Free GED Math Exam Flashcards

Evaluate in this order: Parentheses/grouping symbols → Exponents → Multiplication and Division (left to right) → Addition and Subtraction (left to right). The 5 no-calculator items often hide a trap like 3 + 4 × 2, where solving left-to-right (14) instead of multiplying first (11) is the most common wrong answer.

Two negatives (or two positives) produce a positive result; one negative and one positive produce a negative result. Consequence: (−3) × (−4) = 12, but (−3) × 4 = −12 — a sign slip on negative-number arithmetic is a leading cause of missed no-calculator items.

Keep the first fraction, change ÷ to ×, and flip (take the reciprocal of) the second fraction — Keep-Change-Flip. Example: 2/3 ÷ 3/4 becomes 2/3 × 4/3 = 8/9. Forgetting to flip only the second fraction is the most common fraction-division mistake.

Divide the numerator by the denominator, then multiply by 100. Example: 3/8 = 0.375 = 37.5%. Converting to a decimal first — rather than comparing fractions with different denominators directly — avoids errors on word problems that compare quantities.

The amount of one quantity per single unit of another (dollars per item, miles per hour). Find it by dividing the total by the number of units, then multiplying by the new quantity needed — this two-step method solves nearly every 'at this rate' word problem.

Write two equal ratios with matching units in the same position (part/whole = part/whole), then cross-multiply to solve for the unknown. Putting a quantity in the wrong position between the two ratios — mismatching top and bottom — is the most common setup error.

Percent change = (New Value − Original Value) ÷ Original Value × 100. Always divide by the ORIGINAL value, not the new one — dividing by the new value instead is a common wrong-answer trap on percent increase/decrease problems.

Markup increases a price: New Price = Original × (1 + rate). A discount decreases it: New Price = Original × (1 − rate). Consequence: a 20% markup followed by a 20% discount does NOT return the original price, because the second percent is taken from a different base amount.

If a part is a known percent of an unknown whole, divide the part by the percent written as a decimal: Whole = Part ÷ Percent. Example: 15 is 25% of what number? 15 ÷ 0.25 = 60. Multiplying instead of dividing is the typical error here.

Mean (average) = sum of values ÷ number of values; it is pulled toward extreme outliers. Median = the middle value once data is ordered; it is NOT affected by outliers. A data set with one very large outlier will have a mean noticeably higher than its median.

The value that appears most often in a data set. A set can have no mode, one mode, or several modes (bimodal/multimodal). Unlike mean and median, mode is the only measure of center that also works for non-numeric (categorical) data.

Probability = number of favorable outcomes ÷ total number of possible outcomes, expressed as a fraction, decimal, or percent between 0 and 1. A probability of 0 means an event is impossible and 1 means it's certain — an answer outside that range signals a setup error.

Before calculating mean, median, or mode from a table, check whether it lists individual values or gives frequencies (how many times each value occurs). Treating a frequency count as if it were a data value instead of a weight is a common table-misreading error.

Area = 1/2 × base × height, where height is measured perpendicular to the chosen base — not along a slanted side. Using a slanted, non-perpendicular side as the height gives a wrong, usually too-large, area.

Area = π × r² (the enclosed space, in square units). Circumference = 2 × π × r (the distance around, in linear units). Mixing these up — for example using the circumference formula when a problem asks for area — is a frequent error, so check whether the answer choices use square or linear units.

Volume = length × width × height, giving a result in cubic units (like cm³ or ft³). Multiplying only two of the three dimensions — as if finding area instead of volume — understates the result by a full linear factor.

Volume = π × r² × height (the area of the circular base times the height). Using the diameter in place of the radius without dividing by 2 first is the most common cylinder-volume mistake.

Surface area answers 'how much material covers the outside' (paint, wrapping paper, labels); volume answers 'how much fits inside' (liquid, sand, capacity). Reading the word problem's real-world context — covering vs. filling — tells you which formula applies before you do any math.

Split an irregular shape into simple shapes you already know how to find the area of (rectangles, triangles, circles or semicircles), then add or subtract those areas. Adding an internal cutout, like a hole, instead of subtracting it will overstate the total area.

a² + b² = c², where c is always the hypotenuse — the longest side, opposite the right angle. It only applies to right triangles. Plugging the hypotenuse into 'a' or 'b' instead of solving for it gives an impossible result.

Look for right-angle scenarios described in words — a ladder against a wall, the shortest path across a rectangular field, a TV's diagonal screen size. The two legs are the straight, perpendicular distances; the diagonal path is the hypotenuse you're usually solving for.

A scale factor relates a drawing's dimensions to real-life dimensions (for example, 1 inch = 5 feet). Find an actual length by setting up a proportion between the scale ratio and the measured ratio, then cross-multiplying — treating the scale factor as something to subtract, instead of multiply by, is a common error.

Convert every measurement to the same unit BEFORE calculating (feet to inches, or minutes to hours), since GED problems often mix units on purpose. Calculating with mismatched units — like adding feet and inches directly — produces an answer that matches one of the wrong choices.

Only add or subtract terms with the exact same variable(s) raised to the exact same power (3x and 5x, but not 3x and 5x²). Combining unlike terms — treating 3x + 5x² as 8x³, for example — is one of the most common algebra errors on the test.

a(b + c) = ab + ac — multiply the term outside the parentheses by EVERY term inside. Distributing to only the first term inside the parentheses, a common shortcut mistake, leaves part of the expression unsimplified and produces a wrong answer.

