GED Math cheat sheet

Order of operations: evaluate parentheses, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Slope of a line through (x1,y1) and (x2,y2) is m = (y2 - y1)/(x2 - x1); horizontal lines have slope 0 and vertical lines have undefined slope.

Slope-intercept form y = mx + b gives slope m and y-intercept b; point-slope form is y - y1 = m(x - x1).

Percent change = (new value - old value) / old value x 100; a percent increase uses a positive result, a decrease a negative one.

Simple interest is I = Prt, where P is principal, r is the annual rate as a decimal, and t is time in years.

Pythagorean theorem for right triangles: a^2 + b^2 = c^2, where c is the hypotenuse opposite the right angle.

Area formulas: rectangle A = lw, triangle A = (1/2)bh, circle A = pi r^2; circumference C = 2 pi r.

Probability of a single event = number of favorable outcomes / total number of equally likely outcomes; probabilities range from 0 to 1.

Mean = sum of values / count of values; median is the middle value of an ordered list, or the average of the two middle values when the count is even.

Density equals mass divided by volume (D = m/V); an object denser than its fluid sinks, one less dense floats.

Speed = distance / time; acceleration is the change in velocity divided by the time over which the change occurs.

In a chemical equation, coefficients balance atoms on each side; subscripts within a formula are never changed to balance.

Photosynthesis converts carbon dioxide and water into glucose and oxygen using light energy: 6CO2 + 6H2O -> C6H12O6 + 6O2.

Cellular respiration releases energy from glucose and oxygen, producing carbon dioxide and water; it is roughly the reverse of photosynthesis.

In genetics, dominant alleles mask recessive alleles; a recessive trait appears only when both alleles are recessive.

In a food chain, energy decreases at each higher trophic level, so producers hold the most energy and top predators the least.

Independent variable is the one deliberately changed; dependent variable is the measured outcome; controlled variables are held constant.

The U.S. Constitution separates power into legislative, executive, and judicial branches, with checks and balances among them.

The Bill of Rights is the first ten amendments to the Constitution, protecting freedoms such as speech, religion, and a fair trial.

Federalism divides power between a national government and state governments; the Supremacy Clause makes federal law supreme.

The Three-Fifths Compromise counted three-fifths of enslaved people toward a state's population for representation and taxation.

Supply and demand: as price rises, quantity supplied generally rises and quantity demanded generally falls, all else equal.

Gross Domestic Product measures the total market value of final goods and services produced in a country in a given period.

Primary sources are firsthand records (letters, laws, photographs); secondary sources interpret or analyze primary sources.

A timeline's century is the first two digits plus one for years not ending in 00: 1776 falls in the 18th century.

Main idea is the central point of a passage; supporting details explain, illustrate, or prove that point.

An inference is a conclusion drawn from evidence in the text plus prior knowledge, not a statement the text makes directly.

Fact is verifiable by evidence; opinion expresses a belief or judgment and often uses evaluative words.

A claim is the position an argument defends; evidence supports the claim, and reasoning explains why the evidence is relevant.

Transition words signal relationships: 'however' and 'although' show contrast; 'therefore' and 'consequently' show cause or result.

Subject-verb agreement: a singular subject takes a singular verb; compound subjects joined by 'and' take a plural verb.

A run-on sentence joins two independent clauses without proper punctuation; a comma alone does not join two independent clauses.

In reading charts paired with passages, locate the specific data point the question names rather than averaging all the values.

Convert units by multiplying by a ratio equal to 1, such as 12 inches / 1 foot, arranging units so unwanted ones cancel.

Solve a proportion by cross-multiplying: if a/b = c/d, then ad = bc.

To solve a linear equation, isolate the variable by applying the same inverse operation to both sides.

A negative number multiplied or divided by a negative number yields a positive result; unlike signs yield a negative result.

Exponent rules: x^a times x^b = x^(a+b); x^a divided by x^b = x^(a-b); (x^a)^b = x^(ab).

Volume formulas: rectangular prism V = lwh, cylinder V = pi r^2 h, sphere V = (4/3) pi r^3.

Scientific notation writes a number as a value between 1 and 10 times a power of 10, e.g., 4,500 = 4.5 x 10^3.

Newton's second law: force equals mass times acceleration (F = ma); weight is mass times gravitational acceleration.

