NGSA Mathematics cheat sheet

Order of operations: evaluate parentheses/brackets first, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Divisibility rules: by 3 if digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 9 if digit sum is divisible by 9; by 6 if divisible by both 2 and 3.

A fraction a/b is in simplest form when gcd(a, b) = 1; reduce by dividing numerator and denominator by their greatest common divisor.

To add or subtract fractions, rewrite over the least common denominator, then combine numerators and simplify.

Dividing by a fraction equals multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b) × (d/c), valid when c ≠ 0.

Percent means per hundred: p% = p/100. Percent change = (new − old)/old × 100%, with the original value as the base.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts them.

For integers, a negative exponent gives a reciprocal: a^(−n) = 1/a^n (a ≠ 0), and any nonzero number to the zero power equals 1.

Absolute value |x| is the distance from x to 0 on the number line, so |x| ≥ 0 and |−x| = |x|.

Slope between two points: m = (y₂ − y₁)/(x₂ − x₁); horizontal lines have slope 0, vertical lines have undefined slope.

Slope-intercept form y = mx + b has slope m and y-intercept b; point-slope form is y − y₁ = m(x − x₁).

Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (negative reciprocals).

A function assigns exactly one output to each input; the vertical line test fails a graph that is not a function.

Solve linear systems by substitution or elimination; a system has no solution when slopes are equal but intercepts differ.

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)]/(2a).

The discriminant b² − 4ac gives two real roots if positive, one real root if zero, and no real roots if negative.

Exponent rules: a^m · a^n = a^(m+n), (a^m)^n = a^(mn), and (ab)^n = a^n b^n for real bases with valid exponents.

Direct variation is y = kx (constant ratio y/x); inverse variation is y = k/x (constant product xy).

Area formulas: rectangle A = lw, triangle A = ½bh, parallelogram A = bh, trapezoid A = ½(b₁ + b₂)h, circle A = πr².

Circumference of a circle is C = 2πr = πd; π ≈ 3.14159 or approximately 22/7 for estimation.

Volume formulas: rectangular prism V = lwh, cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³.

Triangle angle sum is 180°; the exterior angle equals the sum of the two nonadjacent interior angles.

Pythagorean theorem: in a right triangle with legs a, b and hypotenuse c, a² + b² = c²; the converse identifies right triangles.

Similar figures have equal corresponding angles and proportional corresponding sides; the ratio of areas equals the square of the scale factor.

Sum of interior angles of an n-gon is (n − 2) × 180°; each interior angle of a regular n-gon is that sum divided by n.

Complementary angles sum to 90°; supplementary angles sum to 180°; vertical angles are equal.

Unit conversion uses a ratio equal to 1, such as 1 km = 1000 m, so units cancel and the value is unchanged.

Mean = sum of values ÷ number of values; median is the middle value of ordered data; mode is the most frequent value.

Range = maximum − minimum; the interquartile range (IQR) = Q₃ − Q₁ and describes the spread of the middle half of the data.

Outliers by the 1.5 × IQR rule lie below Q₁ − 1.5(IQR) or above Q₃ + 1.5(IQR).

Probability of an event = favorable outcomes ÷ total equally likely outcomes, so 0 ≤ P(E) ≤ 1.

Complement rule: P(not E) = 1 − P(E). For independent events, P(A and B) = P(A) × P(B).

Mutually exclusive events satisfy P(A or B) = P(A) + P(B); otherwise subtract P(A and B) to avoid double counting.

The mean is pulled toward skew and outliers; the median is more resistant, so compare them to judge skew direction.

Correlation r ranges from −1 to 1; r near 0 means little linear association, and correlation does not imply causation.

A line of best fit predicts y from x; a residual is the actual value minus the predicted value, and residuals should show no pattern.

Estimate first by rounding to compatible numbers, then compute exactly to check that the estimate is reasonable.

Ratio and proportion problems: set up a/b = c/d and cross-multiply, keeping like units in the same position.

Unit rate = quantity ÷ time (or per item); use it to scale to other amounts by multiplication.

Work-rate problems add rates: combined rate = 1/t₁ + 1/t₂, and time together = 1 ÷ combined rate.

Distance problems use d = rt; for round trips with different speeds, average speed = total distance ÷ total time, not the mean of speeds.

For consecutive integers, represent them as n, n + 1, n + 2; for consecutive even or odd integers use n, n + 2, n + 4.

Check a solution by substituting back into the original equation or by verifying the answer satisfies every stated condition.

When comparing quantities, convert to the same unit and the same base before computing a ratio or percent difference.

Multi-step word problems: identify the unknown, translate each condition into an equation, solve, then confirm the units of the result.