Adding polynomials means combining like terms across both expressions. Multiplying polynomials means distributing every term of one across every term of the other (FOIL for two binomials). Confusing the two operations produces an answer with the wrong degree (highest exponent).

Key words map to operations: 'more than' or 'increased by' → addition, 'less than' → subtraction written in reverse order, 'times'/'product' → multiplication, 'quotient' → division. '5 less than a number' means x − 5, NOT 5 − x — reversing subtraction order is a top translation error.

Substitute the given number for every instance of the variable, then follow order of operations. Forgetting to substitute into EVERY occurrence of the variable when it appears more than once leaves the evaluation only partly correct.

Work in this order: distribute or clear parentheses, combine like terms on each side, move variable terms to one side and constants to the other, then divide by the coefficient. Skipping the distribute step before combining terms is the most frequent multi-step equation error.

Multiply every term on both sides of the equation by the least common denominator (LCD) to clear all fractions at once, then solve the resulting whole-number equation. Multiplying only one side, or only one term, by the LCD unbalances the equation and gives a wrong solution.

Solve one equation for one variable, then substitute that expression into the OTHER equation to solve for the remaining variable. This method works best when one equation is already solved for a variable, or can easily be rearranged into that form.

Add or subtract the two equations, after multiplying one or both by a constant if needed, so one variable cancels out and leaves a single-variable equation. This method works best when both equations are already in the same standard form (Ax + By = C).

When you multiply or divide both sides of an inequality by a NEGATIVE number, you must flip the inequality sign. Forgetting to flip the sign after dividing by a negative — turning −2x < 8 into x < −4 instead of the correct x > −4 — is the single most common inequality error.

Use an open circle for < or > (the boundary value is NOT included) and a closed/filled circle for ≤ or ≥ (the boundary value IS included), then shade the direction the inequality points. Mixing up open vs. closed circles misrepresents whether the boundary value counts as a solution.

Find two numbers that multiply to c and add to b; those numbers become the constants in the two binomial factors: (x + m)(x + n). Finding numbers that add to b but don't correctly multiply to c (or vice versa) is the most common factoring mistake — always check both conditions.

If two factors multiply to zero, at least one of them must equal zero — this is why factoring a quadratic lets you solve it: set each factor equal to zero and solve separately. It only works when one side of the equation is exactly 0, so move all terms to one side first.

a² − b² always factors as (a + b)(a − b). Recognizing this pattern — two perfect squares separated by subtraction — lets you factor instantly without trial and error, but it does NOT apply to a sum of squares (a² + b²), which doesn't factor over real numbers.

A quadratic function's graph is a parabola (a U-shaped curve), not a straight line. A positive leading coefficient opens upward with a minimum point; a negative leading coefficient opens downward with a maximum point. Misreading the sign of the leading coefficient flips your prediction of the graph's shape.

Slope = (y₂ − y₁) ÷ (x₂ − x₁) — the change in y divided by the change in x (rise over run) between two points. Subtracting the x-values and y-values in mismatched order, such as (y₂ − y₁) over (x₁ − x₂), flips the sign of the slope.

A positive slope means the line rises left to right (as x increases, y increases). A negative slope means the line falls left to right (as x increases, y decreases). A real-world rate that decreases over time, like a car's value depreciating, must be represented by a NEGATIVE slope, not a positive one.

m is the slope, or rate of change, and b is the y-intercept — the starting value where the line crosses the y-axis, at x = 0. Reading the constant term as the slope and the coefficient of x as the intercept swaps the two values and misreads the equation or graph.

Set y = 0 in the equation and solve for x — the x-intercept is where the line crosses the x-axis. Setting x = 0 instead, which finds the y-intercept, is a common mix-up when a problem specifically asks where a line 'crosses the x-axis.'

Parallel lines have the SAME slope. Perpendicular lines have slopes that are negative reciprocals of each other, multiplying to −1. Using the same slope for a 'perpendicular' line instead of flipping and negating it is a frequent graphing-relationship error.

In a table of x/y values, the rate of change equals the slope: divide the change in output (y) by the change in input (x) between any two rows. If the rate of change is NOT constant between every pair of rows, the relationship is NOT linear — a common trap in table-to-graph questions.

f(x) is read 'f of x' and represents the output value of function f when the input is x — it does NOT mean f multiplied by x. To evaluate f(3), substitute 3 for every x in the function's rule.

A graph represents a function only if every vertical line crosses it at most once, meaning each input has exactly one output. A graph where a vertical line crosses twice, like a full circle, is NOT a function, even if it looks like a smooth, continuous curve.

Each row pairs one input (x) with its corresponding output, f(x) or y; check whether the SAME input ever produces two different outputs — if so, the table does not represent a valid function. Assuming any table of numbers is automatically a function skips this required check.

A linear function has a constant rate of change and graphs as a straight line (y = mx + b). A nonlinear function's rate of change varies — quadratics, exponentials, and absolute value functions all curve. Assuming a curved relationship can be modeled with a single slope value leads to an incorrect linear equation.

Domain is the complete set of possible input (x) values a function can accept; range is the complete set of possible output (y) values it can produce. Confusing which axis, x or y, each term refers to leads to swapped domain and range answers on graph-reading questions.

Match graph features to real-world meaning: the y-intercept is often a starting amount, the x-intercept is often when a quantity reaches zero, and the slope is a rate (per hour, per item, and so on). Reading only the numeric graph values without connecting them to what they represent leads to picking a mathematically plausible but contextually wrong answer.