Free GRE Quantitative Reasoning Exam Flashcards

Every Quantitative Comparison item asks whether Quantity A is greater, Quantity B is greater, the two quantities are equal, or the relationship cannot be determined from the information given. Select the relationship proved by the information, not the one suggested by appearance.

A select-one item requires exactly one answer choice. A select-one-or-more item requires all and only the correct choices; the stated number of choices may or may not be given. Identify the response format before solving.

Numeric Entry may require an integer or decimal in one box, or a fraction entered with separate numerator and denominator boxes. Attend to the requested form, units, and any rounding instruction rather than entering a merely equivalent-looking response.

Data Interpretation questions share a table, graph, or other data display. ETS says they may use either multiple-choice format or Numeric Entry. Base each answer on the displayed data, stated information, everyday facts, and mathematics—not outside specialist knowledge about the context.

A divisibility test is a shortcut for deciding whether one integer divides another with no remainder. Use it to constrain possibilities before doing full factorisation; the relevant rule depends on the proposed divisor.

Prime factorisation writes a positive integer greater than 1 as a product of primes. It helps reveal divisors, common factors, least common multiples, and perfect-square factors. A prime has exactly two positive divisors: 1 and itself.

When a nonnegative integer n is divided by a positive integer d, it can be written n = dq + r, where q is an integer and 0 ≤ r < d. The remainder r is what remains after accounting for complete groups of d.

For a nonzero base a, multiplying powers with the same base adds exponents: a^m × a^n = a^(m+n). Dividing powers with the same nonzero base subtracts exponents: a^m ÷ a^n = a^(m−n).

If x² = n, then x is a square root of n. The radical √n denotes the principal, nonnegative square root when n is nonnegative. Keep the two ideas distinct: an equation can have two real square roots, while √n is one nonnegative value.

The absolute value |x| is x's distance from 0 on the number line, so it is never negative. For example, |−7| and |7| both equal 7. Translate absolute-value statements into distance or into the appropriate cases.

A percent increase compares the change with the original value: change ÷ original × 100%. A percentage-point increase compares two percentages by subtraction. Moving from 20% to 25% is a 5-percentage-point increase and a 25% relative increase.

A ratio compares quantities, often in the same units, such as 3 red items to 2 blue items. A rate compares quantities with different units, such as miles per hour. Preserve units and the order of a ratio when scaling or interpreting it.

Each place to the left of a decimal point is ten times the value of the place to its right; each place to the right is one-tenth the value of the place to its left. Align decimal points when adding or subtracting decimals.

An arithmetic sequence has a constant difference between successive terms. If the first term is a and the common difference is d, the nth term is a + (n−1)d. Check that the difference—not merely the ratio—is constant.

Estimate with compatible numbers, bounds, or order of magnitude before or after exact work. A useful estimate need not be the final answer; it can catch a misplaced decimal, unreasonable percentage, or calculator-entry error.

Assign variables with clear meanings, preserve comparison words and units, and translate one phrase at a time. For example, 'five more than twice x' is 2x + 5, while 'five less than twice x' is 2x − 5. Check the model against the wording.

Use inverse operations to isolate the variable while applying the same valid operation to both sides. Then substitute the result back into the original equation to verify it, especially after clearing fractions or expanding parentheses.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Adding or subtracting the same expression does not reverse it. Track this rule explicitly whenever a negative factor is involved.

A denominator cannot equal zero. Before simplifying or solving a rational expression, record every value that makes a denominator zero. An algebraic cancellation may simplify an expression but does not make an originally excluded value allowable.

The distributive property states a(b + c) = ab + ac. Use it in both directions: expand to combine like terms or factor out a common factor. Signs must distribute to every term inside the parentheses.

Like terms have the same variable part raised to the same powers, such as 3x and −5x. Their coefficients can be added or subtracted. Terms such as x and x² are not like terms and cannot be combined by addition.

Factoring rewrites a quadratic as a product of simpler expressions. When a product equals zero, at least one factor must equal zero. Verify a factorisation by expanding it, and check any solutions in the original equation when restrictions exist.

In f(x), the symbol f names a function and x is its input; f(x) is the resulting output, not multiplication. To evaluate f(a), substitute a everywhere the input variable appears and respect any domain restriction.

For two distinct points (x₁, y₁) and (x₂, y₂), slope is (y₂ − y₁)/(x₂ − x₁). It represents vertical change per unit horizontal change. A vertical line has undefined slope because its horizontal change is zero.

