Math Rules Algebra
Counting: Basic principles
"and" = multiply "or" = add -begin by listing some possible combinations to get a feel for what you are dealing with
The Fundamental Counting Principle
"and" = multiply If a task can be divided into stages, and if stage 1 can be done in n1 ways, stage 2 can be done in n2 ways, then the complete task can be done in: N = (n1) * (n2) * (n3) * etc. If we have to arrange a set of n different items in order, the number of possible orders is the product of n times all of the positive integers less than n. Ex: Company of 25 employees wants to see how many steering committees it can create of 3 different jobs for 3 different people. Solution: 25 * 24 *23
Probability "or" rule (mutually exclusive)
"or" means to ADD P(A or B) = P(A) + P(B) One event cannot occur if the other event occurs. Ex: flipping a coin - you cannot flip a heads and a tails simultaneously P(A and B) = 0
% Change with Given Values
% Change = (new - old) / old Ex: After a student discount, the price of the gym membership went from 120$ a month to 108$ a month. What was the student discount? % change = (108-120) / 120 = -0.1 So discount is 10% off.
(xy)^a = (x^a) (y^a)
(2*3)^2 = (2^2)(3^2) = 4 * 9 = 36
(x^a)^b = x^ab
(3^3)^2 = 27^2 = 729
(x/y) ^a = (x^a) / (y^a)
(4/5)^3 = (4^3) / (5^3)
Set of Consecutive Integers
-In a set of "n" integers, it will always contain one integer divisible by "n" -If "n" is odd, then the sum of the integers will always be divisible by "n" Ex: (n-2)(n-1)n(n+1) = 4 consecutive integers, -So, these integers must be divisible by 4. -If, since we have 4 integers, two must be even and two must be odd. So 2 even factors are divisible by 2 and at least one odd factor has to be divisible by 3, so the whole set is divisible by 6.
Find All the Factors of a Big Number
1. Find the prime factors of a big number: Ex: 8400 = 2⁴×3¹×5²×7¹ 2. Make a list of the exponents for these factors: (4,1,2,1) 2. Add 1 to each exponent in the list: (5,2,3,2). 3. Multiply all of these numbers together to get the total number of factors. 5*2*3*2 = 60 = all factors for 8400
Number of Equations and Variables
1. Solve for 1 variable you need only 1 equation 2. Solve to 2 variables you need 2 equations 3. Solve for 3 variables you need 3 equations
Perfect Squares
1x1= 1, 2x2=4, 3x3=9, 4x4=16, 5x5=25, 6x6=36, 7x7=49, 8x8=64, 9x9=81, 10x10=100, 11x11=121, 12x12=144, 13x13=169, 14x14=196, 15x15=225 If all of the exponents for the prime factorization of a big number are even, then we know that factor must be a prime number. Why? There are two of each factor for every perfect square. Ex: 144 = 12 x 12. 12 = 4x3 = 3x2x2 12= 4x3 = 3x2x2 *Perfect squares are the only integers with an odd number of factors. *A perfect square always has an odd number of factors. How? Every factor for a number comes in pairs. For example, 42 = 42x1 and 21x2, 3x7, etc. But perfect squares have a number at the very end that is a single number multiplied by itself. For example, factors of 36 are (1-36, 2-18, 3-12, 4-9, 6).
Substitution Method for Solving Equations with Two Variables
2x+3y = 15 and x + 2y = 11. Solve for one variable in one equation. x = 11-2y. Plug these values in for x in the other equation: 2(11-2y) + 3y = 15 you get 22-4y +3y = 15 -> 22-y=15 --> finally -y=-7 so y=7. Sub this value for 7 into one of the equations to get x = -3. The point where both lines intersect is (-3, 7).
Prime Numbers
A positive integer greater than 1 that is only divisible by the number 1 and itself; in other words, it has only two whole number factors: itself and 1. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... ***NOTICE THAT 1 IS NOT A PRIME NUMBER! IT ONLY HAS ITSELF AS A FACTOR AND NOTHING ELSE
Simple Interest
A=P(1+rt) A= interest P= principle (original amount of money) r= annual rate (expressed as a decimal) t= time in years
Even and Odd Integers
Addition and Subtraction: Even +/- Even = Even Odd +/- Odd = Even Even +/- Odd = Odd Multiplication: Even x Even = Even Odd x Odd = Odd Even x Odd = Even
Intersection of A and B
A∩B = data found in both A and B
Union of A and B
A∪B = data found in A or B
Positive and Negative Numbers Rules
Big Idea Rules: 1. We can always factor out a negative sign for simplification Ex: (-46) - 37 = -(46 + 37) = -83 2. Factoring out a negative sign always reverses the order of subtraction Ex: 23-64 = -(64-23) = -41
Word Problems: Distance, Rate, Time
D=RT
Least Common Multiples
Define: LCM is the smallest multiple that is common to two factors. How? Find the Greatest Common Factor of both numbers. Multiply that number by the another factor that equals the original number. Ex: Find the LCM of 24 and 32. The GCF of 24 and 32 is 8. 8x3=24 and 8x4=32. So 8x4x3 = LCM or 96.
