GMAT Formula Sheet
Quantitative Reasoning and Data Insights — one-page reference for the Focus Edition
Percent change
- Change:
- (new − original) ÷ original. The base is always the original.
- Apply:
- rise r% → × (1 + r/100); fall f% → × (1 − f/100).
- Successive:
- multiply the factors. +25% then −20% → 1.25 × 0.80 = 1.00, i.e. no net change — not +5%.
- Undo a fall:
- a fall of f% needs a rise of f/(100 − f) × 100 percent.
- Trap:
- a bigger percentage rise is not a bigger absolute rise — the bases differ.
Ratios and mixtures
- Split:
- T in a : b gives shares T·a/(a+b) and T·b/(a+b).
- Chain:
- (a/b)(b/c) = a/c.
- Mixtures:
- track the quantity that does not change; adding water leaves the syrup fixed.
Interest
- Simple:
- I = P·r·t.
- Compound:
- A = P(1 + r/n)^(nt); interest alone is A − P.
- Doubling:
- roughly 72 ÷ r periods.
Algebra identities
- Squares:
- (a ± b)² = a² ± 2ab + b²; (a+b)(a−b) = a² − b².
- Rearrange:
- a² + b² = (a+b)² − 2ab.
- Reciprocal:
- if x + 1/x = k then x² + 1/x² = k² − 2.
- Cubes:
- a³ ± b³ = (a ± b)(a² ∓ ab + b²).
- Quadratic:
- x = (−b ± √(b² − 4ac)) ÷ 2a; sum −b/a, product c/a.
Exponents and roots
- Rules:
- xᵃ·xᵇ = xᵃ⁺ᵇ; xᵃ/xᵇ = xᵃ⁻ᵇ; (xᵃ)ᵇ = xᵃᵇ; x⁰ = 1; x⁻ᵃ = 1/xᵃ.
- Roots:
- x^(1/n) = ⁿ√x; √(ab) = √a·√b for a, b ≥ 0.
- Trap:
- √(a + b) ≠ √a + √b.
Rates and work
- Core:
- distance = rate × time. Rates add; times do not.
- Together:
- a and b hours alone → ab/(a+b) hours together.
- Relative:
- same direction subtract speeds, opposite add. Catch-up = gap ÷ speed difference.
- Average speed:
- total distance ÷ total time. Equal distances at u, v → 2uv/(u+v).
- Trap:
- never average the two speeds unless equal times were spent at each.
Number properties
- Divisors:
- from p^a·q^b, count = (a+1)(b+1).
- GCD/LCM:
- GCD × LCM = a × b.
- Tests:
- 3 or 9 — digit sum; 4 — last two digits; 8 — last three; 11 — alternating digit sum.
- Remainder:
- n = dq + r; substitute and reduce.
- Parity:
- odd×odd = odd; anything×even = even. 3n even ⇒ n even.
- Squares:
- all exponents even. Least n with n² divisible by k: halve each exponent, round up.
Statistics
- Mean:
- sum ÷ count — most useful as sum = mean × count.
- Median:
- middle of the ordered list; order first.
- Weighted:
- (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂); lands nearer the larger group.
- SD:
- add c → unchanged; multiply by k → × |k|; identical values → 0.
- Spaced sets:
- mean = median = (first + last) ÷ 2.
- Signal:
- mean well above median means a high outlier.
Probability and counting
- Basic:
- favourable ÷ total; P(not A) = 1 − P(A).
- At least one:
- 1 − P(none) — almost always faster.
- And / or:
- P(A and B) = P(A)·P(B|A); P(A or B) = P(A) + P(B) − P(both).
- No replacement:
- both numerator and denominator drop: (4/10)(3/9) = 2/15.
- Combinations:
- C(n,k) = n! ÷ (k!(n−k)!); permutations P(n,k) = n! ÷ (n−k)!.
- Repeats:
- divide by the factorial of each repeat count — LEVEL → 5!/(2!2!) = 30.
Inequalities
- Sign flip:
- multiplying or dividing by a negative reverses the inequality.
- Ranges:
- for x − y, pair extremes in opposition: least x minus greatest y, and vice versa.
- Powers:
- x² > 4 ⇒ x > 2 or x < −2; x³ > 0 ⇒ x > 0.
- Absolute:
- |x − a| < b ⇔ a − b < x < a + b.
Data Sufficiency method
- Order:
- read the stem, test (1) alone, test (2) alone forgetting (1), then together.
- Answers:
- (1) only A · (2) only B · both needed C · each alone D · neither E.
- Yes/no:
- a consistent NO is sufficient. Insufficient means sometimes yes, sometimes no.
- Disprove:
- find two permitted values giving different answers — try negatives, zero, fractions.
- Assume nothing:
- integer, positive and distinct apply only if stated.