Every high-yield formula and fact in one place. Type to filter.
| Concept | Formula / Fact |
|---|---|
| Number of factors of n | For n = pa·qb·…, factors = (a+1)(b+1)… |
| GCF × LCM | GCF(a,b) × LCM(a,b) = a × b |
| Division algorithm | n = d·q + r, with 0 ≤ r < d |
| Even/odd | even ± even = even; odd ± odd = even; even × anything = even |
| Sign of product | negative if an odd number of negative factors |
| Absolute value | |x| ≥ 0; |x| = k → x = ±k; |ab| = |a||b| |
| Special facts | 0 is even; 1 is not prime; 2 is the only even prime; division by 0 undefined |
| Concept | Formula / Fact |
|---|---|
| Add fractions (bowtie) | a/b + c/d = (ad + cb)/(bd) |
| Divide fractions | (a/b) ÷ (c/d) = (a/b)(d/c) |
| Compare fractions | a/b vs c/d → compare ad vs cb |
| Proportion | a/b = c/d → ad = bc |
| Ratio to whole | parts a:b → a is a/(a+b) of the total |
| Key conversions | 1/8 = 12.5%, 1/6 ≈ 16.7%, 1/3 ≈ 33.3%, 3/8 = 37.5%, 5/8 = 62.5% |
| Concept | Formula / Fact |
|---|---|
| Percent of | x% of N = (x/100)·N |
| Percent change | (new − old) / old × 100 |
| Increase / decrease | multiply by (1 + p/100) or (1 − p/100) |
| Successive changes | multiply the factors (do not add percents) |
| Reverse percent | original = final ÷ multiplier |
| Symmetry | x% of y = y% of x |
| Simple interest | I = P·r·t |
| Compound interest | A = P(1 + r/n)nt |
| Profit percent | (sell − cost) / cost × 100 |
| Concept | Formula / Fact |
|---|---|
| Product / quotient | bm·bn = bm+n; bm/bn = bm−n |
| Power of power | (bm)n = bmn |
| Zero / negative | b0 = 1; b−n = 1/bn |
| Fractional exponent | bm/n = n-th root of bm |
| Radical product | √a · √b = √(ab) |
| 0 < x < 1 | x3 < x2 < x < √x (powers shrink, roots grow) |
| Scientific notation | a × 10n, 1 ≤ a < 10 |
| Concept | Formula / Fact |
|---|---|
| Square of a sum | (a + b)2 = a2 + 2ab + b2 |
| Square of a difference | (a − b)2 = a2 − 2ab + b2 |
| Difference of squares | a2 − b2 = (a + b)(a − b) |
| Quadratic formula | x = (−b ± √(b2 − 4ac)) / 2a |
| Sum / product of roots | r + s = −b/a; r·s = c/a |
| Discriminant | b2 − 4ac: >0 two roots, =0 one, <0 none |
| Concept | Formula / Fact |
|---|---|
| Distance | d = r·t |
| Average speed | total distance ÷ total time (not the average of speeds) |
| Combined work | 1/a + 1/b = 1/T |
| Mixture | c₁V₁ + c₂V₂ = c_final · V_final |
| Consecutive integers | n, n+1, n+2 … ; even/odd use n, n+2, n+4 |
| Concept | Formula / Fact |
|---|---|
| Triangle angles | sum = 180°; exterior = sum of remote interior |
| Triangle area | (1/2)·base·height |
| Pythagorean theorem | a2 + b2 = c2; triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 |
| 45-45-90 triangle | sides 1 : 1 : √2 |
| 30-60-90 triangle | sides 1 : √3 : 2 |
| Triangle inequality | |a − b| < third side < a + b |
| Polygon interior angles | sum = (n − 2)·180° |
| Equilateral area | (√3/4)·s2 |
| Circle | circumference = 2πr; area = πr2 |
| Arc / sector | arc = (θ/360)·2πr; sector = (θ/360)·πr2 |
| Rectangular box | V = lwh; diagonal = √(l2+w2+h2) |
| Cube | V = s3; surface = 6s2; diagonal = s√3 |
| Cylinder | V = πr2h; surface = 2πr2 + 2πrh |
| Similar figures (factor k) | lengths ×k, areas ×k2, volumes ×k3 |
| Concept | Formula / Fact |
|---|---|
| Distance | √((x₂−x₁)2 + (y₂−y₁)2) |
| Midpoint | ((x₁+x₂)/2, (y₁+y₂)/2) |
| Slope | (y₂−y₁)/(x₂−x₁) |
| Slope-intercept | y = mx + b |
| Point-slope | y − y₁ = m(x − x₁) |
| Parallel / perpendicular | equal slopes / negative reciprocals (m₁·m₂ = −1) |
| Parabola vertex form | y = (x − h)2 + k → vertex (h, k) |
| Concept | Formula / Fact |
|---|---|
| Mean | sum ÷ count; total = average × count |
| Median | middle of sorted data (avg of two middles if even count) |
| Range | max − min |
| Weighted average | Σ(value×weight) ÷ Σweight |
| IQR | Q3 − Q1 |
| Standard deviation | add a constant → SD unchanged; ×k → SD ×k |
| Normal distribution | 68% within 1 SD, 95% within 2 SD, 99.7% within 3 SD |
| Concept | Formula / Fact |
|---|---|
| Basic probability | favorable ÷ total (between 0 and 1) |
| Complement | P(not A) = 1 − P(A) |
| At least one | P(at least one) = 1 − P(none) |
| OR (overlap) | P(A or B) = P(A) + P(B) − P(A and B) |
| AND (independent) | P(A and B) = P(A)·P(B) |
| Counting principle | multiply choices at each stage |
| Permutations | nPr = n! / (n−r)! |
| Combinations | nCr = n! / [r!(n−r)!]; nCr = nC(n−r) |
| Two-set total | Neither = Total − (A + B − Both) |