NES Essential Academic Skills Subtest III: Mathematics cheat sheet

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

Divisibility: a number is divisible by 3 if its digit sum is divisible by 3, by 9 if its digit sum is divisible by 9, by 4 if its last two digits form a multiple of 4, and by 6 if it is divisible by both 2 and 3.

Fraction operations: to add or subtract, rewrite over a common denominator; to multiply, multiply numerators and denominators; to divide, multiply by the reciprocal of the divisor.

Percent change = (new value − old value) ÷ old value × 100; a 20% increase followed by a 20% decrease yields a net 4% decrease, not zero change.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; multiplying adds exponents, dividing subtracts exponents, and a negative exponent moves the decimal point left.

Absolute value |x| is the distance from x to 0 on the number line, so |x| ≥ 0 always and |a − b| equals the distance between a and b.

Rational numbers can be written as a ratio of integers; their decimal forms terminate or repeat, while irrational numbers such as √2 and π neither terminate nor repeat.

Prime factorization via repeated division by primes; the greatest common factor is the product of shared prime factors and the least common multiple is the product of the highest power of every prime present.

Rounding decision rule: identify the place value being kept, then round up when the next digit is 5 or greater and keep the digit unchanged when it is 4 or less.

Slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines have equal slopes and perpendicular lines have slopes whose product is −1.

Linear forms: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), and standard Ax + By = C; the x-intercept occurs where y = 0.

Solving linear equations: perform the same operation on both sides to isolate the variable; when multiplying or dividing an inequality by a negative number, reverse the inequality symbol.

Quadratic formula x = (−b ± √(b² − 4ac))/(2a); the discriminant b² − 4ac gives two real roots when positive, one when zero, and no real roots when negative.

Factoring patterns: a² − b² = (a − b)(a + b), a² + 2ab + b² = (a + b)², and for x² + bx + c find two numbers whose product is c and whose sum is b.

Function notation: f(x) names the output for input x; the domain is the set of legal inputs (denominators nonzero, radicands of even roots nonnegative) and the range is the set of outputs.

Exponent rules: a^m · a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^0 = 1 for a ≠ 0, and a^(−n) = 1/a^n.

Direct variation y = kx graphs as a line through the origin; inverse variation y = k/x graphs as a hyperbola with k = xy constant.

Systems of two linear equations: substitution isolates one variable, elimination adds or subtracts multiples of the equations; parallel lines give no solution and identical lines give infinitely many.

Angle relationships: vertical angles are equal, complementary angles sum to 90°, supplementary angles sum to 180°, and the interior angles of any triangle sum to 180°.

Polygon interior angle sum = (n − 2) × 180°; each interior angle of a regular n-gon measures (n − 2) × 180° ÷ n.

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

Circumference C = 2πr = πd; arc length equals (central angle ÷ 360°) × 2πr and sector area equals (central angle ÷ 360°) × πr².

Volume: rectangular prism V = lwh, cylinder V = πr²h, cone V = ⅓πr²h, sphere V = (4/3)πr³; surface area of a prism is the sum of all face areas.

Pythagorean theorem a² + b² = c² for right triangles with hypotenuse c; the converse identifies right triangles, and common triples include 3-4-5 and 5-12-13.

Similar figures have equal corresponding angles and proportional corresponding sides; scale factor k multiplies lengths by k, areas by k², and volumes by k³.

Unit conversion by dimensional analysis: multiply by conversion ratios equal to 1 so unwanted units cancel; 1 inch = 2.54 cm, 1 km ≈ 0.621 mi, 1 L = 1000 mL.

Coordinate geometry: distance = √((x₂ − x₁)² + (y₂ − y₁)²) and midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Mean = sum of values ÷ number of values; median is the middle value of an ordered data set (average of the two middle values when n is even); mode is the most frequent value.

Range = maximum − minimum; the interquartile range IQR = Q3 − Q1 describes the spread of the middle 50% and outliers are commonly flagged beyond 1.5 × IQR from the quartiles.

A normal distribution is symmetric about its mean; roughly 68% of values fall within one standard deviation, 95% within two, and 99.7% within three.

Probability of an event P(E) = favorable outcomes ÷ total equally likely outcomes, so 0 ≤ P(E) ≤ 1 and P(not E) = 1 − P(E).

For independent events P(A and B) = P(A) × P(B); for mutually exclusive events P(A or B) = P(A) + P(B), otherwise subtract P(A and B).

Conditional probability P(A|B) = P(A and B) ÷ P(B); events are independent exactly when P(A|B) = P(A).

Counting: fundamental counting principle multiplies choices across stages; permutations nPr = n!/(n − r)! count ordered arrangements and combinations nCr = n!/(r!(n − r)!) count unordered selections.

Misleading-graph rule: truncated vertical axes exaggerate differences, and a mean is pulled toward outliers while the median resists them.

Correlation measures the strength and direction of a linear association between −1 and 1; correlation does not establish causation.

Estimation check: round each quantity to one or two significant digits, compute mentally, and compare the estimate to the exact answer to catch place-value or operation errors.

Work-rate rule: rates add as reciprocals of completion times, so if one worker finishes in a hours and another in b hours, together they finish in ab/(a + b) hours.

Distance problems use d = rt; for two objects moving toward each other, add speeds, and for same-direction pursuit, subtract speeds.

Weighted average = Σ(value × weight) ÷ Σ(weights); mixture problems balance the total amount of a component before and after combining.

Percent problems translate to equations: part = percent × whole, with the percent written as a decimal (35% = 0.35).

Backsolving substitutes answer choices into the original equation or conditions, and plugging in numbers replaces abstract variables with convenient values to test algebraic claims.

Proportional reasoning sets two ratios equal and cross-multiplies; confirm the units of the unknown match the units requested by the question.

Answer reasonableness filters: a probability must lie between 0 and 1, a length must be positive, and a triangle's third side must be less than the sum and greater than the difference of the other two sides.