1 year ≈ 365 days.

US length conversions

1 ft = 12 in; 1 yd = 3 ft; 1 mile = 5,280 ft.

US volume conversions

1 cup = 8 fl oz; 1 pint = 2 cups; 1 quart = 2 pints; 1 gallon = 4 quarts.

US weight conversions

1 lb = 16 oz; 1 ton = 2,000 lb.

Measurement & Conversion

Celsius &harr; Fahrenheit?

F = 95C + 32C = 59(F − 32)

Common US &harr; metric conversions?

1 in ≈ 2.54 cm · 1 lb ≈ 0.45 kg · 1 mi ≈ 1.61 km · 1 gal ≈ 3.79 L.

Convert with a conversion factor (dimensional analysis)?

Multiply by a fraction equal to 1 so unwanted units cancel: 5 ft × 12 in1 ft = 60 in.

Convert within the metric system?

Move the decimal by powers of 10: km, hm, dam, m, dm, cm, mm. 1 m = 100 cm = 1000 mm.

Number Sense

Negative-number sense

A negative sign may describe direction, loss, or a value below zero. Interpret the context before applying sign rules.

Percent as a multiplier

An increase of r% multiplies by 1+r/100; a decrease multiplies by 1-r/100. Repeated changes use repeated multiplication.

Rate and unit rate

A rate compares quantities with different units. Divide to find the amount for one unit, then scale as needed.

Ratio as a relationship

A ratio compares quantities in a stated order. Equivalent ratios scale both terms by the same nonzero factor.

Number Sense & Proportions

Absolute value as distance

|x-a|=distance between x and a

What it means: Absolute value measures distance, so it is never negative.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Absolute value equation

|x-a|=d, d>0 x-a=d or x-a=-d

What it means: If d > 0, split into two equations. If d = 0, there is one solution. If d < 0, there is no solution.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Order of operations

Work in this order: grouping symbols, exponents, multiplication/division from left to right, then addition/subtraction from left to right.

Study move

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Percent change

percent change=(new-old) / (old) x 100%

What it means: A positive result is increase. A negative result is decrease.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Percent formula

part=(percent) / (100) x whole

What it means: Use this for percent word problems. First identify the base, or whole, before calculating.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Proportion cross products

(a) / (b)=(c) / (d) ad=bc

What it means: Use cross products to solve equivalent-ratio problems.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Simple interest

I=Prt

What it means: P is principal, r is annual rate as a decimal, and t is time in years.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Unit rate

unit rate=(amount) / (number of units)

What it means: Divide the total amount by the number of units to find the amount per 1 unit.

How to use it

1. Name the part, whole, rate, or unit before calculating.
2. Convert percents to decimals when multiplying.
3. Check whether the answer is reasonable compared with the original amount.

Number Theory

Divisibility tests

÷3 if the digit sum is divisible by 3; ÷9 if the digit sum is divisible by 9; ÷4 if the last two digits are; ÷8 if the last three digits are.

Fundamental theorem of arithmetic

Every integer greater than 1 has a unique prime factorization (apart from order). Example: 60 = 2² · 3 · 5.

GCD and LCM relationship

For positive integers a and b: GCD(a,b) · LCM(a,b) = a · b.

Imaginary unit

i = √(−1), so i² = −1. A complex number is a + bi with real part a and imaginary part b. I⁴ⁿ = 1 cycles every 4 powers.

Mathematical induction

Prove a statement for all n ≥ 1 by (1) base case n = 1 and (2) assuming it holds for n = k and proving it for n = k + 1.

Modular arithmetic

a ≡ b (mod n) means n divides (a − b); a and b have the same remainder when divided by n. Example: 17 ≡ 2 (mod 5).

Rational vs. Irrational

Rational = a ratio p/q of integers (q ≠ 0), with a terminating or repeating decimal. Irrational = nonrepeating, nonterminating: √2, π, e.

Numbers

Absolute value

A number’s distance from 0 on the number line, so it is never negative. |−22| = 22. Also: |x| < n means −n < x < n.

Absolute-value inequalities

|x| < n means −n < x < n. |x| > n means x < −n or x > n.

Coefficient

The number multiplied by a variable. In 3x, the coefficient is 3.

Complex numbers and i

i is the imaginary unit with i² = −1, so √(−1) = i. A complex number has the form a + bi.

Divisibility rules

A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, and by 3 if its digits add to a multiple of 3.

Factorial (n!)

The product of n and every counting number below it: 5! = 5×4×3×2×1 = 120. By definition 0! = 1.

Factoring a number

Breaking a number into numbers that multiply to give it. 12 = 2 × 2 × 3, so 2 and 3 are factors of 12.

Factors of a number

Numbers that multiply together to give it. 12 = 2 × 2 × 3, so 2 and 3 are factors of 12.

Greatest Common Factor (GCF)

The largest number that divides two numbers evenly. Multiply the prime factors they share. GCF(200, 60) = 2 × 2 × 5 = 20.

Imaginary unit i

i = √(−1), so i² = −1. A complex number has the form a + bi. The ACT tests basic i² = −1 simplifications.

Integers

The set …, −3, −2, −1, 0, 1, 2, 3, …, the counting numbers, their negatives, and zero.

Least Common Multiple (LCM)

The smallest number both numbers divide into. Take every prime to its highest power across the numbers.

Powers of i

i¹ = i, i² = −1, i³ = −i, i⁴ = 1, then the pattern repeats every 4 powers.

Prime number

A whole number greater than 1 whose only factors are 1 and itself: 2, 3, 5, 7, 11, 13, …

Real numbers

Every number on the number line: integers, fractions, decimals, and irrationals like √2, √3, and π.

Rounding a number

Move to the nearest place value. 64 rounded to the nearest ten is 60, because 64 is closer to 60 than to 70.

Whole numbers

The set 0, 1, 2, 3, …, zero and the counting numbers, with no negatives or fractions.

Numbers & Operations

Absolute value | x |?

The distance of x from 0 on the number line; always ≥ 0. |−7| = 7, |7| = 7.

Distributive property?

a(b + c) = ab + acMultiply the outside term by each term inside the parentheses.

LCM vs GCF?

GCF = greatest factor shared by numbers. LCM = smallest multiple shared by numbers.

Order of operations (PEMDAS)?

Parentheses, Exponents, Multiply/Divide (left→right), Add/Subtract (left→right).

Prime number?

A whole number greater than 1 with exactly two factors: 1 and itself (2, 3, 5, 7, 11, …). 2 is the only even prime.

Rules for multiplying/dividing signed numbers?

Same signs → positive. Different signs → negative. (−)(−)=+,  (−)(+)=−

What is an integer?

Any whole number and its negative, including zero: …, −2, −1, 0, 1, 2, …

Percent

Compound interest?

A = P(1 + rn)^ntn = times compounded per year, t = years.

Find the original after a percent change?

Divide by the multiplier. After a 25% increase: original = new1.25.

Percent change?

% change = new − oldold × 100Positive = increase, negative = decrease.

Percent of a number?

part = percent × wholee.g. 20% of 80 = 0.20 × 80 = 16.

Simple interest?

I = P r tP = principal, r = rate (decimal), t = time. Total = P + I.

What percent is one number of another?

percent = partwhole × 100

Polynomial Functions

End behavior

A polynomial’s leading term determines what the graph does far to the left and right.

Multiplicity and graph behavior

At a zero with odd multiplicity a polynomial crosses the x-axis; at an even multiplicity it touches and turns.

Polynomials & Factoring

Binomial theorem

(a+b)^n= sum _k=0^nnka^n-kb^k

What it means: Each term uses a binomial coefficient from Pascal’s triangle.

How to use it

1. Find the term where the exponent matches.
2. Compute the binomial coefficient nCk.
3. Multiply by the powers of a and b.

Combine like terms

ax+bx=(a+b)x

What it means: Only combine terms with the same variable part.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Degree of a polynomial

The degree is the highest exponent after the polynomial is simplified.

Study move

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Difference and sum of cubes

a^3 +/- b^3=(a +/- b)(a^2 -/+ ab+b^2)

What it means: Memorize the SOAP sign pattern: Same, Opposite, Always Positive.

How to use it

1. Write each term as a perfect cube.
2. Apply the cube pattern.
3. Check signs with SOAP.

Difference of squares

a^2-b^2=(a+b)(a-b)

What it means: Works for subtraction of two squares, not addition.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Distributive property

a(b+c)=ab+ac

What it means: Use it forward to multiply and backward to factor.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

End behavior of a polynomial

leading term a_nx^n controls the ends

What it means: Even degree: both ends match. Odd degree: ends go opposite. Sign of leading coefficient sets direction.

How to use it

1. Find the degree and leading coefficient.
2. Even degree → ends together; odd → opposite.
3. Negative leading coefficient flips the ends down.

Factor out the GCF

ab+ac=a(b+c)

What it means: Always look for a common factor first.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Factor theorem

(x-a) is a factor P(a)=0

What it means: If a is a root, then (x−a) divides the polynomial evenly.

How to use it

1. Test a candidate a by computing P(a).
2. If P(a)=0, then (x−a) is a factor.
3. Divide out the factor and repeat.

Factoring ax^2 + bx + c

ax^2+bx+c: find p and q such that p+q=b and pq=ac

What it means: After checking for a GCF, use the AC method: find two numbers with product ac and sum b, rewrite the middle term, then factor by grouping.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Factoring by grouping

ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)

What it means: Group terms so a common binomial appears.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Factoring x^2 + bx + c

x^2+bx+c=(x+m)(x+n)

What it means: Find m and n so m + n = b and mn = c.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

FOIL / binomial multiplication

(a+b)(c+d)=ac+ad+bc+bd

What it means: Multiply every term in the first binomial by every term in the second.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Monomial times polynomial

a(b+c+d)=ab+ac+ad

What it means: Distribute the monomial to every term.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Multiplicity from a graph

(x-r)^m: m even touches, m odd crosses

What it means: How a graph meets the x-axis reveals the multiplicity of each root.

