Praxis 5003 cheat sheet

Place value is base-ten: each position to the left is ten times the value of the position to its right, so 10 × 0.01 = 0.1 and 0.1 = 1/10.

Divisibility: by 2 if the last digit is even; by 5 if the last digit is 0 or 5; by 10 if the last digit is 0; by 4 if the last two digits form a multiple of 4.

Divisibility by 3 or 9 is tested with the digit sum: divisible by 3 if the digit sum is a multiple of 3, by 9 if the digit sum is a multiple of 9.

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

a/b ÷ c/d = a/b × d/c - dividing by a fraction is the same as multiplying by its reciprocal.

For any two positive integers, LCM(a, b) × GCF(a, b) = a × b.

Percent change = (new value − original value) ÷ original value × 100; the result is positive for an increase and negative for a decrease.

Multiplying a decimal by 10ⁿ shifts the decimal point n places to the right; dividing by 10ⁿ shifts it n places to the left.

Cross products: a/b = c/d exactly when ad = bc, which is the basis for solving proportions.

Slope m = (y₂ − y₁)/(x₂ − x₁); a horizontal line has slope 0 and a vertical line has undefined slope.

y = mx + b gives slope m and y-intercept b; y − y₁ = m(x − x₁) is the line through (x₁, y₁) with slope m.

A linear equation is solved by applying the same inverse operation to both sides until the variable stands alone.

A system of two linear equations has no solution when the lines are parallel (same slope, different intercepts), infinitely many solutions when both equations describe the same line, and exactly one solution otherwise.

Exponent rules: xᵃ · xᵇ = xᵃ⁺ᵇ, xᵃ ÷ xᵇ = xᵃ⁻ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁰ = 1 for x ≠ 0, and x⁻ᵃ = 1/xᵃ.

Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ (2a); the discriminant b² − 4ac yields two, one, or zero real solutions when it is positive, zero, or negative.

A relation is a function when each input has exactly one output; the vertical line test identifies a function from its graph.

A proportional relationship has the form y = kx, a constant ratio y/x, and a graph that is a straight line through the origin.

Distribute a(b + c) = ab + ac; when factoring a polynomial, first remove the greatest common factor of all terms.

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

Circle: circumference C = 2πr = πd and area A = πr².

Volume formulas: prism or cylinder V = Bh, pyramid or cone V = ⅓Bh, sphere V = (4/3)πr³.

Surface area: rectangular prism = 2(lw + lh + wh); cylinder = 2πr² + 2πrh.

The Pythagorean theorem a² + b² = c² applies only to right triangles, with c the hypotenuse; its converse tests whether a triangle is right.

Interior angles of a polygon with n sides sum to (n − 2) × 180°; each interior angle of a regular n-gon measures (n − 2) × 180° ÷ n.

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

Similar figures have congruent corresponding angles and proportional sides; a linear scale factor k gives area ratio k² and volume ratio k³.

Conversions: 1 in = 2.54 cm, 1 km = 1,000 m, 1 kg = 1,000 g, 1 L = 1,000 mL; convert by multiplying by a ratio equal to 1.

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

The mean is pulled toward outliers while the median resists them, so comparing the two indicates skew.

The five-number summary is minimum, Q1, median, Q3, maximum; the interquartile range IQR = Q3 − Q1 spans the middle 50% of the data.

Probability of an event = favorable outcomes ÷ total equally likely outcomes, and the value always lies between 0 and 1.

Complement and addition rules: P(not A) = 1 − P(A) and P(A or B) = P(A) + P(B) − P(A and B).

For independent events P(A and B) = P(A) × P(B); for dependent events multiply by the conditional probability P(B|A).

Bar graphs display categorical data, histograms display numerical data grouped into intervals, and line graphs show change over time.

A y-axis that does not start at zero exaggerates differences; axis scale, labels, and intervals determine whether a graph is misleading.

Theoretical probability comes from equally likely outcomes, while experimental probability is the observed relative frequency and approaches the theoretical value as trials increase.