Word Problems: Division
Divided by, quotient of, per, out of, each, ratio of _ to _ ...
Divisibility Rules
Divisible by 2: last digit must be even Divisible by 5: last digit must be 0 or 5 ***Divisible by 3: if all of the digits add up to a number that is divisible by 3, the whole number is divisible by 3*** Divisible by 4: if last two digits are divisible by 4, then the entire number is divisible by 4 Divisible by 6: a number must be divisible by 2 and 3 Divisible by 9: if all of the digits add up to a number that is divisible by 9, the whole number is divisible by 9 If a number is divisible by both 3 and 4, then it is also divisible by 12
Functions: Domain and Range
Domain = the set of input numbers for a function. Ex: -3<x>3 Domain = x Range = set of output numbers for a given domain Range = y
(x^a)(x^b) = x^a+b
Ex: (3^2)(3^3) = 3^2+3 = 3^5
(x^a) / (x^b) = x^(a-b)
Ex: (3^3) / (3^2) = 3^(3-2) = 3^1
x^0 = 1
Ex: -6^0 = 1
Relationship Between GCF and LCM
Ex: 18 and 24. Find their prime factorization: 18= 2x3x3 24=2x2x2x3 Both have 2x3. So 2x3=6 which is their GCF Now, build the LCM by combining all the prime factors of one of the numbers (say 18) and add the non-GCF factors of the other number (24). So you get: 2x3x3x2x2 2x2x2x3x3= = 8x9=72. 72 is the LCM.
x^-a = 1/(x^a)
Ex: 2^-3 = 1/(2^3)
% Increase/Decrease with only multipliers
Ex: At the beginning of the year, the price of an item increased by 30%. After the increase, an employee purchased it with a 40% discount. The price the employee paid what what % below the original price? First multiplier = 1.3 for adding 30% to the price. Second multiplier = .60 (for subtracting 1-%40 = %60). (1.3)(.60) = .78 1 - .78 = .22 so the employee paid 22% below the original price
Finding Prime Factors of Really Messy Big Numbers
Ex: Find the prime factorization of 9975. Well, 9975 = 10,000-25. which equals: (100²)-(5²) = (100+5)(100-5). This equals (105)(95). 95 = 19*5 which are both prime #s so keep those. 105 = 7*15, keep 7, 3*5 = 15. So we have as the prime factors 3*5²*7*19.
Finding Two Numbers to Get a Decimal
Ex: Find two numbers that is the product of 0.9991 Well, 0.9991 = 1-.0009 Which equals (1²)-(.03²) Factor: (1+.03)(1-.03) This equals: (1.03)(.97)
Function Notation
Ex: If f(x) = 3x-7, then what is f(4x) = ? Plug in 4x for x so you get 3(4x) - 7 f(4x)= 12x-7
Simplifying Big Decimals by Division
Ex: Simplify the expression: (0.999951) / (0.993) (1-.000049) / (1-.007) = [(1²) - (.0007²)] / (1-.007) Factor the top to get (1+.007)(1-.007) / (1-.007) Cancel the top (1-.007) and bottom out and you are left with 1+.007 which equals 1.007.
Simplifying Rational Expressions with Two Factors
Ex: Simplifying this expression: (y²+2x-8) / (x-4). You can separate the numerator: y² / x-4 and 2x-8 / x-4 put y²/x-4 aside, and factor 2x-8 = 2(x-4) / x-4. The (x-4) / (x-4) cancel out and get (y² / x-4) + 2
Probability "or" rule (not mutually exclusive)
Ex: for over lapping areas on Venn Diagram GENERALIZED "OR" RULE: P(A or B) = P(A) + P(B) - P(A and B)
x^0 = 1 (tricky rule)
Ex: simplify the following: (3b^0)^4 The ^0 applies only to the variable b and not the coefficient 3. So... (3*b^0) = (3*1)^4 = 81
Elimination Method When Substitution Method Creates Messy Fractions
Ex: solve these two systems of equations: 7x+3y=5 2x-3y=13 Add them together to get: 9x=18, x = 2. Plus this x value into one of the equations to get y=-3 Or you can subtract one equation from the other equation. Or you can multiply both equations by different numbers to eliminate a variable.
