These are the meta-strategies and reference tables that apply across all Quant question types. Mastering these separates 600-level thinking from 700+ thinking.

A) Estimation: Bounding Technique

When exact computation is slow, bracket the answer between a lower and upper bound to eliminate choices. Round one factor down and one up so you know which direction the rounding went, then check if choices fall outside the range.

• Example: is 7.8 × 31.2 closer to 240 or 243? Lower bound: 7 × 31 = 217. Upper bound: 8 × 32 = 256. Answer ≈ 243.4 — confirmed it's between those bounds and closer to 243.
• DS application: if Statement 1 gives a range like 10 ≤ x ≤ 20 and you need to determine if x > 15, bounding shows you can't conclude either way → Insufficient.

B) ZONEF: Test These Five Values First

When a question asks "must be true" or "could be true" for a variable, test these five cases before concluding:

• Zero (x = 0) — breaks many assumptions about sign and positivity
• One (x = 1) — breaks exponent assumptions (x² = x when x=1)
• Negative (x = −2) — flips inequalities, changes sign of products
• Extreme (x = 100 or x = −100) — reveals behavior at scale
• Fraction (x = 1/2, also x = −1/2) — fractions between 0 and 1 are smaller when squared

DS application: to prove a statement insufficient, find two values allowed by that statement that give different answers to the question. ZONEF tells you which values to try first.

• Zero kills it — Is n² > n? | Statement: n ≥ 0. Try n = 0: 0 > 0? No. Try n = 2: 4 > 2? Yes. Two different answers → Insufficient.
• One kills it — Is x² > x? | Statement: x > 0. Try x = 1: 1 > 1? No. Try x = 3: 9 > 3? Yes. → Insufficient.
• Negative kills it — Is √(x²) = x? | Statement: x² = 9. Try x = 3: √9 = 3 ✓. Try x = −3: √9 = 3 ≠ −3 ✗. → Insufficient.
• Fraction kills it — Is x² > x? | Statement: 0 < x < 2. Try x = 1/2: 1/4 > 1/2? No. Try x = 3/2: 9/4 > 3/2? Yes. → Insufficient.
• Negative fraction kills it — Is |x| > x? | Statement: x < 1. Try x = 1/2: |1/2| = 1/2 > 1/2? No. Try x = −1/2: |−1/2| = 1/2 > −1/2? Yes. → Insufficient.

Two values from the same statement → two different answers → statement is insufficient. You only need one such pair to eliminate it.

C) Backsolving

For word problems with numeric answer choices: plug in answers and work backwards. Start with choice C (middle value). If C is too large, try A or B; too small, try D or E. Saves setup time on complex word problems.

D) Smart Numbers

• Percent problems: use 100 as the base value
• Fraction problems: use the LCM of all denominators
• Ratio problems: use the ratio values directly as actual values
• Avoid 0 and 1 when testing general rules (they have special properties). Use ZONEF to deliberately test them for must-be-true questions.

E) Estimation Benchmarks

• √2 ≈ 1.414   √3 ≈ 1.732   √5 ≈ 2.236   π ≈ 3.14
• 1/7 ≈ 0.143   1/9 ≈ 0.111   1/11 ≈ 0.091   1/12 ≈ 0.083
• 1/6 ≈ 0.167   1/8 = 0.125   3/8 = 0.375   5/8 = 0.625   7/8 = 0.875

F) Perfect Squares — with Square Roots

Memorize these. On the GMAT you will need to recognize perfect squares instantly and know their square roots without computing.

• 1² = 1  → √1 = 1
• 2² = 4  → √4 = 2
• 3² = 9  → √9 = 3
• 4² = 16 → √16 = 4
• 5² = 25 → √25 = 5
• 6² = 36 → √36 = 6
• 7² = 49 → √49 = 7
• 8² = 64 → √64 = 8
• 9² = 81 → √81 = 9
• 10² = 100 → √100 = 10
• 11² = 121 → √121 = 11
• 12² = 144 → √144 = 12
• 13² = 169 → √169 = 13
• 14² = 196 → √196 = 14
• 15² = 225 → √225 = 15
• 16² = 256 → √256 = 16
• 17² = 289 → √289 = 17
• 18² = 324 → √324 = 18
• 19² = 361 → √361 = 19
• 20² = 400 → √400 = 20
• 21² = 441 → √441 = 21
• 22² = 484 → √484 = 22
• 23² = 529 → √529 = 23
• 24² = 576 → √576 = 24
• 25² = 625 → √625 = 25

Tip: If you see a large number and need to check if it's a perfect square, check if its last digit is 0, 1, 4, 5, 6, or 9. Numbers ending in 2, 3, 7, or 8 are never perfect squares.

G) Perfect Cubes — with Cube Roots

• 1³ = 1  → ∛1 = 1
• 2³ = 8  → ∛8 = 2
• 3³ = 27 → ∛27 = 3
• 4³ = 64 → ∛64 = 4
• 5³ = 125 → ∛125 = 5
• 6³ = 216 → ∛216 = 6
• 7³ = 343 → ∛343 = 7
• 8³ = 512 → ∛512 = 8
• 9³ = 729 → ∛729 = 9
• 10³ = 1000 → ∛1000 = 10

H) Powers of 2 and 3

• Powers of 2: 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024
• Powers of 3: 3¹=3, 3²=9, 3³=27, 3⁴=81, 3⁵=243, 3⁶=729

I) Factorials

1! = 1, 2! = 2, 3! = 6, 4! = 24, 5! = 120, 6! = 720, 7! = 5040, 8! = 40320, 9! = 362880, 10! = 3628800

J) Key Number Property Facts

• Any integer squared has remainder 0 or 1 when divided by 4 (x² mod 4 ∈ {0,1})
• If n is prime and n > 3, then n² − 1 is divisible by 24
• Product of any k consecutive integers is divisible by k!
• If n = p^a × q^b × …, number of factors = (a+1)(b+1)…
• Trailing zeros in n! = ⌊n/5⌋ + ⌊n/25⌋ + ⌊n/125⌋ + …