GMAT math ultimate accurate questions and
answers with solutions 2024
Common Factors - ANSWER 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 - ANSWER Gross profit = Selling Price - Cost
Combined Events - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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) - ANSWER 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) - ANSWER 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) - ANSWER 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 - ANSWER The number of ways to arrange n distinct objects along a fixed circle is:
(n - 1)!
Slope of a Line - ANSWER y = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
30-60-90 Triangle - ANSWER 30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle - ANSWER 45-45-90
,x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles - ANSWER 3-4-5 or 6-8-10 or 9-12-15
5-12-13
Number Added or Deleted - ANSWER 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 - ANSWER Odd numbers have only odd factors
Quadratic Formula - ANSWER To find roots of quadratic equation: ax^2+ bx + c = 0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant - ANSWER 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 - ANSWER (x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) - ANSWER 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) - ANSWER 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 - ANSWER 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) - ANSWER 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 - ANSWER (5^k)−(5^k−1)
(5^k)-(1/5)(5^k)
(5^k)(1 - 1/5)
answers with solutions 2024
Common Factors - ANSWER 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 - ANSWER Gross profit = Selling Price - Cost
Combined Events - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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 - ANSWER 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) - ANSWER 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) - ANSWER 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) - ANSWER 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 - ANSWER The number of ways to arrange n distinct objects along a fixed circle is:
(n - 1)!
Slope of a Line - ANSWER y = mx + b
m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
30-60-90 Triangle - ANSWER 30-60-90
x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
45-45-90 Triangle - ANSWER 45-45-90
,x (shorter legs), x(sqrt 2) (hypotenuse)
Common Right Triangles - ANSWER 3-4-5 or 6-8-10 or 9-12-15
5-12-13
Number Added or Deleted - ANSWER 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 - ANSWER Odd numbers have only odd factors
Quadratic Formula - ANSWER To find roots of quadratic equation: ax^2+ bx + c = 0
x = [−b ± √(b^2 − 4ac)]/2a
Discriminant - ANSWER 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 - ANSWER (x^r)(y^r)=(xy)^r
(3^3)(4^3)=12^3 = 1728
Prime Factorization: Greatest Common Factor (GCF) - ANSWER 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) - ANSWER 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 - ANSWER 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) - ANSWER 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 - ANSWER (5^k)−(5^k−1)
(5^k)-(1/5)(5^k)
(5^k)(1 - 1/5)
