GMAT Quantitative Reasoning cheat sheet

Divisibility: divisible by 3 iff digit sum is divisible by 3; by 9 iff digit sum is divisible by 9; by 4 iff last two digits form a multiple of 4; by 8 iff last three digits do.

Divisibility by 6 requires divisibility by both 2 and 3; by 12 requires both 3 and 4; by 15 requires both 3 and 5.

Prime factorization: divide repeatedly by primes 2, 3, 5, 7, 11, ... up to the square root of the remaining number to test primality.

LCM times GCF of two positive integers equals the product of the two integers: LCM(a,b) x GCF(a,b) = a x b.

Percent change = (new - old) / old x 100. A p% increase multiplies by (1 + p/100); a p% decrease multiplies by (1 - p/100).

Successive percent changes multiply, not add: a 20% rise then a 20% fall yields a factor of 1.2 x 0.8 = 0.96, a net 4% decrease.

Ratios: if a:b = 3:5, write parts as 3x and 5x; the total is 8x and the difference is 2x, so any one piece determines all others.

Average of evenly spaced values equals the median, which is (first + last)/2; the count of terms is ((last - first)/step) + 1.

Powers and roots: x^a times x^b = x^(a+b); x^a / x^b = x^(a-b); (x^a)^b = x^(ab); x^0 = 1 for nonzero x; x^(-a) = 1/x^a.

Quadratic formula: for ax^2 + bx + c = 0 with a not 0, x = [-b +/- sqrt(b^2 - 4ac)] / (2a).

Discriminant b^2 - 4ac: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Sum of roots = -b/a and product of roots = c/a for ax^2 + bx + c = 0.

Difference of squares: a^2 - b^2 = (a - b)(a + b). Perfect square: (a +/- b)^2 = a^2 +/- 2ab + b^2.

Slope of a line through (x1,y1) and (x2,y2) is (y2 - y1)/(x2 - x1); parallel lines share slope, perpendicular slopes multiply to -1.

Two linear equations in two variables have a unique solution when their slopes differ; none when slopes are equal but intercepts differ; infinitely many when the equations are equivalent.

Absolute value: |x| = k with k > 0 gives x = k or x = -k; |x| < k means -k < x < k; |x| > k means x > k or x < -k.

Exponential growth/decay: A = P(1 + r)^t for growth and A = P(1 - r)^t for decay, where r is the per-period rate and t the number of periods.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.

Area formulas: rectangle lw; triangle (1/2)bh; parallelogram bh; trapezoid (1/2)(b1 + b2)h; circle pi r^2; circumference 2 pi r.

Pythagorean theorem: a^2 + b^2 = c^2 for right triangles; common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Special right triangles: 45-45-90 has sides in ratio x : x : x sqrt(2); 30-60-90 has sides in ratio x : x sqrt(3) : 2x.

Sum of interior angles of an n-sided polygon is (n - 2) x 180 degrees; each exterior angle of a regular n-gon is 360/n degrees.

Volume: rectangular solid lwh; cylinder pi r^2 h; sphere (4/3) pi r^3; cone (1/3) pi r^2 h. Surface area of a sphere is 4 pi r^2.

Inscribed angle equals half its intercepted arc; a triangle inscribed in a semicircle is right-angled at the circumference.

Similar figures: corresponding lengths scale by k, areas by k^2, and volumes by k^3.

Coordinate geometry: distance between points = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint = ((x1 + x2)/2, (y1 + y2)/2).

An equation of a circle centered at (h,k) with radius r is (x - h)^2 + (y - k)^2 = r^2.

Rate problems: distance = rate x time. For combined work, add rates: 1/t_total = 1/t1 + 1/t2 for two workers.

Mixture problems: track the amount of each component; (volume A)(concentration A) + (volume B)(concentration B) = (total volume)(final concentration).

Weighted average = sum of (value x weight) / sum of weights; it lies between the smallest and largest values, closer to the larger weight.

Simple interest: I = P r t. Compound interest: A = P(1 + r/n)^(nt), where n is the number of compounding periods per year.

Profit = revenue - cost; markup and discount are percentages of cost and original price respectively, applied multiplicatively in sequence.

Set problems: total = group A + group B - both + neither. For three sets, inclusion-exclusion subtracts pairwise overlaps and adds back the triple overlap.

Consecutive integers n, n+1, n+2: the sum is 3n + 3, so it is always divisible by 3; the product of any k consecutive integers is divisible by k!.

Age problems: set up equations using a fixed time offset, expressing each person's age at the target time as current age plus or minus the same number of years.

Answer choices: (A) statement 1 alone suffices; (B) statement 2 alone suffices; (C) both together suffice but neither alone; (D) each alone suffices; (E) together still insufficient.

A single value is required for sufficiency; if a statement yields two or more possible values for the target, it is insufficient.

Never carry information from one statement into the other; evaluate each statement independently against the question stem.

Do not solve the problem fully; determine only whether the data fixes a unique answer, stopping as soon as sufficiency is decided.

Yes/no questions require a definite yes or definite no; a statement that allows both a yes and a no case is insufficient.

Rephrase the question before evaluating statements, simplifying the stem to the minimal quantity needed.

Test edge cases: zero, negative numbers, fractions between 0 and 1, and equal values, since these frequently break an apparently sufficient statement.

In geometry figures, do not assume lengths, right angles, or parallelism unless stated; a statement that fixes shape but not scale may still be insufficient for a length.