A solution to a system satisfies every equation in the system simultaneously. Graphically, it is a shared point or shared set of points. Use substitution, elimination, or a graph, then check the candidate in all original equations.

In a proportional relationship y = kx, the ratio y/x is constant for nonzero x and the graph passes through the origin. Do not assume any straight-line relationship is proportional; y = mx + b with nonzero b is linear but not proportional.

Substitution replaces a variable with a known or proposed value. It can verify an equation solution, evaluate a function, compare expressions, or test answer choices. Maintain parentheses when substituting a negative value into an expression.

Vertical angles are equal, and adjacent angles that form a straight line sum to 180°. Use only relationships supported by the diagram's markings or stated conditions; do not infer an unmarked right angle or equal length from appearance.

When parallel lines are cut by a transversal, corresponding angles are equal, alternate interior angles are equal, and same-side interior angles are supplementary. These conclusions require that parallelism be stated or marked.

The interior angles of a plane triangle sum to 180°. An exterior angle equals the sum of the two remote interior angles. Use the stated angle information and triangle type; do not infer side or angle equality from an unmarked drawing.

For a right triangle with legs a and b and hypotenuse c, a² + b² = c². The theorem applies only when the triangle is known to be right, and c is the side opposite the right angle.

The sum of the interior angles of a simple n-sided polygon is (n−2) × 180°. This total concerns all interior angles; equal angles or equal side lengths require additional information such as regularity.

A radius connects the centre of a circle to a point on the circle. A diameter passes through the centre and has length twice the radius. Circumference is 2πr and area is πr², so keep linear and square units distinct.

Perimeter measures boundary length in linear units. Area measures the amount of surface inside a plane figure in square units. Figures can share a perimeter but have different areas, or share an area but have different perimeters.

Volume measures three-dimensional space and uses cubic units. For a right rectangular prism, volume equals length × width × height. Match each dimension's units before multiplying, and distinguish volume from surface area.

A horizontal or vertical distance is the absolute difference of the relevant coordinates. For a slanted segment, use a justified geometric relationship such as the Pythagorean theorem after identifying horizontal and vertical changes, rather than estimating from the diagram.

ETS says geometric figures are not necessarily drawn to scale. Do not use apparent side lengths or angle sizes as evidence. You may rely on stated or marked facts, straight lines, point order on a line, and relative positions shown unless the question says otherwise.

ETS permits reading, estimating, or comparing by sight from coordinate systems such as xy-planes and number lines, and from graphical data presentations such as bar, circle, and line graphs. Apply the displayed scale, labels, and units.

Mean equals total of the values divided by the number of values. Therefore total equals mean × number of values. Adding the same amount to every value raises the mean by that amount; changing one value affects the total and therefore the mean.

The median is the middle value after values are ordered, or the average of the two middle values for an even count. The mean uses every value and is often more affected by an extreme value than the median is.

Range is maximum minus minimum. Interquartile range is Q3 minus Q1 and describes the spread of the middle half of ordered data. Both measure spread, but the range is especially sensitive to extremes.

Standard deviation describes how dispersed values are around their mean. A smaller standard deviation indicates values more tightly clustered around the mean; a larger one indicates greater spread. Compare it only when the distributions and units make the comparison meaningful.

Before interpreting a graph, identify axes, scale increments, units, legend, title, and any break in the scale. A bar's visual height alone can mislead if the axis does not start at zero or if values use a stated multiplier.

For any event, probability is between 0 and 1 inclusive. A probability of 0 means impossible under the model; 1 means certain. Use complements when easier: P(not A) = 1 − P(A).

If events A and B are independent, P(A and B) = P(A) × P(B). Independence is a condition, not an assumption: the occurrence of one event must not change the probability of the other.

Conditional probability restricts attention to outcomes where a given condition holds. P(A given B) asks for the proportion of B outcomes that are also A outcomes, provided P(B) is not zero. Read the condition before selecting the denominator.

A permutation counts ordered arrangements, so changing the order creates a different outcome. A combination counts unordered selections, so the same selected group is counted once regardless of order. Decide whether order matters before choosing a method.

A Venn diagram represents sets and their overlap. When adding counts from overlapping sets, subtract the intersection once to avoid double-counting. Use the universal set and any 'neither' region to check that all categories are accounted for.

A normal distribution is symmetric about its mean, so the mean and median coincide. Its shape is determined by centre and spread. Do not assume a data set is normally distributed unless the problem states or shows that model.

Round only when the question explicitly instructs you to do so, and use the requested precision. Keep intermediate computations exact when feasible, then round the final value; if no rounding instruction is given, enter the exact answer required by the question.