GRE Formula Sheet

Quantitative Reasoning — one-page reference for the GRE General Test

Medharu

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% → 1.00, i.e. no net change.
Undo a fall:
a fall of f% needs a rise of f/(100 − f) × 100 percent.

Ratios, mixtures, interest

Split:
T in a : b gives T·a/(a+b) and T·b/(a+b).
Chain:
(a/b)(b/c) = a/c.
Mixtures:
track the quantity that does not change.
Simple:
I = P·r·t.
Compound:
A = P(1 + r/n)^(nt).

Algebra identities

Squares:
(a ± b)² = a² ± 2ab + b²; (a+b)(a−b) = a² − b².
Rearrange:
a² + b² = (a+b)² − 2ab; x + 1/x = k ⇒ x² + 1/x² = k² − 2.
Quadratic:
x = (−b ± √(b² − 4ac)) ÷ 2a; sum −b/a, product c/a.
Exponents:
xᵃ·xᵇ = xᵃ⁺ᵇ; (xᵃ)ᵇ = xᵃᵇ; x⁻ᵃ = 1/xᵃ; x^(1/n) = ⁿ√x.
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.
Relative:
same direction subtract, opposite add; catch-up = gap ÷ difference.
Average speed:
total distance ÷ total time; equal distances → 2uv/(u+v).

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; 11 — alternating digit sum.
Remainder:
n = dq + r; substitute and reduce.
Parity:
odd×odd = odd; anything×even = even.

Statistics

Mean:
sum ÷ count; use sum = mean × count.
Median:
middle of the ordered list — order first.
SD:
add c → unchanged; multiply by k → × |k|.
Quartiles:
IQR = Q3 − Q1; the kth percentile has k% at or below it.
Spaced sets:
mean = median = (first + last) ÷ 2.

Probability and counting

Basic:
favourable ÷ total; P(not A) = 1 − P(A).
At least one:
1 − P(none).
Dependent:
P(A and B) = P(A)·P(B|A); without replacement both parts drop.
Choose:
C(n,k) = n! ÷ (k!(n−k)!); order matters → P(n,k) = n! ÷ (n−k)!.
Repeats:
divide by the factorial of each repeat count.

Geometry — triangles

Area:
½ × base × height; angles sum to 180°.
Pythagoras:
a² + b² = c².
Triples:
3-4-5, 5-12-13, 8-15-17, 7-24-25 and multiples.
Special:
45-45-90 → 1 : 1 : √2; 30-60-90 → 1 : √3 : 2.
Inequality:
each side is between the sum and the difference of the other two.
Similar:
sides in proportion, areas in the square of that proportion.

Geometry — figures and solids

Rectangle:
area lw; perimeter 2(l + w). Square diagonal s√2.
Parallelogram:
area = base × height. Trapezoid: ½(b₁ + b₂)h.
Circle:
circumference 2πr; area πr².
Arc / sector:
(θ/360)·2πr and (θ/360)·πr².
Angles:
inscribed angle is half the central angle; an angle in a semicircle is 90°.
Solids:
box lwh; cylinder πr²h; sphere (4/3)πr³, surface 4πr².
Polygons:
interior angles sum to (n − 2)·180°.
Trap:
figures are not to scale unless stated — reason from the values.

Coordinate geometry

Distance:
√((x₂−x₁)² + (y₂−y₁)²).
Midpoint:
((x₁+x₂)/2, (y₁+y₂)/2) — average, do not just add.
Slope:
m = (y₂−y₁)/(x₂−x₁); parallel equal, perpendicular multiply to −1.
Line:
y = mx + b; y − y₁ = m(x − x₁).
Circle:
(x−h)² + (y−k)² = r².
Parabola:
y = ax² + bx + c; vertex at x = −b/2a.

Inequalities

Sign flip:
multiplying or dividing by a negative reverses the inequality.
Ranges:
for x − y pair extremes in opposition.
Powers:
x² > 4 ⇒ x > 2 or x < −2; x³ > 0 ⇒ x > 0.
Absolute:
|x − a| < b ⇔ a − b < x < a + b.

Quantitative Comparison method

Choices:
A — A greater · B — B greater · C — equal · D — cannot be determined.
Method:
simplify both sides the same way rather than computing each in full.
Test values:
try 0, 1, a negative, and a fraction. Two different verdicts ⇒ D.
Careful:
if a variable may be negative or fractional, the answer is very often D.
Note:
D is never right when both quantities are pure numbers — those always compare.