How to use it

1. Find each x-intercept.
2. Crosses straight through → odd multiplicity.
3. Touches and turns → even multiplicity.

Remainder theorem

P(x) / (x-a) has remainder P(a)

What it means: Evaluate the polynomial at a to get the remainder without long division.

How to use it

1. Set the divisor x−a equal to zero to find a.
2. Substitute a into P(x).
3. The value P(a) is the remainder.

Square of a difference

(a-b)^2=a^2-2ab+b^2

What it means: Watch the sign of the middle term.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Square of a sum

(a+b)^2=a^2+2ab+b^2

What it means: The middle term is twice the product.

How to use it

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Standard form of a polynomial

Write terms from highest degree to lowest degree.

Study move

1. Look for a greatest common factor first.
2. Choose the factoring pattern that matches the expression.
3. Multiply the factors mentally to check that you get the original polynomial.

Probability

Conditional probability

P(A given B) restricts the sample space to outcomes in B, then asks what fraction also belong to A.

Probability & Statistics

Addition rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B). For mutually exclusive events, P(A ∩ B) = 0.

Correlation vs. Causation

A correlation coefficient r (from −1 to 1) measures linear association. Correlation alone does not prove that one variable causes the other.

Expected value

E(X) = Σ x·P(x): multiply each outcome by its probability and add.

Measures of center

Mean = sum ÷ count. Median = middle value (ordered). Mode = most frequent value.

Multiplication rule

P(A ∩ B) = P(A)·P(B | A). If A and B are independent, P(A ∩ B) = P(A)·P(B).

Spread: range, variance, SD

Range = max − min. Variance = average of squared deviations from the mean. Standard deviation σ = √variance.

Quadratic Functions

Completing the square

x^2+bx=(x+b2)^2-(b2)^2

What it means: Adds (b/2)² to form a perfect square, used to solve or to find the vertex.

How to use it

1. Take half of b and square it.
2. Add and subtract that value.
3. Write the perfect-square trinomial.

Sum and product of roots

x_1+x_2=-ba, x_1x_2=ca

What it means: Relates the roots of ax²+bx+c=0 to its coefficients.

How to use it

1. Identify a, b, c.
2. Sum of roots is −b/a.
3. Product of roots is c/a.

Vertex form of a parabola

y=a(x-h)^2+k

What it means: The vertex is (h, k). A controls direction and width.

How to use it

1. Read the vertex (h, k) directly.
2. Use the sign of a for opens up or down.
3. Plot the vertex, then a point on each side.

Quadratics

Axis of symmetry

x=-(b) / (2a), a != 0

What it means: This vertical line passes through the vertex.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Complete the square pattern

x^2+bx+((b) / (2))^2=(x+(b) / (2))^2

What it means: Add half of b squared to create a perfect-square trinomial.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Discriminant

D=b^2-4ac

What it means: D > 0: two real solutions. D = 0: one real solution, also called a double root. D < 0: no real solutions.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Discriminant & what it tells you?

D = b² − 4acD > 0 → 2 real roots; D = 0 → 1; D < 0 → none (real).

Factored form

y=a(x-r)(x-s), a != 0

What it means: The x-intercepts are r and s.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Minimum or maximum of a quadratic

If a>0, the vertex is a minimum. If a<0, the vertex is a maximum.

Study move

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Quadratic formula

x=(-b +/- sqrt(b^2-4ac)) / (2a), a != 0

What it means: Solves ax^2 + bx + c = 0, where a is not 0.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Quadratic formula?

x = −b ± √(b² − 4ac)2aSolves ax² + bx + c = 0.

Quadratic standard form

y=ax^2+bx+c, a != 0

What it means: The graph is a parabola. If a > 0 it opens up; if a < 0 it opens down.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Root relationships

x^2+bx+c=0: r+s=-b, rs=c

What it means: For monic quadratics, roots add to -b and multiply to c. Do not confuse the roots with the numbers in the factors: if x^2 + bx + c = (x + m)(x + n), then the roots are -m and -n.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Vertex form

y=a(x-h)^2+k, a != 0

What it means: The vertex is (h, k).

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Vertex form?

y = a(x − h)² + kVertex is (h, k).

Vertex from standard form

h=-(b) / (2a), a != 0, k=f(h)

What it means: The vertex is (h, k). Substitute h into the function to find k.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Vertex of a parabola y = ax² + bx + c?

x = −b2aPlug back in to get the y-value of the vertex.

Y-intercept of a quadratic

y=ax^2+bx+c y-intercept=(0,c)

What it means: Set x = 0 to find where the parabola crosses the y-axis.

How to use it

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Zero-product property

If ab=0, then a=0 or b=0. Use this after factoring.

Study move

1. Choose the form that reveals what you need: standard, vertex, or factored.
2. Use the graph to connect roots, vertex, axis of symmetry, and y-intercept.
3. Check solutions by substituting them back into the quadratic.

Zero-product property?

If (x − r)(x − s) = 0, then x = r or x = s. Used to solve factored quadratics.

Radicals

Extraneous solution

An algebraic step such as squaring can introduce an answer that fails the original equation, so every candidate must be checked.

Rational & Radical

Add or subtract rational expressions

ab +/- cd=ad +/- bcbd

What it means: Build a common denominator, combine numerators, then simplify.

How to use it

1. Find the least common denominator.
2. Rewrite each fraction over the LCD.
3. Combine and simplify the numerator.

Rational exponents

a^m/n=sqrt([)n]a^m=(sqrt([)n]a)^m

What it means: The denominator is the root; the numerator is the power.

How to use it

1. Read n as the index of the root.
2. Read m as the power.
3. Take the root first to keep numbers small.

Rational function asymptotes

VA: Q(x)=0; HA from degree compare

What it means: Vertical asymptotes occur where the denominator is zero (after canceling).

How to use it

1. Factor and cancel to find holes vs. Asymptotes.
2. Set the remaining denominator to 0 for VAs.
3. Compare degrees for the horizontal asymptote.

Rationalize a denominator

1sqrt(a)=sqrt(a)a

What it means: Multiply by a form of 1 to remove the radical from the denominator.

How to use it

1. Multiply top and bottom by the radical.
2. For binomials, use the conjugate.
3. Simplify the resulting expression.

Simplify a rational expression

P(x)Q(x)=factored Pfactored Q (cancel)

What it means: Factor top and bottom, cancel common factors, and state restrictions.

How to use it

1. Factor the numerator and denominator.
2. Cancel any common factors.
3. Exclude values that made the original denominator 0.

Solve a radical equation

sqrt(f(x))=g(x) f(x)=g(x)^2 (check!)

What it means: Squaring can introduce extraneous solutions, so always check answers.

How to use it

1. Isolate the radical on one side.
2. Square both sides to remove it.
3. Check each solution in the original equation.

Rational Functions

Hole versus vertical asymptote

A canceled common factor creates a hole; an uncanceled denominator zero creates a vertical asymptote.

Horizontal asymptote by degree

Compare numerator and denominator degrees: lower gives y=0, equal gives the leading-coefficient ratio, and higher needs other analysis.

Ratios & Proportions

Average speed

Average speed = total distance ÷ total time.

Convert units

Multiply by the conversion factor. 1 lb = 16 oz, so 2.75 lb = 2.75 × 16 = 44 oz.

Direct vs inverse variation?

Direct: y = kx (both rise together). Inverse: y = kx (one rises as the other falls).

Direct vs. Inverse variation

Direct: y = kx (y grows as x grows). Inverse: y = k/x (y shrinks as x grows). K is the constant.

Directly proportional

y = kx for a constant k. As x grows, y grows by the same factor. If x doubles, y doubles.

Distance / rate / time?

d = r tso r = d/t and t = d/r.

Proportion

Two equal ratios: a/b = c/d, also written a: b = c: d. Solve by cross-multiplying: a·d = b·c.

Rate

A ratio that compares two quantities with different units, such as 60 miles per hour or 3 dollars per pound.

Ratio

A comparison of two numbers by division, written 3: 5 or 3/5.

Solve a proportion?

ab = cd → ad = bcCross-multiply, then solve.

Unit rate?

unit rate = total amountnumber of unitse.g. Miles ÷ hours = miles per hour.

What is a ratio?

A comparison of two quantities, written a: b, a to b, or ab.

Reasoning

Check with an inverse operation

Undo the operation when possible: addition with subtraction, multiplication with division, and squaring with a square root plus a sign check.

Distinguish exact and approximate

Keep exact forms such as fractions and radicals during the work; round only when the question or context requires it.

Eliminate impossible choices

Reject answers with the wrong sign, scale, unit, domain, or relationship before doing detailed arithmetic.

Estimate before calculating

Predict the size and sign of an answer first. The estimate helps catch a misplaced decimal, wrong operation, or impossible result.

Explain the deciding step

After solving, state why the method fits. A short explanation reveals whether the answer came from understanding or a lucky pattern.

Find the hidden constraint

Check for limits such as positive length, nonzero denominator, integer count, defined radical, or realistic time.

Read every label

For a table, graph, diagram, or formula, identify the title, variables, units, scale, and restrictions before calculating.

Test a simple case

Use an easy valid number to check whether a rule or algebraic statement behaves the way you expect.

