GMAT Math Ultimate (Correct Answers)
Common Factors Correct AnswersBreak down both numbers to their prime factors to
see what factors they have in common. Multiply all combinations of shared prime factors
to find all common factors.
Gross Profit Correct AnswersGross profit = Selling Price - Cost
Combined Events Correct AnswersFor events E and F:
• not E = P(not E) = 1 - P(E)
• E or F = P(E or F) = P(E) + P(F) - P(E and F)
• E and F = P(E and F) = P(E)P(F)
Multiplication Principle Correct AnswersThe number of ways independent events can
occur together can be determined by multiplying together the number of possible
outcomes for each event.
1st Rule of Probability: Likelihood of A Correct AnswersBasic rule: The probability of
event A occurring is the number of outcomes that result in A divided by the total number
of possible outcomes.
2nd Rule of Probability: Complementary events Correct AnswersComplementary
Events: The probability of an event occurring plus the probability of the event not
occurring = 1.
P(E) = 1 - P(not E)
3rd Rule of Probability: Conditional Probability Correct AnswersConditional Probability:
The probability of event A AND event B occurring is the probability of event A times the
probability of event B, given that A has already occurred.
P(A and B) = P(A) × P(B|A)
4th Rule of Probability: Probability of A OR B Correct AnswersThe probability of event A
OR event B occurring is: the probability of event A occurring *plus* the probability of
event B occurring *minus* the probability of both
events occurring.
P(A or B) = P(A) + P(B) - P(A and B)
Probability of Multiple Events Correct AnswersRules:
• A *and* B < A *or* B
• A *or* B > Individual probabilities of A, B
• P(A and B) = P(A) x P(B) ← "fewer options"
• P(A or B) = P(A) + P(B) ← "more options"
Indistinguishable Events (i.e., anagrams with repeating letters) Correct AnswersTo find
the number of distinct permutations of a set of items with indistinguishable ("repeat")
items, divide the factorial of the items in the set by the product of the factorials of the
number of indistinguishable elements.
Example: How many ways can the letters in TRUST be arranged? (5!)/(2!) = 60
5! is the factorial of items in the set, 2! is the factorial of the number of repeat items
("T"s)
Combinations (Order Does Not Matter) Correct AnswersnCr = n! / (r! (n - r)!)
Where n is the total number of items in the set and r is the number of chosen items.
Permutations (Order Does Matter) Correct AnswersnPr = n! / (n - r)!
Where n is the total number of items in the set and r is the number of chosen items.
,Circular Permutations Correct AnswersThe number of ways to arrange n distinct objects
along a fixed circle is: (n - 1)!
Slope of a Line Correct Answersy = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
30-60-90 Triangle Correct Answers30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle Correct Answers45-45-90
x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles Correct Answers3-4-5 or 6-8-10 or 9-12-15
5-12-13
Number Added or Deleted Correct AnswersUse the mean to find number that was
added or deleted.
• Total = mean x (number of terms)
• Number deleted = (original total) - (new total)
• Number added = (new total) - (original total)
Factors of Odd Numbers Correct AnswersOdd numbers have only odd factors
Quadratic Formula Correct AnswersTo find roots of quadratic equation: ax^2+ bx + c =
0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant Correct AnswersQuadratic equation: ax^2+ bx + c = 0
Dicriminant = b^2 - 4ac
If discriminiant > 0, there are two roots (and two x-intercepts)
If discriminant = 0, there is one root (and one x-intercept)
If discriminant < 0, there are no (real) roots
Exponents Correct Answers(x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) Correct Answers1. Start by writing
each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Greatest Common Factor (GCF) is found by multiplying all factors appearing on
BOTH lists.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
GCF = 2 x 2 x 3 = 12
Prime Factorization: Lowest Common Multiple (LCM) Correct Answers1. Start by writing
each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Lowest common multiple found by multiplying all factors in EITHER list.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
Check for Prime Correct Answers1. Pick a number n.
2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is, your
number is not prime.
3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your number is
not prime.
, 4. Keep trying the next prime number until you reach one that is a factor (in which case
n is not prime), or you reach a prime number that is *equal to or greater than the square
root of n.*
5. If you have not found a number less than or equal to the square root of n, you can be
sure that your number is prime.
