TSIA2 Math cheat sheet

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

Percent change = (new value − original value) ÷ original value × 100%; positive results are increases, negative results are decreases.

Part = percent × whole, so percent = part ÷ whole × 100 and whole = part ÷ percent (used as a decimal).

To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c.

A proportion a/b = c/d is solved by cross-multiplying: ad = bc; then divide to isolate the unknown.

A unit rate expresses a comparison per one unit of the second quantity, found by dividing the first quantity by the second.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10 and n an integer; moving the decimal left increases n.

Increasing a quantity by p percent and then decreasing by p percent gives a net factor of (1 + p)(1 − p) = 1 − p², which is less than 1.

|x| is the distance from x to 0 on the number line, so |x| = k (k > 0) has two solutions, x = k and x = −k.

Slope: m = (y₂ − y₁)/(x₂ − x₁); slope-intercept form is y = mx + b with y-intercept (0, b).

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

Point-slope form: y − y₁ = m(x − x₁) writes the line through (x₁, y₁) with slope m.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol.

Exponent rules: x^a · x^b = x^(a+b), (x^a)^b = x^(ab), x^0 = 1 for x ≠ 0, and x^(−a) = 1/x^a.

Quadratic formula: x = (−b ± √(b² − 4ac))/(2a) solves ax² + bx + c = 0 when a ≠ 0.

The discriminant b² − 4ac gives the number of real solutions: positive → two, zero → one, negative → none.

Factoring patterns: x² + (p + q)x + pq = (x + p)(x + q), and a² − b² = (a − b)(a + b).

Systems of two linear equations are solved by substitution (solve one for a variable) or elimination (add multiples to cancel a variable).

Area formulas: rectangle lw, triangle ½bh (height perpendicular to base), trapezoid ½(b₁ + b₂)h, circle πr².

Perimeter and circumference: rectangle 2l + 2w, circle C = 2πr = πd.

Volume: rectangular prism lwh, cylinder πr²h, cone (1/3)πr²h, sphere (4/3)πr³.

Surface area of a rectangular prism = 2(lw + lh + wh); add the areas of all faces for any solid.

Pythagorean theorem: a² + b² = c² in a right triangle, where c is the hypotenuse; common triples include 3-4-5, 5-12-13, 8-15-17.

Similar figures have proportional corresponding sides with scale factor k; areas scale by k² and volumes by k³.

Angle facts: triangle interior angles sum to 180°, angles on a straight line sum to 180°, vertical angles are equal, complementary angles sum to 90°.

Distance between two points = √((x₂ − x₁)² + (y₂ − y₁)²); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

An inscribed angle in a circle is half the measure of the central angle that intercepts the same arc.

Mean = sum of values ÷ number of values; median = middle value of ordered data; mode = most frequent value; range = maximum − minimum.

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

Complement rule: P(not A) = 1 − P(A).

Independent events: P(A and B) = P(A) × P(B); mutually exclusive events: P(A or B) = P(A) + P(B).

Expected value = sum of (each outcome × its probability).

Counting principle: multiply the number of options at each stage; permutations nPr = n!/(n − r)! order matters; combinations nCr = n!/(r!(n − r)!) order does not.

An outlier pulls the mean toward it far more than the median, so the median better describes the center of skewed data.

Standard deviation measures typical distance of values from the mean; a larger value means greater spread, not a larger center.

Random sampling best represents a population, and larger samples generally produce less variable statistics; correlation does not imply causation.