Math Guides
Comprehensive guides for algebra, arithmetic, geometry, and more.
What Are Math Guides?
Math guides are structured references that explain concepts step by step — from basic arithmetic to algebra and geometry. Each section below teaches the idea, shows the formula, walks through an example, and links to a free calculator so you can practice immediately.
Arithmetic & Number Sense
Fractions
A fraction represents parts of a whole: a/b means a parts out of b equal pieces. To add fractions, find a common denominator. To multiply, multiply numerators and denominators: (2/3) × (4/5) = 8/15.
Example: Simplify 12/18 → GCF is 6 → 2/3
Percentages
Percent means "per hundred." 25% = 25/100 = 0.25. To find X% of Y: (X/100) × Y.
Example: 15% of 240 = 0.15 × 240 = 36
Ratios & Proportions
A ratio compares two quantities (3:2). A proportion states two ratios are equal: a/b = c/d. Cross-multiply to solve: ad = bc.
Example: If 3/4 = x/12, then x = (3×12)/4 = 9
Fraction Calculator · Percentage Calculator · Ratio Calculator
Algebra Essentials
Linear Equations
Form: ax + b = 0. Solution: x = −b/a (when a ≠ 0). Isolate the variable by doing the same operation to both sides.
Example: 2x + 6 = 0 → x = −6/2 = −3
Quadratic Equations
Form: ax² + bx + c = 0. Use the quadratic formula when factoring is difficult.
Example: x² − 5x + 6 = 0 → x = 2 or 3
Geometry Fundamentals
| Shape | Area Formula | Example |
|---|---|---|
| Rectangle | A = l × w | 5×8 = 40 sq units |
| Triangle | A = ½ × b × h | ½×6×4 = 12 |
| Circle | A = πr² | π×5² ≈ 78.54 |
| Right triangle | a² + b² = c² | 3²+4² = 5² |
Area Calculator · Triangle Calculator · Pythagorean Theorem Guide
Scientific Notation & Precision
Scientific notation writes numbers as a × 10ⁿ where 1 ≤ a < 10. Significant figures reflect measurement precision — round your final answer to match the least precise input.
- Non-zero digits are always significant
- Zeros between non-zero digits are significant (1002 has 4 SF)
- Leading zeros are NOT significant (0.0045 has 2 SF)
Common Mistakes in Math
- Order of operations: PEMDAS — Parentheses, Exponents, Multiply/Divide, Add/Subtract
- Sign errors: −(−3) = +3, but −3² = −9 while (−3)² = 9
- Rounding too early: keep extra digits until the final step
- Wrong percentage base: "increase by 20%" multiplies by 1.20, not 0.20