Ex: the number n=19 has a square root of ~4.35. Test 2, 3, 4 --> you know 19 is prime
because none of them are factors, and any other factor would be greater than sqrt(19).
Rate x Time = Distance (rt = d) Correct AnswersFor a fixed distance, the average speed
is inversely related to the amount of time required to make the trip.
Ex: Since Mieko's average speed was 3/4 of
Chan's, her time was 4/3 as long.
(3/4)r(4/3)t = d
Factoring Exponents Correct Answers(5^k)−(5^k−1)
(5^k)-(1/5)(5^k)
(5^k)(1 - 1/5)
(4/5)(5^k)
Squaring Fractions Correct AnswersWhen positive fractions between 0 and 1 are
squared, they get smaller.
Ex: (1/2)^2 = 1/4
Approximations of Common Square Roots Correct AnswersSquare root of 2 = 1.4
Square root of 3 = 1.7
Square root of 5 = 2.25
Inscribed Angle, Minor Arc Correct AnswersAn inscribed angle = two chords that have a
vertex on the circle
Inscribed angle with one chord as diameter = 35 degrees
Minor arc = 2 x inscribed angle = 70 degrees
Area of Trapezoid Correct AnswersA = (sum of bases)(height)/2
A = {[(b1 + b2)/2](height)}/2
Area of a Rhombus Correct AnswersA = bh OR
A = [(d1)(d2)]/2
Compound Interest Formula - Compounding Annually Correct AnswersTo compound
annually:
P = principal
r = rate of interest (in decimal form)
y = number of years
New value = P (1 + r)^y
Compound Interest Formula - Compounding More Than Annually Correct AnswersTo
compound multiple times per year:
P = principal
r = rate of interest (in decimal form)
y = number of years
n = number of times per year (i.e., compounded every 3 months would be n = 4)
FV = P (1 + r/n)^ny
Interest Problem: If $10,000 is invested at 10% annual interest, compounded semi-
annually, what is the balance after 1 year? Correct AnswersP = 10,000 r = .10 y = 1 n =
2
Common Factors Correct AnswersBreak down both numbers to their prime factors to
see what factors they have in common. Multiply all combinations of shared prime factors
to find all common factors.
Gross Profit Correct AnswersGross profit = Selling Price - Cost
Combined Events Correct AnswersFor events E and F:
• not E = P(not E) = 1 - P(E)
• E or F = P(E or F) = P(E) + P(F) - P(E and F)
• E and F = P(E and F) = P(E)P(F)
Multiplication Principle Correct AnswersThe number of ways independent events can
occur together can be determined by multiplying together the number of possible
outcomes for each event.
1st Rule of Probability: Likelihood of A Correct AnswersBasic rule: The probability of
event A occurring is the number of outcomes that result in A divided by the total number
of possible outcomes.
2nd Rule of Probability: Complementary events Correct AnswersComplementary
Events: The probability of an event occurring plus the probability of the event not
occurring = 1.
P(E) = 1 - P(not E)
3rd Rule of Probability: Conditional Probability Correct AnswersConditional Probability:
The probability of event A AND event B occurring is the probability of event A times the
probability of event B, given that A has already occurred.
P(A and B) = P(A) × P(B|A)
4th Rule of Probability: Probability of A OR B Correct AnswersThe probability of event A
OR event B occurring is: the probability of event A occurring *plus* the probability of
event B occurring *minus* the probability of both
events occurring.
P(A or B) = P(A) + P(B) - P(A and B)
Probability of Multiple Events Correct AnswersRules:
• A *and* B < A *or* B
• A *or* B > Individual probabilities of A, B
• P(A and B) = P(A) x P(B) ← "fewer options"
• P(A or B) = P(A) + P(B) ← "more options"
Indistinguishable Events (i.e., anagrams with repeating letters) Correct AnswersTo find
the number of distinct permutations of a set of items with indistinguishable ("repeat")
items, divide the factorial of the items in the set by the product of the factorials of the
number of indistinguishable elements.
Example: How many ways can the letters in TRUST be arranged? (5!)/(2!) = 60
5! is the factorial of items in the set, 2! is the factorial of the number of repeat items
("T"s)
Combinations (Order Does Not Matter) Correct AnswersnCr = n! / (r! (n - r)!)