Translate one phrase at a time

Name the unknown, translate each relationship, and only then combine the pieces into an equation or expression.

Use units as evidence

Carry units through the work. A final unit that does not match the question often reveals an incorrect setup.

Sequences & Discrete

Arithmetic series

Sum of n terms: Sₙ = (n/2)(a₁ + aₙ) = (n/2)(2a₁ + (n − 1)d).

Combinations

Unordered selections: nCr = n! / (r!(n − r)!).

Geometric series

Finite: Sₙ = a₁(1 − rⁿ)/(1 − r). Infinite (|r| < 1): S = a₁/(1 − r).

Permutations

Ordered arrangements: nPr = n! / (n − r)!.

Sets and logic

Union A ∪ B = in either set; intersection A ∩ B = in both. Contrapositive of ‘if p then q’ is ‘if not q then not p’ (logically equivalent).

Sequences & Series

Arithmetic sequence

a_n=a_1+(n-1)d

What it means: Each term adds a constant common difference d.

How to use it

1. Find a₁ and the common difference d.
2. Substitute the term number n.
3. Evaluate to get the nth term.

Arithmetic series sum

S_n=n2(a_1+a_n)

What it means: Average the first and last term, then multiply by the number of terms.

How to use it

1. Find a₁, aₙ, and n.
2. Average the first and last terms.
3. Multiply by n to get the sum.

Finite geometric series

S_n=a_1 1-r^n1-r, r != 1

What it means: Sums the first n terms of a geometric sequence.

How to use it

1. Find a₁, r, and n.
2. Substitute into the formula.
3. Simplify carefully with the exponent n.

Geometric sequence

a_n=a_1 r^ n-1

What it means: Each term multiplies by a constant ratio r.

How to use it

1. Find a₁ and the common ratio r.
2. Raise r to the power n−1.
3. Multiply by a₁.

Infinite geometric series

S=a_11-r, |r|<1

What it means: Converges only when the ratio’s absolute value is less than 1.

How to use it

1. Check that |r| < 1.
2. Divide a₁ by (1−r).
3. If |r| ≥ 1, the series diverges.

Statistics & Probability

Mean (average)?

mean = sum of valuesnumber of values

Mean vs median, which to use?

Median resists outliers; mean is pulled toward extreme values. Skewed data → median is often more representative.

Median, mode, range?

Median = middle value (ordered). Mode = most frequent. Range = max − min.

Probability of an event?

P = favorable outcomestotal outcomesAlways between 0 and 1.

Probability of independent events (both)?

P(A and B) = P(A) × P(B)

Systems & Inequalities

Absolute value inequality: greater than

|x-a|>d, d>0 x<a-d or x>a+d

What it means: Greater than means the solution is outside the distance interval. Use the same pattern with greater-than-or-equal when the inequality includes equality.

How to use it

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Absolute value inequality: less than

|x-a|<d, d>0 a-d<x<a+d

What it means: Less than means the solution is inside the distance interval. Use the same pattern with less-than-or-equal when the inequality includes equality.

How to use it

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Compound inequality with AND

a<x <= b

What it means: AND means the overlap of the two conditions.

How to use it

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Compound inequality with OR

x<a or x>b

What it means: OR often creates two separated rays on a number line.

How to use it

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Elimination method

Add or subtract equations so one variable cancels. Multiply an equation first if needed.

Study move

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Graph a linear inequality

Use a solid boundary for <= or >=. Use a dashed boundary for < or >. Then shade the solution region.

Study move

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Inequality flip rule

When multiplying or dividing both sides by a negative number, reverse the inequality sign.

Study move

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

One, none, or infinitely many system solutions

Intersecting lines: one solution. Parallel lines: no solution. Same line: infinitely many solutions.

Study move

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Solution to a system

casesy=m_1x+b_1=m_2x+b_2cases

What it means: The solution is the point where both equations are true at the same time.

How to use it

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Substitution method

Solve one equation for one variable, substitute into the other equation, then solve.

Study move

1. Turn each sentence or equation into a boundary, line, or interval.
2. Graph or solve each part carefully, then look for the overlap or intersection.
3. Check the solution in the original statement, especially for inequalities.

Systems & Matrices

2×2 determinant

bmatrixa&b&dbmatrix=ad-bc

What it means: The determinant tells whether a matrix is invertible (nonzero).

How to use it

1. Multiply the main diagonal a·d.
2. Multiply the other diagonal b·c.
3. Subtract: ad − bc.

Matrix multiplication

(AB)_ij= sum _k A_ikB_kj

What it means: Multiply rows of A by columns of B; inner dimensions must match.

How to use it

1. Check inner dimensions agree.
2. Dot each row of A with each column of B.
3. Place each result in row i, column j.

Nonlinear system

solve by substitution: set equations equal

What it means: Solutions are the intersection points of the two graphs.

How to use it

1. Solve one equation for a variable.
2. Substitute into the other equation.
3. Solve and back-substitute for all points.

Three-variable system

eliminate one variable, reduce to 2×2

What it means: Combine equations to remove a variable, then solve the smaller system.

How to use it

1. Pair equations to cancel one variable.
2. Solve the resulting two-variable system.
3. Back-substitute to find the third variable.

Test Strategy

Keep an error log

For each miss, record the topic, the deciding idea, the error type, and one action that would prevent the same mistake.

Reread the actual question

Before choosing, identify exactly what is requested: a value, expression, reason, comparison, or best estimate.

Review guesses as well as misses

A correct guess can hide the same weakness as a missed question. Review every answer you could not fully justify.

Use a two-pass approach

Answer clear questions first, mark slower items, and return with the remaining time and a calmer view.

Verify in the original problem

Substitute or interpret the result in the original statement, in a transformed equation.

Trigonometry

30-60-90 triangle ratios?

Sides opposite 30°-60°-90° are in ratio 1: √3: 2.

45-45-90 triangle ratios?

Sides are in ratio 1: 1: √2 (legs equal; hypotenuse = leg·√2).

Amplitude & period

For y = a·sin(bx), amplitude = |a| (height) and period = 2π/b (length of one cycle).

Complementary angles

sin θ = cos(90° − θ). The sine of an angle equals the cosine of its complement.

Degrees ↔ radians

radians = degrees × π/180, and degrees = radians × 180/π. 180° = π radians.

Double-angle formulas

sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ.

Inverse trig functions

sin⁻¹, cos⁻¹, tan⁻¹ return the angle with a given ratio, on restricted ranges so each output is unique.

Law of cosines

c² = a² + b² − 2ab·cos C. Use it for two sides and the included angle, or all three sides.

Law of Cosines?

c² = a² + b² − 2ab·cos C

Law of sines

a/sin A = b/sin B = c/sin C. Use when you know two angles and a side, or two sides and a non-included angle.

Law of Sines?

asin A = bsin B = csin C

Pythagorean identity

sin²θ + cos²θ = 1. Dividing gives 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

Pythagorean identity?

sin²θ + cos²θ = 1

Radian measure

π radians = 180°. To convert: radians = degrees · π/180. Arc length s = rθ (θ in radians).

Radians ↔ degrees?

π radians = 180°Multiply by π180 to go deg→rad.

Radians and degrees

180° = π radians. To convert degrees to radians, multiply by π ÷ 180.

Reciprocal identities

csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ, tan θ = sin θ/cos θ.

Reciprocal trig functions

csc = 1/sin, sec = 1/cos, cot = 1/tan.

SOH CAH TOA

In a right triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.

SOH-CAH-TOA

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent (right triangles).

SOHCAHTOA?

sin = opphyp,  cos = adjhyp,  tan = oppadjRight triangles.

Sum and difference formulas

sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

tan in terms of sin and cos?

tan θ = sin θcos θ

Tangent in terms of sine and cosine

tan θ = sin θ ÷ cos θ.

Unit circle

Circle of radius 1; a point at angle θ is (cos θ, sin θ). Key angles 0, π/6, π/4, π/3, π/2 give the standard sine and cosine values.

Unit circle basics

On a circle of radius 1, a point at angle θ is (cos θ, sin θ). One full turn is 360° = 2π radians.

235 more terms from the exam-specific decks

Terms from the ATI TEAS, HiSET, GED, SSAT, ISEE, DAT, CLEP, FTCE, ParaPro, OAR and pre-algebra decks that are not already listed above. Duplicates across those decks have been collapsed, so each term appears once.

Test Facts

DAT QR cover

Numerical calculations, algebra, geometry, trigonometry, unit conversions, and probability/statistics, plus applied word problems.

How is ISEE Lower Level math structured?

Quantitative Reasoning: 38 questions in 35 minutes. Mathematics Achievement: 30 questions in 30 minutes.

How is ISEE Middle Level math structured?

Quantitative Reasoning: 37 questions in 35 minutes. Mathematics Achievement: 47 questions in 40 minutes.

How is ISEE Upper Level math structured?

Quantitative Reasoning: 37 questions in 35 minutes. Mathematics Achievement: 47 questions in 40 minutes.

How is the DAT Quantitative Reasoning section structured?

40 questions in 45 minutes.

How is the DAT scored?

On a 1 to 30 standard scale; competitive applicants often score around 19 to 21.

How is the ISEE scored?

Each section gets a scaled score and a 1 to 9 stanine comparing you to other applicants.

How is the OAR Math Skills section structured?

About 30 questions in 40 minutes.

How is the OAR scored?

On a 20 to 80 scaled score; many aviation programs look for 40+ or higher.

How is the SSAT Lower Level quantitative section structured?

Two quantitative sections of about 30 questions in 30 minutes each.

