GMAT Math Ultimate exam review EXAM
QUESTIONS AND WELL DETAILLED ANSWERS
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.
QUESTIONS AND WELL DETAILLED ANSWERS
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.