Where n is the total number of items in the set and r is the number of chosen items.
Permutations (Order Does Matter) Correct AnswersnPr = n! / (n - r)!
Where n is the total number of items in the set and r is the number of chosen items.
,Circular Permutations Correct AnswersThe number of ways to arrange n distinct objects
along a fixed circle is: (n - 1)!
Slope of a Line Correct Answersy = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
30-60-90 Triangle Correct Answers30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle Correct Answers45-45-90
x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles Correct Answers3-4-5 or 6-8-10 or 9-12-15
5-12-13
Number Added or Deleted Correct AnswersUse the mean to find number that was
added or deleted.
• Total = mean x (number of terms)
• Number deleted = (original total) - (new total)
• Number added = (new total) - (original total)
Factors of Odd Numbers Correct AnswersOdd numbers have only odd factors
Quadratic Formula Correct AnswersTo find roots of quadratic equation: ax^2+ bx + c =
0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant Correct AnswersQuadratic equation: ax^2+ bx + c = 0
Dicriminant = b^2 - 4ac
If discriminiant > 0, there are two roots (and two x-intercepts)
If discriminant = 0, there is one root (and one x-intercept)
If discriminant < 0, there are no (real) roots
Exponents Correct Answers(x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) Correct Answers1. Start by writing
each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Greatest Common Factor (GCF) is found by multiplying all factors appearing on
BOTH lists.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
GCF = 2 x 2 x 3 = 12
Prime Factorization: Lowest Common Multiple (LCM) Correct Answers1. Start by writing
each number as product of its prime factors.
2. Write so that each new prime factor begins in same place.
3. Lowest common multiple found by multiplying all factors in EITHER list.
60 = 2 x 2 x 3 x 5
72 = 2 x 2 x 2 x 3 x 3
LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
Check for Prime Correct Answers1. Pick a number n.
2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is, your
number is not prime.
3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your number is
not prime.
, 4. Keep trying the next prime number until you reach one that is a factor (in which case
n is not prime), or you reach a prime number that is *equal to or greater than the square
root of n.*
5. If you have not found a number less than or equal to the square root of n, you can be
sure that your number is prime.
Ex: the number n=19 has a square root of ~4.35. Test 2, 3, 4 --> you know 19 is prime
because none of them are factors, and any other factor would be greater than sqrt(19).
Rate x Time = Distance (rt = d) Correct AnswersFor a fixed distance, the average speed
is inversely related to the amount of time required to make the trip.
Ex: Since Mieko's average speed was 3/4 of
Chan's, her time was 4/3 as long.
(3/4)r(4/3)t = d
Factoring Exponents Correct Answers(5^k)−(5^k−1)
(5^k)-(1/5)(5^k)
(5^k)(1 - 1/5)
(4/5)(5^k)
Squaring Fractions Correct AnswersWhen positive fractions between 0 and 1 are
squared, they get smaller.
Ex: (1/2)^2 = 1/4
Approximations of Common Square Roots Correct AnswersSquare root of 2 = 1.4
Square root of 3 = 1.7
Square root of 5 = 2.25
Inscribed Angle, Minor Arc Correct AnswersAn inscribed angle = two chords that have a
vertex on the circle
Inscribed angle with one chord as diameter = 35 degrees
Minor arc = 2 x inscribed angle = 70 degrees
Area of Trapezoid Correct AnswersA = (sum of bases)(height)/2
A = {[(b1 + b2)/2](height)}/2
Area of a Rhombus Correct AnswersA = bh OR
A = [(d1)(d2)]/2
Compound Interest Formula - Compounding Annually Correct AnswersTo compound
annually:
P = principal
r = rate of interest (in decimal form)
y = number of years
New value = P (1 + r)^y
Compound Interest Formula - Compounding More Than Annually Correct AnswersTo
compound multiple times per year:
P = principal
r = rate of interest (in decimal form)
y = number of years
n = number of times per year (i.e., compounded every 3 months would be n = 4)
FV = P (1 + r/n)^ny
Interest Problem: If $10,000 is invested at 10% annual interest, compounded semi-
annually, what is the balance after 1 year? Correct AnswersP = 10,000 r = .10 y = 1 n =
2