How is the SSAT Middle Level quantitative section structured?

Two quantitative sections totaling 50 questions (25 each), 30 minutes per section.

How is the SSAT scored?

Scaled scores plus percentile ranks comparing you with other applicants of the same grade and gender.

How is the SSAT Upper Level quantitative section structured?

Two quantitative sections totaling 50 questions (25 each), 30 minutes per section.

How long is the CLEP College Mathematics exam?

60 questions in 90 minutes (some are unscored pretest items).

How long is the FTCE GK Math subtest?

About 45 multiple-choice questions in 100 minutes.

How many math questions are on the ParaPro?

30 math questions (of 90 total). The full assessment runs about 2.5 hours.

Is a calculator allowed on CLEP College Math?

Yes, an on-screen TI-30XS MultiView scientific calculator is provided.

Is a calculator allowed on the ISEE?

No. Calculators are not permitted on either math section.

Is a calculator allowed on the OAR?

No. Calculators are not permitted; scratch paper is provided.

Is a calculator allowed on the ParaPro?

No. Calculators are not permitted.

Is a calculator allowed on the SSAT?

No. Calculators are not permitted.

it cover

Number sense, algebraic reasoning, geometry and measurement, and data analysis/probability.

OAR math cover

Arithmetic, solving for variables, fractions, exponents and roots, and geometry (angles, area, perimeter).

ParaPro math cover

Number sense, algebra basics, geometry and measurement, and data analysis, plus applying math in instructional settings.

What math does it cover?

Algebra, functions, geometry and coordinate geometry, exponents, and data analysis/probability.

What score do I need?

Passing scores are set by each state or district (commonly around 455 to 467).

What score earns credit?

The American Council on Education recommends 50; many colleges grant credit at 50.

What topics are on CLEP College Math?

Algebra and functions, counting and probability, data analysis and statistics, financial math, geometry, logic and sets, and numbers.

What’s a passing score?

FTCE subtests are reported pass/fail with a scaled passing score of 200.

What’s special about Quantitative Reasoning?

It includes quantitative-comparison questions (compare Column A and Column B) as well as word problems.

Who takes the ISEE Lower Level?

Students currently in grades 4 to 5 applying for admission to grades 5 to 6.

Who takes the ISEE Middle Level?

Students currently in grades 6 to 7 applying for admission to grades 7 to 8.

Who takes the ISEE Upper Level?

Students currently in grades 8 to 11 applying for admission to grades 9 to 12.

Who takes the SSAT Lower Level?

Younger students applying for independent-school admission in the lower grades.

Who takes the SSAT Middle Level?

Students in grades 5 to 7 applying for admission to grades 6 to 8.

Who takes the SSAT Upper Level?

Students in grades 8 to 11 applying for admission to grades 9 to 12.

Algebra

A bike rental company charges a flat fee plus an hourly cost. The function T(h) gives the total cost, in dollars, for renting a bike for h hours. What does T(4)=47 mean?<br><br>A) Renting for 47 hours costs 4 dollars.<br>B) The rental company has 4 bikes and earns 47 dollars.<br>C) The hourly cost is 47 dollars after a 4 dollar fee.<br>D) Renting a bike for 4 hours costs 47 dollars.

Answer: D) Renting a bike for 4 hours costs 47 dollars.<br><br>In function notation, the number inside the parentheses is the input and the value on the other side is the output. Here, h=4 is the number of rental hours, and T(4)=47 is the total cost for that input. So the statement means renting the bike for 4 hours costs 47 dollars.

A florist uses 8 roses per bouquet. She makes between 1 and 5 bouquets for a wedding. Let x = number of bouquets. Which set best represents the domain of the function that describes the number of roses she could use?<br><br>A) All real numbers from 0 to 40<br>B) {0, 8, 16, 24, 32, 40}<br>C) {1, 2, 3, 4, 5}<br>D) All real numbers from 1 to 5

Answer: C) {1, 2, 3, 4, 5}<br><br>The variable x represents the number of bouquets, so the domain is about possible bouquet counts, not rose counts. Since she makes between 1 and 5 bouquets and cannot make a fractional bouquet, the domain is {1,2,3,4,5}.

A garden plot has dimensions 4m+1 by 4m-1. Which expression represents the area?<br><br>A) 16m²+1<br>B) 16m²-1<br>C) 16m²+8m+1<br>D) 16m²-8m-1

Answer: B) 16m²-1<br><br>Area is length times width, so multiply (4m+1)(4m-1). These are conjugates, so the middle terms cancel. The result is (4m)²-1²=16m²-1.

A plant was 3 inches tall at the start of spring. After 5 weeks, it was 28 inches tall. If it grew the same amount each week, how many inches did it grow per week?<br><br>A) 4<br>B) 5<br>C) 6<br>D) 25

Answer: B) 5<br><br>Let n be the weekly growth in inches. The plant starts at 3 inches and grows for 5 weeks, so the model is 5n+3=28. Subtract 3 to get 5n=25, then divide by 5 to get n=5. Check: 3+5(5)=28, so the plant grew 5 inches each week.

A rectangle has area 10x²+19x+6. Which pair of binomials could represent its side lengths?<br><br>A) 5x+3 and 2x+2<br>B) 10x+3 and x+2<br>C) 5x+2 and 2x+3<br>D) 5x-2 and 2x-3

Answer: C) 5x+2 and 2x+3<br><br>Factor the area expression to find possible side lengths. The AC product is 10·6=60, and the numbers 15 and 4 multiply to 60 and add to 19. So 10x²+19x+6=10x²+15x+4x+6=5x(2x+3)+2(2x+3)=(5x+2)(2x+3).

A school orders x art kits. Each kit has 8 regular markers and 2 metallic markers. The supplier also sends 3 extra packs of 8 regular markers. The total number of markers is represented by 8(x+3)+2x. Which equivalent expression best separates the markers in the kits from the extra markers?<br><br>A) 10x+24<br>B) 8(x+3)+2x<br>C) 2(5x+12)<br>D) 8x+24

Answer: A) 10x+24<br><br>Distribute first: 8(x+3)+2x=8x+24+2x. Then combine the marker terms from the kits: 8x+2x=10x, so the expression becomes 10x+24. The expressions 8(x+3)+2x and 2(5x+12) are equivalent forms, but they do not best separate kit markers from extra markers; the expression 8x+24 is not equivalent because it leaves out the 2x metallic markers. The form 10x+24 is most useful here because it clearly shows 10 markers in each kit and 24 extra regular markers from the 3 extra packs.

A student solves x²+6x+8=0 and says the solutions are x=2 and x=4. What mistake did the student make?<br><br>A) The student forgot that (x+2)(x+4)=0 gives negative solutions.<br>B) The student factored the trinomial incorrectly.<br>C) The student should have divided by x first.<br>D) The student found only one solution.

Answer: A) The student forgot that (x+2)(x+4)=0 gives negative solutions.<br><br>The factorization x²+6x+8=(x+2)(x+4) is correct, but the zero product property means x+2=0 or x+4=0. Those equations give x=-2 and x=-4, not positive 2 and 4. The mistake is changing the signs of the solutions.

Base

A base is the number or expression that is repeatedly multiplied when using an exponent.

Binomial

A binomial is a polynomial with exactly two terms.

Constant

A constant is a fixed value that does not change.

Equation

An equation is a statement that two expressions are equal.

Expression

An expression is a number, variable, or combination of numbers and variables without an equal sign.

Factoring

Factoring rewrites a number or expression as a product of factors.

Inequality

An inequality compares values using symbols such as <, >, <=, or >=.

Like Terms

Like terms have the same variable parts raised to the same powers.

Point, slope form

(y – y_1 = m(x – x_1))

Polynomial

A polynomial is an expression made from constants, variables, and non-negative whole-number exponents using addition, subtraction, and multiplication.

Quadratic Equation

A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a is not 0.

Sarah has $200 to spend on gifts. She buys 3 gifts that cost $35 each. She wants to buy additional gifts that cost $18 each. Which inequality represents the total-cost constraint before solving for the number of additional gifts g she can buy?<br><br>A) 35 + 18g ≤ 200<br>B) 105 + 18g ≤ 200<br>C) 53g ≤ 200<br>D) 18g ≤ 200

Answer: B) 105 + 18g ≤ 200<br><br>Cost of 3 gifts: 3 × 35 = 105. Additional gifts: 18g. Total must be ≤ 200: 105 + 18g ≤ 200. Solving: 18g ≤ 95 ⇒ g ≤ 5.28. So Sarah can buy at most 5 additional gifts.

Slope

Slope measures how steep a line is: the change in y divided by the change in x.

Slope, intercept form

(y = mx + b)

Solution

A solution is a value that makes an equation or inequality true.

System of Equations

A system of equations is a set of equations solved together.

The four side lengths of a shape are 2x+3, x²-x+1, x²+4x-5, and 3x-2. What polynomial represents the perimeter?<br><br>A) 2x²+6x-3<br>B) x²+8x-3<br>C) 2x²+8x-3<br>D) 2x²+8x+7

Answer: C) 2x²+8x-3<br><br>Perimeter is the sum of all side lengths. The quadratic terms are x²+x²=2x², the linear terms are 2x-x+4x+3x=8x, and the constants are 3+1-5-2=-3. Therefore, the perimeter is 2x²+8x-3.

