Math Formulas Flashcards
Probability of an event
P ( A ) = number of outcomes where A occurs /
total number of outcomes
Probability: The complement of an event
P (event happens) + P (event does not happen) = 1
Probability: Mutually exclusive events
Two events are mutually exclusive if they can happen together: P (A and B) = 0
Probability: Events A and B (if they are independent events)
P (A and B) = P(A) x P(B)
Probability: Events A or B (A happens, B happens, or both A and B happen
P (A or B) = P(A) + P(B) - P(A and B)
Probability: Events A and B (if A and B are dependent events)
P (A and B) = P(A) x P B|A )
Fundamental Counting Principal
If a task is comprised of stages, where…
- One stage can be accomplished in A ways
- Another can be accomplished in B ways
- Another can be accomplished in C ways
… and so on, then the total number of ways to accomplish the task is:
A x B x C x …..
Factorial Notation: n! = n x (n - 1) x (n - 2)
Example: n unique objects can be arranged in n! ways. Example: There are 9 unique letters in the word ‘wonderful’, so we can arrange its letters in 9x8x7x… = 362,880 ways
Fundamental Counting Principal: arranging objects when some are alike
n! / (A!) (B!) (C!)….
Example: how many unique ways can the word MISSISSIPPI be arranged?
11! / (1!) (4!) (4!) (2!)
Fundamental Counting Principal: Combinations (when the order DOES NOT matter)
nCr = n! / r! (n-r)!
Example: when the order does not matter - for example picking any 3 friends from a group of 5
5C3 = 5! / 3! (2!) = 10
Fundamental Counting Principal: Combinations (when the order DOES matter)
nPr = n! / ( n-r )!
Example: when the order does matter - for example how many ways you could order 3 letters from the word PARTY?
Prime Numbers
A prime number is one that is divisible by itself and 1.
1 is NOT a prime number. 2 is the ONLY even prime number.
Prime numbers below 60: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59
Greatest Common Factor (GCF)
The greatest common factor of two numbers is the biggest factor shared by two numbers.
Example: The GCF of 12 and 30 = 6
If 2 numbers share no primes, the GCF is 1
Least Common Multiple (LCM)
The least common multiple of two numbers is the smallest positive integer with both numbers as a factor. Example: The LCM of 4 and 6 is 12 - it is the smallest number that has both 4 and 6 in its divisors.
To find the LCM of two numbers, take the prime factorization of each number, find what prime factors appear in both and multiply one of each of the shared primes and then by all the unshared primes.
Example: 12 = 2 x 2 x 3 and 56 = 2 x 2 x 2 x 7, so the LCM of 12 and 56 is (2 x 2) [shared primes] x 3 [12’s unshared primes] x (2 x 7) [56’s unshared primes] = 168
Divisibility of Numbers
3 - sum of digits divisible by 3
4 - the last two digits of numbers are divisible by 4
5 - the last digit is either 0 or 5
6 - even number and sum is divisible by 3
8 - if the last 3 digits are divisible by 8
9 - sum of digits is divisible by 9
Adding Fractions
1/x + 1/y = x + y / xy
Example: 1/2 + 1/5 = 2 + 5 / 2 x 5 = 7/10
Multiplying Fractions
a / b x c / d = a x c / b x d
Dividing Fractions
(a / b) / (c / d) = a x d / b x c
Note: Cross multiple when dividing fractions
Example (1/2) ÷ (1/5) = 5/2
Percentages: Percent of some part of a whole
part / whole = percent / 100
Percentages: Percent change
% change = change / original value
Example: If the price of something goes from $40 to $52, the percent change is (52-40) / 40 = 12/40 = 3/10 = 30/100 or 30%
Also can be written as (change x 100) / original value
Example: (52-40) x 100 / 40 = 1200 / 40 = 30%
Powers & Roots: Exponent Laws
x^a * x^b = x^(a + b)
x^a / x^b = x^(a - b)
(x^a)^b = x^(a*b)
1 and 0 as bases:
- 1 raised to any power is 1. 0 raised to any nonzero power is 0
- Any nonzero number to the power of 0 is 1: 7^0 = 1
Powers and Roots: Fractions as exponents
x^(1/2) = √x : x^(2/3) = 3√x^2
Powers and Roots: Negative exponents
x^(-1) = 1/x; x^(-2) = 1/(x^2)
Sum of Integers (inclusive)
1 + 2 + 3 +….+n = n (n + 1) / 2
Example: The sum of integers from 1 to 40 inclusive = 40 (40 + 1) / 2 = 820
Motion Questions: Distance Formula
Distance = Rate x Time
Motion Questions: Rate Formula
Rate = Distance / Time
Motion Questions: Time Formula
Time = Distance / Rate
Motion Question: DiRT Formula
Distance / Rate x Time
Average Speed Formula
Average Speed = Total Distance Traveled / Total Time Traveled
Speed & Distance: Shrinking gaps
Shrink rate = R1 + R2
Example: Bob travels at 3mph and Yolanda travels at 4mph. How long will it take to meet if they are 42 miles apart?
Shrink rate = 3mph + 4mph = 7mph
Total Time = Total Distance / Shrink rate = 42/7 = 6h
Speed & Distance: Expanding gaps
Expansion rate = R1 - R2
Example: Bob travels to Houston at a rate of 40mph. Yolanda travels to Houston at a rate of 30mph. How far apart will they be in 4 hours?
Expansion rate = 40mph - 30mph = 10mph
Distance = Rate x Time = 10mph x 4 = 40miles
Work Questions: Output rate
Output Rate = Rate x Time = O/RT
Triangles: Area
Area = 1/2bxh
Triangles: Isosceles right triangle (45-45-90) side ratio
Side ratio = x : x : x√2
Triangles: 30-60-90 triangle side ratio
Side ratio = x: x√3 : 2x
Triangles: Right triangle Pythagorean Theoem
A^2 x B^2 = C^2
Triangles: Pythagorean triples
3-4-5 5-12-13 8-15-17 7-24-25 *any multiple of a pythagorean triple constitutes a pythagorean triple
Triangles: Basic rules
- The length of the longest side can never be greater than the sum of the two other sides
- The length of the shortest side can never be less than the positive difference of the other two sides
Polygons:
- Any figure with 3 or more sides (triangles, squares, octagons, etc.
- Total degrees = 180 (n-2), where n=# of sides
- Average degrees per side or degree measure of congruent polygon = 180 (n - 2) / n
