ACT Math: Integrating Essential Skills (Core Quantitative Tools)

Ratio

A comparison of two quantities using division, written as a:b, a/b, or “a to b.”

Rate

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

Unit rate

A rate with a denominator of 1, used to compare prices, speeds, or other quantities easily.

Proportional relationship

A relationship in which two quantities change at a constant multiplicative rate, so y/x stays constant and the model is y = kx.

Constant of proportionality

The constant k in the equation y = kx, found by k = y/x.

Proportion

An equation showing that two ratios are equal, such as a/b = c/d.

Cross-multiplication

A method for solving proportions by using the fact that if a/b = c/d, then ad = bc.

Percent

A ratio out of 100; for example, 35% = 35/100 = 0.35.

Percent change

The amount of change compared to the original value, calculated by ((new - original) / original) × 100%.

Multiplier method

A percent-change method that uses 1 + r/100 for an increase and 1 - r/100 for a decrease.

Area

The amount of 2D space inside a figure, measured in square units.

Surface area

The total area of all the outside faces or surfaces of a 3D object.

Volume

The amount of 3D space inside a solid, measured in cubic units.

Scale factor

The number k by which every linear dimension is multiplied; perimeter scales by k, area by k^2, and volume by k^3.

Mean

The average of a data set, found by dividing the sum of the values by the number of values.

Median

The middle value in an ordered data set, or the average of the two middle values if there is an even number of data points.

Weighted mean

An average in which some values count more than others, found by dividing the weighted sum by the total of the weights.

Unit conversion

Changing a measurement to different units without changing its actual quantity by using a conversion factor equal to 1.

Dimensional analysis

A unit-canceling method that treats units like algebraic symbols to set up conversions correctly.

Scientific notation

A way to write very large or very small numbers as a × 10^n, where 1 <= a < 10 and n is an integer.

Prime factorization

Writing a whole number as a product of prime numbers.

Simplifying radicals

Rewriting a radical by factoring out perfect squares, such as sqrt(72) = 6sqrt(2).

Multi-step problem

A problem that requires several connected steps, often including an intermediate result before finding the final answer.

Mathematical modeling

Translating a real-world situation into math, such as an expression, equation, inequality, or function.

Linear model

An equation of the form y = mx + b, where m is the rate of change and b is the starting value.