Which is the correct quadratic-formula setup for 5x²+2x-3=0?<br><br>A) x=2±√(64)/10<br>B) x=-2±√(-56)/10<br>C) x=-2±√(64)/10<br>D) x=-2±√(64)/5

Answer: C) x=-2±√(64)/10<br><br>For 5x²+2x-3=0, a=5, b=2, and c=-3. The quadratic formula gives x=-b±√(b²-4ac)/2a=-2±√(2²-4(5)(-3))/10. The discriminant is 64, so the correct setup is x=-2±√(64)/10.

Y-Intercept

The y-intercept is the point where a graph crosses the y-axis.

Conversions

Capacity (US customary)

1 gal = 4 qt<br>1 qt = 2 pt<br>1 pt = 2 cups = 16 fl oz

Length (US customary)

1 ft = 12 in<br>1 yd = 3 ft<br>1 mi = 5280 ft

Metric mass &amp; volume

1 kg = 1000 g<br>1 g = 1000 mg<br>1 L = 1000 mL

Temperature

(F = tfrac95 C + 32)<br>(C = tfrac59 (F – 32))

US &harr; Metric

1 in &asymp; 2.54 cm<br>1 kg &asymp; 2.2 lb<br>1 L &asymp; 1.06 qt

Weight (US customary)

1 lb = 16 oz<br>1 ton = 2000 lb

Coordinate

Distance between points

(d=sqrt{(x_2-x_1)^2+(y_2-y_1)^2})

Distance between two points

(d = sqrt{(x_2-x_1)^2 + (y_2-y_1)^2})

Midpoint

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

Midpoint of a segment

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

Coordinate Geometry

A candle is 12 inches tall at the start. After 2 hours, it is 9 inches tall. Assume it burns at a constant rate. Which point-slope equation models the candle height h after t hours using the point (2,9)?<br><br>A) h-12=-1.5(t-2)<br>B) h-9=1.5(t-2)<br>C) h+9=-1.5(t+2)<br>D) h-9=-1.5(t-2)

Answer: D) h-9=-1.5(t-2)<br><br>The height changes from 12 inches to 9 inches in 2 hours, so the rate is 9-12/2-0=-1.5 inches per hour. Using the point (2,9) in point-slope form gives h-9=-1.5(t-2). The negative slope makes sense because the candle is getting shorter.

A school sells adult tickets for 12 dollars and student tickets for 8 dollars. If x is the number of adult tickets and y is the number of student tickets, which standard form equation represents a total of 240 dollars in ticket sales?<br><br>A) 8x+12y=240<br>B) 12x+8y=20<br>C) 20x+y=240<br>D) 12x+8y=240

Answer: D) 12x+8y=240<br><br>Each adult ticket contributes 12 dollars, so adult-ticket revenue is 12x. Each student ticket contributes 8 dollars, so student-ticket revenue is 8y. The total revenue is 240 dollars, giving the standard form equation 12x+8y=240.

A theater sells adult tickets and child tickets. A group buys 10 tickets total for 104. Adult tickets cost 12 each, and child tickets cost 8 each. If a is adult tickets and c is child tickets, which ordered pair solves the system? A+c=10; 12a+8c=104<br><br>A) (6,4)<br>B) (4,6)<br>C) (5,5)<br>D) (7,3)

Answer: A) (6,4)<br><br>From a+c=10, write c=10-a. Substitute into the money equation: 12a+8(10-a)=104. This gives 12a+80-8a=104, so 4a=24 and a=6. Then c=10-6=4, so the solution is (6,4).

Adult tickets cost 10 and student tickets cost 6. A group buys 30 tickets for 220. Which ordered pair (a,s) gives the number of adult and student tickets? A+s=30; 10a+6s=220<br><br>A) (20,10)<br>B) (15,15)<br>C) (10,20)<br>D) (12,18)

Answer: C) (10,20)<br><br>Multiply a+s=30 by -6 to get -6a-6s=-180. Add this to 10a+6s=220, and the s-terms cancel: 4a=40, so a=10. Then s=30-10=20, so the ordered pair is (10,20).

Maya invests a total of $5,000 in two separate accounts. The first account earns 4% annual simple interest, and the second account earns 6% annual simple interest. After one year, the two accounts together earn $260 in interest. How much did Maya invest in each account?<br><br>A) $2,000 at 4% and $3,000 at 6%<br>B) $3,000 at 4% and $2,000 at 6%<br>C) $2,500 at 4% and $2,500 at 6%<br>D) $1,500 at 4% and $3,500 at 6%

Answer: A) $2,000 at 4% and $3,000 at 6%<br><br>Let x be the amount invested at 4% and y the amount at 6%. The total invested gives x + y = 5000. The total interest after one year gives 0.04x + 0.06y = 260. Multiply the second equation by 100: 4x + 6y = 26000. From the first, x = 5000 – y; substitute: 4(5000 – y) + 6y = 26000 ⇒ 20000 + 2y = 26000 ⇒ y = 3000. Then x = 5000 – 3000 = 2000. Therefore, Maya invested $2,000 at 4% and $3,000 at 6%.

The segment with endpoints (-5,2) and (7,2) is horizontal. Which equation represents the line perpendicular to this segment through (1,0)?<br><br>A) x=1<br>B) y=1<br>C) x=2<br>D) y=0

Answer: A) x=1<br><br>The endpoints have the same y-value, so the segment is horizontal. A line perpendicular to a horizontal segment is vertical. Through (1,0), the vertical line has equation x=1.

Data & Probability

A company surveyed 120 employees about remote work and job satisfaction. 72 employees work remotely, and 85 are satisfied. 60 employees both work remotely and are satisfied. If an employee is selected at random, what is the probability they either work remotely or are satisfied?<br><br>A) 97/120<br>B) 85/120<br>C) 72/120<br>D) 60/120

Answer: A) 97/120<br><br>Using the addition rule: Remote OR satisfied = Remote + Satisfied – Both = 72 + 85 – 60 = 97. Probability: 97/120.

A restaurant tracked lunch orders by type. Boys ordered 16 sandwiches and 14 salads. Girls ordered 12 sandwiches and 8 salads. How many students ordered sandwiches in total?<br><br>A) 26 students<br>B) 28 students<br>C) 34 students<br>D) 50 students

Answer: B) 28 students<br><br>Sandwiches: Boys (16) + Girls (12) = 28 students ordered sandwiches.

Histogram A has frequencies 2, 4, 8, 4, 2 across equal-width intervals. Histogram B has frequencies 1, 2, 4, 7, 10 across equal-width intervals from left to right. Which statement is best supported?<br><br>A) Histogram A is skewed right, and Histogram B is symmetric.<br>B) Histogram A is roughly symmetric, and Histogram B has more data in larger intervals.<br>C) Both histograms are uniform.<br>D) Histogram B has no spread.

Answer: B) Histogram A is roughly symmetric, and Histogram B has more data in larger intervals.<br><br>Histogram A rises to a middle bar and then falls in the same pattern, 2, 4, 8, 4, 2, so it is roughly symmetric. Histogram B’s frequencies increase from left to right, so more data are in the larger intervals. That makes choice B the best description.

Exponents & Roots

A deep-space station receives a signal that travels 3.2×10⁸ meters in one second. In another test, the signal travels for 4.5×10² seconds. What total distance does the signal travel, written in scientific notation?<br><br>A) 1.44×10¹⁰<br>B) 1.44×10¹¹<br>C) 1.44×10⁹<br>D) 14.4×10¹⁰

Answer: B) 1.44×10¹¹<br><br>Multiply the distance traveled each second by the number of seconds: (3.2×10⁸)(4.5×10²)=(3.2·4.5)×10⁸⁺²=14.4×10¹⁰. To write this in proper scientific notation, move the decimal one place to the left: 14.4×10¹⁰=1.44×10¹¹. Therefore, the total distance is 1.44×10¹¹ meters.

Earth’s mass is approximately 5.97 × 10²⁴ kg, and the Moon’s mass is approximately 7.35 × 10²² kg. How many times more massive is Earth than the Moon? Round your answer to the nearest whole number.<br><br>A) About 81<br>B) About 8.1<br>C) About 810<br>D) About 0.81

Answer: A) About 81<br><br>To find how many times more massive, divide Earth’s mass by the Moon’s mass: 5.97 × 10²⁴/7.35 × 10²². Separate coefficients and exponents: 5.97/7.35 × 10²⁴⁻²² = 5.97/7.35 × 10². Compute the coefficient ratio: 5.97 ÷ 7.35 ≈ 0.8122. Multiply by the power of 10: 0.8122 × 100 = 81.22. Rounding to the nearest whole number gives 81. Therefore, Earth is approximately 81 times more massive than the Moon, and the answer is A.

Order the numbers π, √(10), 3.15, 22/7 from least to greatest without using a calculator. Use the facts π ≈ 3.14159…, (227) ≈ 3.1428…, and 3² = 9 < 10 < 16 = 4².<br><br>A) π < 22/7 < 3.15 < √(10)<br>B) 3.15 < π < 22/7 < √(10)<br>C) √(10) < π < 22/7 < 3.15<br>D) π < 3.15 < 22/7 < √(10)

Answer: A) π < 227 < 3.15 < √(10)<br><br>Compare each to three decimal places: π ≈ 3.1416, 227 ≈ 3.1429 (the classical rational approximation of π, slightly larger than π), 3.15 = 3.1500 exactly, and √(10). Because 9 < 10 < 16, we have 3 < √(10) < 4; refining with 3.16² = 9.9856 and 3.17² = 10.0489, we see 3.16 < √(10) < 3.17. So the ordering is 3.1416 < 3.1429 < 3.1500 < 3.16…, i.e., π < 227 < 3.15 < √(10). Therefore, the answer is A.