Divisibility: divisible by 3 iff digit sum is divisible by 3; by 9 iff digit sum is divisible by 9; by 4 iff last two digits form a multiple of 4; by 8 iff last three digits do.

Divisibility by 6 requires divisibility by both 2 and 3; by 12 requires both 3 and 4; by 15 requires both 3 and 5.

Prime factorization: divide repeatedly by primes 2, 3, 5, 7, 11, ... up to the square root of the remaining number to test primality.

LCM times GCF of two positive integers equals the product of the two integers: LCM(a,b) x GCF(a,b) = a x b.

Percent change = (new - old) / old x 100. A p% increase multiplies by (1 + p/100); a p% decrease multiplies by (1 - p/100).

Successive percent changes multiply, not add: a 20% rise then a 20% fall yields a factor of 1.2 x 0.8 = 0.96, a net 4% decrease.

Ratios: if a:b = 3:5, write parts as 3x and 5x; the total is 8x and the difference is 2x, so any one piece determines all others.

Average of evenly spaced values equals the median, which is (first + last)/2; the count of terms is ((last - first)/step) + 1.

Powers and roots: x^a times x^b = x^(a+b); x^a / x^b = x^(a-b); (x^a)^b = x^(ab); x^0 = 1 for nonzero x; x^(-a) = 1/x^a.

Quadratic formula: for ax^2 + bx + c = 0 with a not 0, x = [-b +/- sqrt(b^2 - 4ac)] / (2a).

Discriminant b^2 - 4ac: positive gives two distinct real roots, zero gives one repeated root, negative gives no real roots.

Sum of roots = -b/a and product of roots = c/a for ax^2 + bx + c = 0.

Difference of squares: a^2 - b^2 = (a - b)(a + b). Perfect square: (a +/- b)^2 = a^2 +/- 2ab + b^2.

Slope of a line through (x1,y1) and (x2,y2) is (y2 - y1)/(x2 - x1); parallel lines share slope, perpendicular slopes multiply to -1.

Two linear equations in two variables have a unique solution when their slopes differ; none when slopes are equal but intercepts differ; infinitely many when the equations are equivalent.

Absolute value: |x| = k with k > 0 gives x = k or x = -k; |x| < k means -k < x < k; |x| > k means x > k or x < -k.

Exponential growth/decay: A = P(1 + r)^t for growth and A = P(1 - r)^t for decay, where r is the per-period rate and t the number of periods.

Inequalities: multiplying or dividing both sides by a negative number reverses the inequality sign.

Area formulas: rectangle lw; triangle (1/2)bh; parallelogram bh; trapezoid (1/2)(b1 + b2)h; circle pi r^2; circumference 2 pi r.

Pythagorean theorem: a^2 + b^2 = c^2 for right triangles; common triples include 3-4-5, 5-12-13, 8-15-17, and 7-24-25.

Special right triangles: 45-45-90 has sides in ratio x : x : x sqrt(2); 30-60-90 has sides in ratio x : x sqrt(3) : 2x.

Sum of interior angles of an n-sided polygon is (n - 2) x 180 degrees; each exterior angle of a regular n-gon is 360/n degrees.

Volume: rectangular solid lwh; cylinder pi r^2 h; sphere (4/3) pi r^3; cone (1/3) pi r^2 h. Surface area of a sphere is 4 pi r^2.

Inscribed angle equals half its intercepted arc; a triangle inscribed in a semicircle is right-angled at the circumference.

Similar figures: corresponding lengths scale by k, areas by k^2, and volumes by k^3.

Coordinate geometry: distance between points = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint = ((x1 + x2)/2, (y1 + y2)/2).

An equation of a circle centered at (h,k) with radius r is (x - h)^2 + (y - k)^2 = r^2.

Rate problems: distance = rate x time. For combined work, add rates: 1/t_total = 1/t1 + 1/t2 for two workers.

Mixture problems: track the amount of each component; (volume A)(concentration A) + (volume B)(concentration B) = (total volume)(final concentration).

Weighted average = sum of (value x weight) / sum of weights; it lies between the smallest and largest values, closer to the larger weight.

Simple interest: I = P r t. Compound interest: A = P(1 + r/n)^(nt), where n is the number of compounding periods per year.

Profit = revenue - cost; markup and discount are percentages of cost and original price respectively, applied multiplicatively in sequence.

Set problems: total = group A + group B - both + neither. For three sets, inclusion-exclusion subtracts pairwise overlaps and adds back the triple overlap.

Consecutive integers n, n+1, n+2: the sum is 3n + 3, so it is always divisible by 3; the product of any k consecutive integers is divisible by k!.

Age problems: set up equations using a fixed time offset, expressing each person's age at the target time as current age plus or minus the same number of years.

Answer choices: (A) statement 1 alone suffices; (B) statement 2 alone suffices; (C) both together suffice but neither alone; (D) each alone suffices; (E) together still insufficient.

A single value is required for sufficiency; if a statement yields two or more possible values for the target, it is insufficient.

Never carry information from one statement into the other; evaluate each statement independently against the question stem.

Do not solve the problem fully; determine only whether the data fixes a unique answer, stopping as soon as sufficiency is decided.

Yes/no questions require a definite yes or definite no; a statement that allows both a yes and a no case is insufficient.

Rephrase the question before evaluating statements, simplifying the stem to the minimal quantity needed.

Test edge cases: zero, negative numbers, fractions between 0 and 1, and equal values, since these frequently break an apparently sufficient statement.

In geometry figures, do not assume lengths, right angles, or parallelism unless stated; a statement that fixes shape but not scale may still be insufficient for a length.