OAT Quantitative Reasoning key terms

The agreed sequence for evaluating an expression: grouping symbols first, then exponents, then multiplication and division left to right, then addition and subtraction left to right. Following it guarantees one correct value.

The distance of a number from zero on the number line, always nonnegative. Written with vertical bars, it strips a negative sign, so the absolute value of both 7 and negative 7 is 7.

Writing a whole number greater than one as a product of primes only, such as 84 equals 2 times 2 times 3 times 7. It underlies finding greatest common factors, least common multiples, and simplifying radicals.

The largest whole number dividing two or more given numbers without a remainder. For 36 and 60 it is 12. Used to reduce fractions and factor algebraic expressions.

The smallest positive number that is a multiple of two or more given numbers. For 6 and 8 it is 24. Used to build common denominators and to solve rate problems involving repeating events.

Any whole number, positive, negative, or zero, with no fractional or decimal part. Examples include negative 4, 0, and 17. Distinguishing integers from other rational numbers is a common test point.

Any number expressible as a fraction of two integers with a nonzero denominator. Its decimal form either terminates or repeats. Examples include one half, negative 3, and 0.75.

Fractions naming the same value, produced by multiplying or dividing numerator and denominator by the same nonzero number. Three quarters and six eighths are equivalent; this idea justifies common denominators.

Rewriting a fraction with the smallest possible whole-number numerator and denominator by dividing both by their greatest common factor. Twenty-four over thirty-six simplifies to two thirds.

The number that multiplies a given nonzero number to give one; found by flipping a fraction. The reciprocal of five sevenths is seven fifths. Dividing by a fraction is multiplying by its reciprocal.

A decimal in which a digit or block of digits repeats forever, marked with a bar over the repeating part. One third equals 0.333 repeating. Such decimals represent rational numbers.

A ratio comparing a quantity to 100. Forty percent means 40 per 100, or 0.40. Percent problems are solved by converting to decimals or fractions and using the part-whole relationship.

The change from an original amount to a larger amount, expressed as a percentage of the original. Compute the difference, divide by the original value, then multiply by 100.

A comparison of two quantities by division, often written with a colon or as a fraction. A ratio of 3 to 5 means for every 3 of one item there are 5 of the other.

A statement that two ratios are equal, such as 3 over 4 equals x over 12. Solved by cross-multiplying: multiply each numerator by the opposite denominator and set the products equal.

A rate expressed per single unit of one quantity, found by dividing. If 5 pounds cost 12 dollars, the unit rate is 2.40 dollars per pound. Useful for comparing prices or speeds.

A letter or symbol standing for an unknown or changing number in an expression or equation. In 3x plus 5, the x is the variable and 3 is its coefficient.

The numerical factor multiplying a variable in a term. In 7y the coefficient is 7; in negative x the coefficient is negative 1. Coefficients are combined when like terms are added.

A combination of numbers, variables, and operations without an equals sign, such as 2x squared minus 5x plus 1. It can be simplified or evaluated but not solved.

Terms with identical variable parts, including exponents. In 4x plus 3y minus x, the 4x and negative x are like terms and combine to 3x, while 3y stays separate.

An equation in which every variable appears to the first power only, so its graph is a straight line. Example: 2x plus 7 equals 15. Solved by isolating the variable through inverse operations.

A statement comparing expressions with less than, greater than, or their or-equal-to forms. Solving resembles equations, but multiplying or dividing by a negative number reverses the inequality symbol.

Shortcuts for combining powers: add exponents when multiplying like bases, subtract when dividing, multiply when raising a power to a power. A negative exponent means the reciprocal of the positive power.

An equation containing a squared variable as its highest power, generally written ax squared plus bx plus c equals zero. Solved by factoring, completing the square, or the quadratic formula.

Two or more equations sharing the same variables, solved together to find values satisfying all of them. Common methods are substitution and elimination; the solution is the intersection point of their graphs.

The formula distance equals rate multiplied by time, rearranged to find any of the three. Essential for motion problems, including two travelers moving toward or away from each other.

A word problem combining two quantities with different concentrations or prices to find the result. Solved by tracking total amount of each component, often with a table or a single-variable equation.

A problem about people or machines completing a job at different rates. Rates add: if one finishes in a hours and another in b hours, together they finish in ab over a plus b hours.

A mean in which each value contributes in proportion to its assigned weight rather than equally. Computed as the sum of value times weight divided by the sum of weights.

The arithmetic average: the sum of all data values divided by how many there are. Sensitive to extreme values, unlike the median.

The middle value of a data set arranged in order; with an even count, the average of the two middle values. Resistant to outliers, making it useful for skewed data.

The value or values appearing most often in a data set. A set may have one mode, several, or none. Useful for categorical data where averages are meaningless.

The difference between the largest and smallest values in a data set. A simple measure of spread, but it depends on only two data points and ignores everything between.

A measure of how far data values typically lie from the mean. A larger value means greater spread. Roughly, most values fall within one standard deviation of the mean in bell-shaped data.

A number from 0 to 1 expressing how likely an event is, calculated for equally likely outcomes as favorable outcomes divided by total outcomes. Zero means impossible; one means certain.

Events whose outcomes do not affect each other, such as separate coin flips. The probability both occur equals the product of their individual probabilities.

An arrangement of items in which order matters. The number of ways to arrange r items chosen from n distinct items is n factorial divided by n minus r factorial.

A selection of items in which order does not matter, such as choosing a committee. The count is n factorial divided by r factorial times n minus r factorial.

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. Used to find missing lengths and distances between points on a coordinate plane.

Triangles with equal corresponding angles and proportional corresponding sides. Their side lengths form a proportion, allowing indirect measurement of heights and distances.

The number of square units covering a two-dimensional region. Common formulas: rectangle length times width, triangle one half base times height, circle pi times radius squared.

The total distance around the boundary of a plane figure, found by adding side lengths. For a circle the equivalent is circumference, pi times diameter.

The number of cubic units a solid occupies. For a rectangular prism it is length times width times height; for a cylinder it is pi times radius squared times height.

The total area of all outer faces of a three-dimensional solid, measured in square units. For a rectangular prism, sum the areas of its six rectangular faces.

A number that, multiplied by itself, gives the given value. The square root of 49 is 7. Simplifying roots often uses prime factors to pull out perfect squares.

The steepness of a line, computed as the change in y divided by the change in x between two points. Horizontal lines have slope zero; vertical lines have undefined slope.