The distance from Earth to the Sun is approximately 1.496 × 10⁸ kilometers. The distance from Earth to Mars is approximately 2.25 × 10⁸ kilometers. How many times farther is Mars from Earth than the Sun from Earth? Express your answer to the nearest tenth.<br><br>A) 1.2 times<br>B) 2.0 times<br>C) 1.7 times<br>D) 1.5 times

Answer: D) 1.5 times<br><br>Divide the distance to Mars by the distance to the Sun: 2.25 × 10⁸/1.496 × 10⁸ = 2.25/1.496 ≈ 1.504. Rounded to the nearest tenth: 1.5. Therefore, Mars is approximately 1.5 times farther from Earth than the Sun is from Earth.

Which list shows the numbers in order from least to greatest? √(50), 7, 22/3, π²<br><br>A) 22/3, √(50), 7, π²<br>B) 7, √(50), 22/3, π²<br>C) √(50), 7, 22/3, π²<br>D) 7, 22/3, √(50), π²

Answer: B) 7, √(50), 22/3, π²<br><br>Convert all numbers to decimal form for comparison: 7 = 7.0, √(50) ≈ 7.07, 22/3 ≈ 7.33, and π² ≈ 9.87. Ordering from least to greatest gives 7.0 < 7.07 < 7.33 < 9.87, so the correct order is 7, √(50), 22/3, π².

Factoring

FOIL

((a+b)(c+d)=ac+ad+bc+bd)

Perfect-square trinomial

((apm b)^2=a^2pm 2ab+b^2)

Foundations

Study decision 15: Identify a counterexample

To disprove an always statement, find one allowed case where it fails.

Study decision 16: Distinguish example and proof

An example illustrates a claim; a proof must justify every allowed case.

Study decision 17: Interpret zero

Decide what a zero means in context: none, a boundary, an intercept, or no change.

Study decision 18: Interpret a negative

A negative value may mean direction, debt, decrease, or an invalid contextual result.

Study decision 19: Interpret a fraction

A fraction can represent part-whole, quotient, ratio, rate, operator, or probability.

Study decision 20: Interpret a variable

A variable may represent one unknown, a changing input, or every value in a set.

Study decision 21: Use benchmark fractions

Compare with 0, one-half, and 1 to estimate and order fractions.

Study decision 22: Use place value

Each move one place left multiplies by 10; each move right divides by 10.

Study decision 23: Use proportional reasoning

Equivalent ratios preserve a multiplicative relationship, not a fixed additive difference.

Study decision 24: Use a unit rate

Find the amount for one unit, then multiply by the requested number of units.

Study decision 25: Separate percent points

A change from 40% to 50% is 10 percentage points but a 25% relative increase.

Study decision 26: Read graph scale

Unequal intervals or a truncated axis can make a visual change look larger or smaller.

Study decision 27: Read table structure

Identify row and column labels before selecting or comparing values.

Study decision 28: Read diagram marks

Tick marks, arrows, right-angle boxes, and labels carry mathematical information.

Study decision 29: Choose a formula

Match the formula’s quantities and assumptions to the situation before substituting.

Study decision 30: Rearrange before substituting

Solving a formula symbolically first can reduce arithmetic errors and clarify units.

Study decision 31: Keep full precision

Delay rounding until the final step unless intermediate rounding is explicitly required.

Study decision 32: State a conclusion

Answer in a complete phrase with the correct unit and contextual meaning.

Study decision 33: Review the distractors

Understand why each tempting wrong choice fails so the misconception does not return.

Study decision 34: Classify the error

Label a miss as concept, setup, calculation, reading, or time-management error.

Study decision 35: Build a repair card

Turn a missed idea into a short prompt that asks for the deciding relationship.

Study decision 36: Space the review

Revisit difficult cards after increasing delays instead of repeating them only in one sitting.

Study decision 37: Mix related skills

Interleaving forces you to choose the method instead of following a repeated pattern.

Study decision 38: Retrieve before revealing

Say or write the answer before flipping the card; recognition alone is weaker practice.

Study decision 39: Mark confidence

Separate known, uncertain, and guessed answers so review targets actual weakness.

Study decision 40: Retest with new numbers

Transfer is stronger evidence of learning than remembering an old answer.

Study decision 41: Use a worked example

Cover the solution, predict each next step, then compare the reasoning.

Study decision 42: Connect words and symbols

Explain every symbol in ordinary language and translate the verbal relationship back into notation.

Study decision 43: Check boundary cases

Test values near a restriction or endpoint to understand the rule’s limits.

Study decision 44: Check symmetry

Look for balanced structure that can simplify algebra, geometry, graphs, or probability.

Study decision 45: Organize cases

When outcomes differ by condition, list mutually exclusive cases and handle each once.

Study decision 46: Avoid hidden assumptions

Use only information stated or logically guaranteed by definitions and markings.

Study decision 47: Use calculator judgment

Estimate first, enter grouping carefully, and interpret the displayed result.

Study decision 48: Budget time

Set a reasonable checkpoint, move past one stubborn item, and return after collecting easier points.

Study decision 49: Reset after a miss

Take one slow breath, restate the task, and begin from verified information.

Study decision 50: Teach it aloud

If you can explain the idea clearly without notes, you are more likely to retrieve it under pressure.

Fractions & Percents

A baker has 51/2 cups of flour. Each batch of muffins uses 3/4 cup. What is the greatest number of complete batches the baker can make, and how much flour will be left?<br><br>A) 7 batches with 1/4 cup left<br>B) 7 batches with 1/2 cup left<br>C) 8 batches with no flour left<br>D) 6 batches with 1 cup left

Answer: A) 7 batches with 1/4 cup left<br><br>Divide to estimate the number of batches: 512÷34=11/2·43=44/6=713, so only 7 complete batches can be made. Seven batches use 7·34=21/4=514 cups, leaving 512-514=14 cup.

A food truck bought 18.5 pounds of rice. It used 2/5 of the rice for lunch and then used the remaining rice for dinner bowls. Each dinner bowl needs 0.75 pound of rice and sells for $8.25. What is the greatest possible revenue from complete dinner bowls?<br><br>A) $99.00<br>B) $115.50<br>C) $122.10<br>D) $152.63

Answer: B) $115.50<br><br>The truck used 25 of 18.5, so 35 remained for dinner: 35·18.5=11.1 pounds. Since 11.1÷0.75=14.8, the truck can make 14 complete dinner bowls; the greatest possible revenue is 14·8.25=115.50 dollars.

A student tries to multiply 21/2×11/3 by multiplying the whole numbers and then multiplying the fractional parts. The student gets 21/6. Which statement correctly identifies the mistake and the product?<br><br>A) The mixed numbers should be converted to improper fractions first; the product is 31/3.<br>B) The student is correct because 2·1=2 and 12·13=16.<br>C) The product should be 10/6 because only the numerators need to be multiplied.<br>D) The product must be less than 212 because 113 is a fraction.

Answer: A) The mixed numbers should be converted to improper fractions first; the product is 31/3.<br><br>Mixed numbers should be converted to improper fractions before multiplying: 212=52 and 113=43. Then 52·43=20/6=10/3=313, so the student’s method missed part of the multiplication.

Mia and Noah each have a budget of $25.00 for supplies. Mia buys items costing $6.85, $9.47, and $3.29. Noah buys items costing $8.60 and $12.45. Which statement correctly compares the amounts they have left?<br><br>A) Mia has $1.44 more left than Noah.<br>B) Noah has $1.44 more left than Mia.<br>C) Mia has $5.39 more left than Noah.<br>D) They have the same amount left.

Answer: A) Mia has $1.44 more left than Noah.<br><br>Mia spent 6.85+9.47+3.29=19.61, so she has 25.00-19.61=5.39 dollars left. Noah spent 8.60+12.45=21.05, so he has 25.00-21.05=3.95 dollars left; 5.39-3.95=1.44, so Mia has $1.44 more left than Noah.

Sarah deposits $1,500 in an account earning 6% simple interest per year. If she wants to earn $360 in interest, how many years must she leave the money in the account?<br><br>A) 2 years<br>B) 3 years<br>C) 4 years<br>D) 5 years

Answer: C) 4 years<br><br>Using I = P × r × t: 360 = 1500 × 0.06 × t. Solve: 360 = 90t, so t = 4 years.

Store A sells 2.4 pounds of rice for $6.60. Store B sells 3.5 pounds of rice for $9.45. Which store has the lower price per pound, and by how much?<br><br>A) Store A, by $0.05 per pound<br>B) Store B, by $0.05 per pound<br>C) Store A, by $0.10 per pound<br>D) Store B, by $0.10 per pound

Answer: B) Store B, by $0.05 per pound<br><br>Find each unit price by dividing cost by pounds. Store A costs 6.60÷2.4=2.75 dollars per pound, and Store B costs 9.45÷3.5=2.70 dollars per pound, so Store B is cheaper by 2.75-2.70=0.05 dollar per pound.

Two accounts each start with $1,000 and earn 5% interest each year for 2 years. Account A earns simple interest. Account B earns interest compounded once per year. How much more money does Account B have after 2 years?<br><br>A) $0.00<br>B) $2.50<br>C) $50.00<br>D) $102.50

Answer: B) $2.50<br><br>With simple interest, the balance is 1000+2(0.05·1000)=1100 dollars. With compound interest, the balance is 1000(1.05)²=1102.50 dollars. That small extra amount is the interest earned on the first year’s interest, so Account B has 1102.50-1100=2.50 dollars more.

