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.
1
,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)!
2
,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)
3
, 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.
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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.
1
,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)!
2
,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)
3
, 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.
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