GMAT Math Ultimate Exam with
Accurate Solutions
Common Factors - ANS-Break 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 - ANS-Gross profit = Selling Price - Cost
Combined Events - ANS-For 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 - ANS-The 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 - ANS-Basic 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 - ANS-Complementary 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 - ANS-Conditional 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 - ANS-The 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 - ANS-Rules:
• 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) - ANS-To 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) - ANS-nCr = 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) - ANS-nPr = n! / (n - r)!
Where n is the total number of items in the set and r is the number of chosen items.
Circular Permutations - ANS-The number of ways to arrange n distinct objects along a
fixed circle is: (n - 1)!
Slope of a Line - ANS-y = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
30-60-90 Triangle - ANS-30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle - ANS-45-45-90
x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles - ANS-3-4-5 or 6-8-10 or 9-12-15
5-12-13
Number Added or Deleted - ANS-Use 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 - ANS-Odd numbers have only odd factors
Quadratic Formula - ANS-To find roots of quadratic equation: ax^2+ bx + c = 0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant - ANS-Quadratic 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 - ANS-(x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) - ANS-1. 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) - ANS-1. 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 - ANS-1. 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) - ANS-For 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 - ANS-(5^k)−(5^k−1)
(5^k)-(1/5)(5^k)
(5^k)(1 - 1/5)
(4/5)(5^k)
Accurate Solutions
Common Factors - ANS-Break 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 - ANS-Gross profit = Selling Price - Cost
Combined Events - ANS-For 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 - ANS-The 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 - ANS-Basic 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 - ANS-Complementary 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 - ANS-Conditional 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 - ANS-The 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 - ANS-Rules:
• 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) - ANS-To 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) - ANS-nCr = 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) - ANS-nPr = n! / (n - r)!
Where n is the total number of items in the set and r is the number of chosen items.
Circular Permutations - ANS-The number of ways to arrange n distinct objects along a
fixed circle is: (n - 1)!
Slope of a Line - ANS-y = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
30-60-90 Triangle - ANS-30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle - ANS-45-45-90
x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles - ANS-3-4-5 or 6-8-10 or 9-12-15
5-12-13
Number Added or Deleted - ANS-Use 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 - ANS-Odd numbers have only odd factors
Quadratic Formula - ANS-To find roots of quadratic equation: ax^2+ bx + c = 0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant - ANS-Quadratic 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 - ANS-(x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) - ANS-1. 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) - ANS-1. 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 - ANS-1. 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) - ANS-For 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 - ANS-(5^k)−(5^k−1)
(5^k)-(1/5)(5^k)
(5^k)(1 - 1/5)
(4/5)(5^k)