Two stores sell identical shirts originally priced at $40. Store X marks down the shirt by 30%. Store Y marks down by 20% and then marks down the sale price by an additional 10%. Which store offers the better deal, and by how much?<br><br>A) Store X; saves $0.80<br>B) Store Y; saves $0.80<br>C) Both cost the same<br>D) Store X; saves $2.80

Answer: A) Store X; saves $0.80<br><br>Store X (single 30% markdown): Discount: 0.30 × 40 = $12. Final price: 40 – 12 = $28. Store Y (20% then 10% on sale): First markdown: 0.20 × 40 = $8. Price: 40 – 8 = $32. Second markdown: 0.10 × 32 = $3.20. Final price: 32 – 3.20 = $28.80. Comparison: Store X final price is $28 and Store Y final price is $28.80. Store X offers the better deal, saving $0.80 compared to Store Y. Therefore, the answer is A.

Functions

a function

A rule assigning exactly one output to each input. Passes the vertical-line test.

Dependent Variable

A dependent variable is the output value that changes based on the input.

Graph

A graph is a visual display of numbers, relationships, or data.

Independent Variable

An independent variable is an input value that can be chosen or changed.

Input

An input is the value put into a function or rule.

Output

An output is the value produced by a function or rule.

Geometry

A cylindrical fuel tank has a radius of 2 feet and a height of 5 feet. Diesel fuel costs $3 per cubic foot. To the nearest dollar, what is the total cost to fill the tank completely with diesel? (Use π ≈ 3.14.)<br><br>A) $94<br>B) $120<br>C) $188<br>D) $282

Answer: C) $188<br><br>Step 1: compute the cylinder volume. V = π r² h = π (2)² (5) = 20π cubic feet, so V ≈ 20 × 3.14 = 62.8 cubic feet. Step 2: multiply by the price per cubic foot. Total cost = 62.8 × $3 = $188.40. Rounded to the nearest dollar, that is $188. Therefore, it costs about $188 to fill the tank.

A ramp is shaped like a triangular prism. Its triangular cross-section has base 10 ft and height 6 ft, and the ramp is 9 ft long. A student says the volume is 10·6·9=540 ft³. What mistake did the student make?<br><br>A) The student forgot to divide by 2 when finding the triangular base area.<br>B) The student used the wrong unit for length.<br>C) The student should have added the three dimensions instead of multiplying.<br>D) The student found surface area instead of volume.

Answer: A) The student forgot to divide by 2 when finding the triangular base area.<br><br>The cross-section is a triangle, so its area is 12(10)(6)=30 ft², not 10·6=60 ft². The correct volume is 30·9=270 ft³. The student doubled the volume by treating the triangular base like a rectangle.

A rectangular swimming pool is surrounded by a wooden deck. The pool is 25 feet long and 15 feet wide. The deck is 2 feet wide on all sides. What is the perimeter of the outer edge of the deck?<br><br>A) 90 feet<br>B) 92 feet<br>C) 96 feet<br>D) 98 feet

Answer: C) 96 feet<br><br>Add 2 feet on each side: outer length = 25 + 2(2) = 29 ft, outer width = 15 + 2(2) = 19 ft. Perimeter = 2(29+19) = 96 feet.

A student is finding the exterior angle of a triangle. The remote interior angles measure 38° and 74°. The student writes 180-(38+74)=68 and says the exterior angle is 68°. Which statement best explains the student’s mistake?<br><br>A) The student added the two remote interior angles instead of subtracting them.<br>B) The student found the adjacent interior angle, not the exterior angle.<br>C) The student should have divided 180° by 3.<br>D) The student used the two angles that are adjacent to the exterior angle.

Answer: B) The student found the adjacent interior angle, not the exterior angle.<br><br>The exterior angle equals the sum of the two remote interior angles: 38+74=112°. The calculation 180-(38+74)=68° finds the third interior angle of the triangle, which is adjacent to the exterior angle. The student’s arithmetic is useful, but it answers a different question.

A triangular prism has triangular bases with area 54 cm² and side lengths 9 cm, 12 cm, and 15 cm. The prism length is 6 cm. A student calculates 54+(9+12+15)(6)=270 and says the surface area is 270 cm². What mistake did the student make?<br><br>A) The student included only one triangular base instead of two.<br>B) The student multiplied the perimeter by the wrong prism length.<br>C) The student should have used volume instead of surface area.<br>D) The student added the side lengths incorrectly.

Answer: A) The student included only one triangular base instead of two.<br><br>The lateral area part, (9+12+15)(6), is correct because the perimeter is 36 cm and the prism length is 6 cm. The mistake is that a triangular prism has two congruent triangular bases. The correct calculation is 2(54)+36(6)=108+216=324 cm².

A walking path on a city grid goes from the school at (2, -1) directly to the library at (11, 11). Each grid unit represents 100 meters. To the nearest meter, how long is the straight-line walk from the school to the library?<br><br>A) 1,300 meters<br>B) 1,400 meters<br>C) 1,700 meters<br>D) 1,500 meters

Answer: D) 1,500 meters<br><br>First find the distance in grid units: d = √((11 – 2)² + (11 – (-1))²) = √(9² + 12²) = √(81 + 144) = √(225) = 15 units. Then convert: each unit is 100 meters, so the walk is 15 × 100 = 1500 meters. Therefore, the straight-line walk is 1,500 meters.

Angle

An angle is formed by two rays that share the same endpoint.

Area

Area measures the amount of flat space inside a two-dimensional shape.

Area of a circle

(A = pi r^2)

Area of a parallelogram

(A = b h)

Area of a rectangle

(A = l times w)

Area of a trapezoid

(A = tfrac12 (b_1 + b_2) h)

Area of a triangle

(A = tfrac12 b h)

Circumference

Circumference is the distance around a circle.

Circumference of a circle

(C = 2pi r)

Congruent Figures

Congruent figures have the same shape and the same size.

Coordinate Plane

The coordinate plane is a flat grid formed by a horizontal x-axis and vertical y-axis.

Diameter

The diameter is a line segment that passes through the center of a circle and touches both sides.

Hypotenuse

The hypotenuse is the longest side of a right triangle.

Ordered Pair

An ordered pair is two numbers written to locate a point on the coordinate plane.

Parallel Lines

Parallel lines are lines in the same plane that never intersect.

Perimeter

Perimeter is the total distance around a two-dimensional shape.

Perpendicular Lines

Perpendicular lines intersect to form right angles.

Radius

The radius is a line segment from the center of a circle to any point on the circle.

Surface Area

Surface area is the total area of all outside faces of a three-dimensional figure.

The exterior angle of a triangle measures (6x – 5)°. The two non-adjacent interior angles measure (3x + 10)° and (2x – 5)°. What is the value of x?<br><br>A) 7<br>B) 8<br>C) 9<br>D) 10

Answer: D) 10<br><br>An exterior angle equals the sum of the two non-adjacent interior angles: 6x – 5 = (3x + 10) + (2x – 5) ⇒ 6x – 5 = 5x + 5 ⇒ x = 10. Therefore, x = 10.

Transformation

A transformation moves or changes a figure on the coordinate plane.

Triangle

A triangle is a polygon with three sides and three angles.

Two rectangles are similar. The dimensions of the smaller rectangle are 4 centimeters by 6 centimeters. If the longer side of the larger rectangle is 15 centimeters, what is the length of the shorter side?<br><br>A) 8 cm<br>B) 10 cm<br>C) 12 cm<br>D) 22.5 cm

Answer: B) 10 cm<br><br>For similar rectangles, sides are proportional: 4/x = 6/15 ⇒ 6x = 60 ⇒ x = 10. Therefore, the shorter side of the larger rectangle is 10 cm.

Vertex

A vertex is a corner point where two or more lines, rays, or sides meet.

Volume

Volume measures the amount of space inside a three-dimensional figure.

X-Axis

The x-axis is the horizontal number line on the coordinate plane.

Lines

Parallel vs. Perpendicular

parallel: equal slopes<br>perpendicular: negative reciprocals

Measurement

Centimeter

A centimeter is a metric unit of length equal to one hundredth of a meter.

Kilometer

A kilometer is a metric unit of length equal to 1,000 meters.

Meter

A meter is a metric unit of length equal to 100 centimeters.

Width

Width is the distance across a shape or object from side to side.

Number System

Additive Inverse

An additive inverse is a number that adds with another number to make 0.

Composite Number

A composite number is a whole number greater than 1 that has more than two factors.

Denominator

The denominator is the bottom number in a fraction.

Equivalent Fractions

Equivalent fractions are fractions that name the same value.

Factor

A factor is a number or expression that divides another number or expression evenly.

Greatest Common Factor

The greatest common factor is the largest factor shared by two or more numbers.

Improper Fraction

An improper fraction has a numerator that is greater than or equal to its denominator.

Integer

An integer is a whole number, its opposite, or zero.

Irrational Number

An irrational number cannot be written as a fraction of two integers.

Least Common Multiple

The least common multiple is the smallest positive multiple shared by two or more numbers.

Multiple

A multiple is the product of a number and a whole number.

Numerator

The numerator is the top number in a fraction.

Opposite

The opposite of a number is the number the same distance from 0 on the other side of the number line.

Percent

A percent is a number out of 100.

Place Value

Place value is the value of a digit based on its position in a number.

Rational Number

A rational number is any number that can be written as a fraction of two integers.

Rounding

Rounding replaces a number with a nearby number that is easier to use.

Whole Number

A whole number is 0 or a positive counting number.

Zero

Zero is the number that represents none or no quantity.

