NLN NEX Math cheat sheet

Order of operations: evaluate grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

To add or subtract fractions, rewrite them over a common denominator; to multiply, multiply numerators and multiply denominators; to divide, multiply by the reciprocal of the divisor.

A percent is a ratio out of 100: p% = p/100. Percent to decimal means divide by 100; decimal to percent means multiply by 100.

In percent and rate word problems, "of" signals multiplication and "per" signals division.

Percent change = (new value − original value) ÷ original value × 100. A positive result is an increase, a negative result is a decrease.

A proportion a/b = c/d is solved by cross-multiplying: ad = bc.

Rounding rule: locate the target place value, then look at the single digit to its right; 5 or greater rounds up, 4 or less leaves the digit unchanged.

Divisibility tests: by 3 if the digit sum is a multiple of 3, by 9 if the digit sum is a multiple of 9, by 4 if the last two digits form a multiple of 4.

To compare two fractions, cross-multiply (multiply each numerator by the opposite denominator); the larger product sits under the larger fraction.

Distributive property: a(b + c) = ab + ac, and a(b − c) = ab − ac.

Combine like terms only when the variable parts match exactly, including exponents: 7x + 4x = 11x, but 7x + 4x² cannot be combined.

Solve a linear equation by undoing operations in reverse order - add or subtract first, then multiply or divide - applying each operation to both sides.

Slope m = (y₂ − y₁)/(x₂ − x₁) for two points; slope-intercept form is y = mx + b, where b is the y-intercept.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol; adding or subtracting a negative does not.

Inequality phrases: "at least" means ≥, "at most" means ≤, "more than" means >, "fewer than" means <.

Translation order matters: "5 less than x" is x − 5, while "5 is less than x" is 5 < x.

Distance, rate, time: d = rt, so r = d/t and t = d/r.

To solve a system by substitution, solve one equation for one variable, substitute that expression into the other equation, then solve for the remaining variable.

Verify a solution by substituting it back into the original equation and confirming both sides simplify to the same value.

Perimeter of a rectangle P = 2l + 2w; perimeter of a square P = 4s; perimeter is measured in linear units.

Area of a rectangle A = lw; area of a triangle A = ½bh, where the height is perpendicular to the base.

Circle formulas: area A = πr², circumference C = 2πr = πd, and diameter d = 2r.

Volume of a rectangular prism V = lwh; volume of a cylinder V = πr²h; volume is measured in cubic units.

The interior angles of any triangle sum to 180°; a right angle is 90°, and angles on a straight line sum to 180°.

Convert units by multiplying by a fraction equal to 1, placing the unwanted unit opposite the unit you are converting from so it cancels.

Customary lengths: 1 foot = 12 inches, 1 yard = 3 feet, 1 mile = 5,280 feet.

Metric and time units: 1 km = 1,000 m, 1 m = 100 cm, 1 kg = 1,000 g, 1 L = 1,000 mL, 1 hour = 60 minutes.

If every length of a figure is multiplied by a scale factor k, its area is multiplied by k² and its volume by k³.

Mean = sum of all values ÷ number of values; it is strongly affected by unusually large or small values.

Median = middle value of an ordered list; with an even number of values, average the two middle values.

Mode = the value that occurs most often; range = maximum value − minimum value.

To recover a total from a known mean, multiply the mean by the number of values; subtract known values to find a missing one.

Weighted average = sum of (value × weight) ÷ sum of the weights.

Average speed for a trip = total distance ÷ total time, which is not the average of the separate speeds unless the times are equal.

A unit rate is a quantity divided by its time or per-item count; compare rates only after converting to the same units.

Percent of a total from a table = (part ÷ total) × 100, using the row or column total that matches the question.

Before comparing bar heights or line positions on a graph, check the scale and interval of each axis.