The law of conservation of mass states that matter is neither created nor destroyed in a chemical reaction.

pH below 7 is acidic, pH of 7 is neutral, and pH above 7 is basic; each unit represents a tenfold change in hydrogen ion concentration.

States of matter: solids have fixed shape and volume, liquids have fixed volume but flow, gases fill their container.

In a Punnett square, a cross of two heterozygotes (Aa x Aa) yields a 3:1 dominant-to-recessive phenotype ratio.

Mitosis produces two genetically identical daughter cells for growth and repair; meiosis produces gametes with half the chromosome number.

The water cycle moves water through evaporation, condensation, precipitation, and collection or runoff.

The Declaration of Independence (1776) states that governments derive power from the consent of the governed.

The Articles of Confederation created a weak national government with no power to tax, leading to the Constitutional Convention of 1787.

The Great Compromise created a bicameral Congress: representation by population in the House and equal state representation in the Senate.

The Emancipation Proclamation (1863) declared enslaved people in Confederate states free; the 13th Amendment abolished slavery.

The 14th Amendment guarantees equal protection and due process; the 15th Amendment protects voting rights regardless of race.

Opportunity cost is the value of the next best alternative given up when a choice is made.

Inflation is a sustained rise in the general price level, which reduces the purchasing power of money.

Author's purpose is usually to inform, persuade, or entertain; the purpose shapes word choice and structure.

Tone is the author's attitude toward the subject, inferred from word choice; mood is the feeling created in the reader.

A thesis or argument must be debatable; a statement of plain fact cannot serve as a claim.

Parallel structure requires items in a list or comparison to share the same grammatical form.

A pronoun must agree with its antecedent in number and gender, and the antecedent must be clear.

Apostrophes show possession (the student's book) or contraction (it's = it is); 'its' without an apostrophe shows possession.

When comparing two passages, identify the point of agreement first, then isolate where the authors diverge.

To find a percent of a number, convert the percent to a decimal and multiply: 15% of 80 = 0.15 x 80 = 12.

The sum of the interior angles of a triangle is 180 degrees; a straight line measures 180 degrees and a full circle 360 degrees.

Similar figures have equal corresponding angles and proportional corresponding side lengths.

To solve a system of two linear equations, substitute one equation into the other or add/subtract to eliminate a variable.

The absolute value of a number is its distance from zero, so it is never negative.

For a right triangle, sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.

A function assigns exactly one output to each input; the vertical line test checks whether a graph represents a function.

Earth's rock types: igneous forms from cooled magma or lava, sedimentary from compacted deposits, metamorphic from heat and pressure.

Plate tectonics: convergent boundaries produce mountains and volcanoes, divergent boundaries produce new crust, transform boundaries cause earthquakes.

Energy flows through an ecosystem as a one-way path, while matter such as carbon and nitrogen cycles repeatedly.

The independent variable is plotted on the x-axis and the dependent variable on the y-axis.

A controlled experiment changes one variable at a time and includes a control group for comparison.

Electric current equals voltage divided by resistance (Ohm's law, I = V/R).

Wavelength and frequency are inversely related for a wave traveling at constant speed: speed = wavelength x frequency.

The president can veto legislation; Congress can override a veto with a two-thirds vote in each chamber.

Judicial review, established in Marbury v. Madison (1803), lets courts declare laws unconstitutional.

The Elastic Clause allows Congress to make laws necessary and proper to carry out its enumerated powers.

The First Amendment protects freedom of speech, religion, press, assembly, and petition.

Voting amendments: the 19th gave women the vote, the 24th banned poll taxes, and the 26th lowered the voting age to 18.

In economics, scarcity means resources are limited relative to wants, forcing choices and trade-offs.

A market economy allocates goods through prices and voluntary exchange rather than government command.

Text structure patterns include cause-effect, compare-contrast, chronological, and problem-solution; identifying the pattern speeds comprehension.

A counterargument acknowledges an opposing view; a strong response then refutes it with evidence.

Loaded language carries emotional connotation and signals bias; neutral wording reports without evaluation.

A semicolon joins two closely related independent clauses; a colon introduces a list or explanation.

Dangling modifiers occur when the introductory phrase does not describe the subject of the main clause.

When two independent clauses are joined by a coordinating conjunction (for, and, nor, but, or, yet, so), place a comma before the conjunction.

Paraphrasing restates a source's idea in your own words and structure, while still requiring citation.