Numbers

A diver starts at an elevation of -18 feet and swims down to an elevation of -47 feet. Which expression gives the change in the diver’s elevation?<br><br>A) -47-(-18)<br>B) -18-(-47)<br>C) -47+(-18)<br>D) 47-18

Answer: A) -47-(-18)<br><br>Change is found by subtracting the starting value from the ending value. The ending elevation is -47 feet and the starting elevation is -18 feet, so the expression is -47-(-18).

A farmer buys bags of seeds for $18 each and sells the crops from each bag for $32. If the farmer used 90 bags of seeds this season, what was the total profit?<br><br>A) $900<br>B) $1,080<br>C) $1,260<br>D) $1,350

Answer: C) $1,260<br><br>Find profit per bag first: 32 – 18 = 14 dollars. Then multiply by the number of bags: 90 × 14 = 1,260 dollars total profit. Students might compute total cost or total revenue instead.

A submarine is at -250 meters below sea level. It descends 80 meters, then ascends 35 meters. At what depth is the submarine now?<br><br>A) -165 m<br>B) -295 m<br>C) -315 m<br>D) -330 m

Answer: B) -295 m<br><br>Start at -250 m. Descend 80 means go deeper (subtract): -250 – 80 = -330 m. Then ascend 35 means go up (add): -330 + 35 = -295 m. The submarine is at -295 meters.

Aisha has a bank account with a balance of -$15 (she owes the bank) and Marcus has a balance of -$8 (he also owes the bank). Who owes more money?<br><br>A) Aisha owes more.<br>B) Marcus owes more.<br>C) They owe the same amount.<br>D) Aisha has more money because -15 < -8.

Answer: A) Aisha owes more.<br><br>A balance of -$15 means Aisha owes $15, while Marcus owes $8. Since -15 < -8, Aisha’s debt is larger.

Removing 8 identical penalty entries changes a team’s score record by +96 points. The removal is represented by dividing by -8. What score change did each original penalty entry represent?<br><br>A) -12 points<br>B) -8 points<br>C) 8 points<br>D) 12 points

Answer: A) -12 points<br><br>The total change from removing the entries is +96 points, and -8 represents the entries being removed. Since 96÷(-8)=-12, each original penalty entry represented a change of -12 points.

Two timers beep at regular intervals. Timer A beeps every 24 seconds, and Timer B beeps every 32 seconds. If both beep together now, after how many seconds will they beep together again?<br><br>A) 96 seconds<br>B) 98 seconds<br>C) 108 seconds<br>D) 120 seconds

Answer: A) 96 seconds<br><br>LCM of 24 equals 2³ × 3 and 32 equals 2⁵ is 2⁵ × 3 = 96 seconds (not 120, which students get from multiplying).

Numbers & Operations

an integer

Any whole number and its negative, including zero: …, &minus;2, &minus;1, 0, 1, 2, …

Percents

Percent of a number

(part = % times whole)

Problem Solving

Estimate

To estimate is to find an answer that is close enough, usually by rounding or using friendly numbers.

Formula

A formula is a rule written with symbols that shows how quantities are related.

Mathematical Model

A mathematical model represents a real situation using numbers, equations, diagrams, or graphs.

Reasonableness

Reasonableness is checking whether an answer makes sense for the problem.

Word Problem

A word problem describes a math situation using words instead of only symbols.

Rates

Average / unit rate

(text{rate} = dfrac{text{amount}}{text{time or unit}})

Distance · rate · time

(d = r t)

Ratios & Proportions

A garden has tomato plants and pepper plants in the ratio 7:5. The garden contains 84 tomato plants. What is the total number of plants in the garden?<br><br>A) 60<br>B) 144<br>C) 175<br>D) 204

Answer: B) 144 plants<br><br>Notice the ratio 7:5 means tomato plants make up 7 of the 12 total parts. Find the size of one part: 84 ÷ 7 = 12 plants. So pepper plants = 5 × 12 = 60, and the total is 84 + 60 = 144 plants.

A manufacturing plant has strict quality-control thresholds. The plant accepts a batch if the defect rate (fraction of defective items per total items) equals or falls below 0.0025. One shift produced a batch where the ratio of defective items to total items in the batch is modeled by the equation 0.0025 = 50/x, where x is the total number of items in the batch. Solve for x to determine the batch size.<br><br>A) 20,000<br>B) 25,000<br>C) 30,000<br>D) 35,000

Answer: A) 20,000<br><br>To solve this proportional relationship, we must isolate x using inverse operations, keeping track of the role of the decimal multiplier. Given: 0.0025 = 50/x Step 1: Identify the structure. This is a proportion where 0.0025 (the defect rate) equals the ratio of 50 defective items to x total items. Step 2: Solve for x. Multiply both sides by x: 0.0025 · x = 50 Step 3: Divide both sides by 0.0025 to isolate x: x = 50/0.0025 Step 4: Compute the division. To divide by a decimal, rewrite as a multiplication by its reciprocal: x = 50 ÷ 0.0025 = 50 × 1/0.0025 = 50 × 400 = 20,000 Alternatively, convert 0.0025 to a fraction: 0.0025 = 2510,000 = 1/400. Then: x = 50/1/400 = 50 × 400 = 20,000 Step 5: Verify. Check that 5020,000 = 1/400 = 0.0025 ✓ Interpretation: A batch of 20,000 items with 50 defectives yields a defect rate of exactly 0.0025 (or 0.25%), which meets the quality threshold. Distractor analysis: – Option B (25,000): Targets the misconception “I divide numerator by denominator” without accounting for the reciprocal. A student might compute 50 ÷ 2 = 25 and append extra zeros. – Option C (30,000): Targets the misconception “I estimate by rounding the decimal.” A student might think 0.0025 ≈ 0.002, leading to a rough estimate that overshoots. – Option D (35,000): Targets the misconception “I can use mental scaling shortcuts.” Attempting 50 × 700 = 35,000 misses the correct factor of 400. Therefore, x = 20,000 items.

A paint mixer blends blue and yellow paint in the ratio 2:3 to make green. If the mixer uses 8 liters of blue paint, how much yellow paint is needed?<br><br>A) 10 liters<br>B) 12 liters<br>C) 16 liters<br>D) 18 liters

Answer: B) 12 liters<br><br>Blue is 2 parts: 8 ÷ 2 = 4 per part. Yellow is 3 parts: 3 × 4 = 12 liters.

a ratio

A comparison of two quantities, written a: b, a to b, or <span class="frac"><span class="n">a</span><span class="d">b</span></span>.

A survey asks students about their favorite sports. The ratio of students who like basketball to students who like soccer is 8:6. Which is the same ratio written in a different form?<br><br>A) 6 to 8<br>B) 6/8<br>C) 8 to 6<br>D) 6:8

Answer: C) 8 to 6<br><br>The ratio 8:6 in word form is “8 to 6.” Choices A, B, and D all reverse the order, which would describe soccer to basketball, not basketball to soccer!

A water tank is being drained at a rate of 600 gallons per hour. If the tank holds 43,200 gallons of water when full, how many days will it take to empty the tank completely?<br><br>A) 2.5 days<br>B) 2.8 days<br>C) 3.0 days<br>D) 3.2 days

Answer: C) 3.0 days<br><br>Divide the total by the rate to get hours: 43,200 ÷ 600 = 72 hours. Then convert hours to days: 72 ÷ 24 = 3 days. Two-step problem—nice!

Ben is preparing a recipe for a cooking class. He needs 2.75 pounds of flour. How many ounces of flour does he need? (Use the conversion 1 pound = 16 ounces.)<br><br>A) 40 ounces<br>B) 42 ounces<br>C) 44 ounces<br>D) 48 ounces

Answer: C) 44 ounces<br><br>Multiply 2.75 pounds by 16 ounces per pound: 2.75 × 16 = 44 ounces.

Ratios and Proportions

Constant of Proportionality

The constant of proportionality is the constant ratio between two proportional quantities.

Scale Factor

A scale factor is the number used to multiply side lengths when making a similar figure larger or smaller.

Unit Price

A unit price is the cost for one item or one unit.

Sets & Logic

Conditional statement and its converse?

&ldquo;If p then q.&rdquo; Converse swaps them: &ldquo;If q then p.&rdquo; The contrapositive (&ldquo;if not q then not p&rdquo;) has the same truth value.

Counting, permutations vs combinations?

<span class="formula">nP r = <span class="frac"><span class="n">n!</span><span class="d">(n&minus;r)!</span></span>,&nbsp; nC r = <span class="frac"><span class="n">n!</span><span class="d">r!(n&minus;r)!</span></span></span>Use permutations when order matters.

Union vs intersection of sets?

Union (A &cup; B) = everything in either set. Intersection (A &cup; B uses ∩) (A &cap; B) = only elements in both.

Statistics

Bar Graph

A bar graph uses bars to compare amounts in different categories.

Box Plot

A box plot displays data using the minimum, first quartile, median, third quartile, and maximum.

Histogram

A histogram uses bars to show how numerical data are grouped into intervals.

Joint Frequency

Joint frequency is the number of data values that fit two categories at the same time.

Line Plot

A line plot shows data values along a number line, often with marks above each value.

Outlier

An outlier is a data value that is far away from most of the other values.

Scatter Plot

A scatter plot displays paired numerical data as points on a coordinate plane.

Statistics & Probability

Expected value?

<span class="formula">E = &Sigma; (value &times; probability)</span>Sum each outcome times its probability.

Volume

Surface area of a cylinder

(SA = 2pi r^2 + 2pi r h)

Volume of a cone

(V = tfrac13 pi r^2 h)

Volume of a cylinder

(V = pi r^2 h)

Volume of a rectangular prism

(V = l w h)

Volume of a sphere

(V = tfrac43 pi r^3)