PEMDAS
FOLLOW THE ORDER! ALWAYS TRY TO CANCEL OR DIVIDE BEFORE YOU MULTIPLY
Find Greatest Common Factors of Large Numbers
Find prime factors of each big number. Find out the highest powers of common numbers that they have (ex. 2² and 2⁵ - the highest common factor = 2². When you have the common one's together. Ex: 2³ × 5¹ = 40. 40 is the greatest common factor of 800 and 360.
Inclusion-Exclusion Principle
For two sets of data, the number of elements in A∪B equals the sum of the number of elements in A and B minus the number of elements in A∩B. A∪B = A + B - A∩B
f(x)=−x²
Graph that looks like an upside down U
f(x) = x²
Graph that looks like the letter U
Rebuilding the Dividend Equation
How do we find the "n" original integer in a remainder question? For example, if know that the divisor is 2, the quotient is 6 and the remainder is 1, what is the original integer "n"? (n/d) = Q + (r/d). (n/2) = 6 +(1/2). Solve for n. n = 6(2) + 1 = 13.
% Combined Increase or Decrease
How to determine combined % increase / decrease Ex: 1. A price rises by 10% one year and then 20% the next year. What's the combined % increase? 64 (1.10) = 70.4 70.4 (1.20) = 84.48. M = new / old M = 84.48/64 M = 1.32 or 32% increase
Remainders
If I have an integer that, when divided by 12 I get a remainder of 5, what could be my possibility of integers? 12 +5 = 17. 17/12 = 1 (5/12) 24+5 = 29. 29/12 = 2 (5/12) ...any multiple of 12 (+5) will get a remainder of 5. ***Tricky: What is the smallest possible integer that, when divided by 12, gives you a remainder of 5? = 5. Why? Because 5/12 = 0 with 5/12 remainder.
The Fundamental Counting Principle with Restrictions
If a problem contains a restriction, start with the most restrictive stage and then move to the less restrictive stage until all you are have non-restrictive stages left.
Combining Ratios
If there are three colors of balls in a bin and the ratio of green to blue is 2:3 and the ratio of blue to red is 5:6 then what is the ratio of all the green balls in the bin? G:B = 2:3 B:R = 5:6 LCM of all the blue balls (since they are in both equations) is 15 so... 2:3 ----- 2(5) = 10 and 3(5) = 15 so you get 10:15 5:6 ----- 5(3) = 15 and 6(3) = 18 so you get 15:18 New ratio of balls for G:B:R = 10:15:18 Total balls in the bin = 10+15+18 = 43 Ratio of Green to all the balls in the bin = 10/43.
Absolute Value Inequalities
If you have a range on a number line from 20-90 a regular inequality would equal 20<x<90. But for absolute value inequalities, we find to find the center of that range = 55. Then find the distance of 20 to 55 and from 55 to 90. This equals +35 and -35. So: |x-55|<35 Example 2: Find the absolute value inequality for: -3≤x≤11. The midpoint between -3 and 11 is 4 so we know that |x-4|≤ something, is the first part. How far is it from 4 to 11? It is 7. So: |x-4|≤7
Ratios With One Known Factor
In a certain company, the ratio of programmers to marketers is 3:8 and the ratio of customer service reps (CSRs) to marketers is 2:3. If there are 27 programmers, how many CSRs are there? P:M = 3:8 ----- 27:M. Cross multiply and you get M=72. Use M=72 in the second ratio of C:M = 2:3 = C:72. Cross multiply and you get C = 48.
Compound Interest
Interest earned not only on the investment but also on the investing plus the interest earned: A=P[1+(r/C)]^tC A= interest P=principle (original amount of money) r= annual rate C= number of times compounded annually
Maximum / Minimum Value of a Function
Max = highest point of graph, when y is biggest number Min = lowest point of the graph, when y is smallest number
Word Problems: Subtraction
Minus, subtracted from, less than, decreased by, difference between
% Increase/Decrease
Multiplier = new value / old value Ex. After decreasing by 5%, the population is now 57,000. What was the original population? Don't put .05 in as the multiplier! Use .95 (1-.05 = .95) instead. .95 = 57,000/x x = 60,000
Permutations with Subsets
Now suppose you have n objects and you only want to order some of them. Let the number for the subset be k. Again you multiply n! but only for k amount of times. This equation equals: nPk = n!/(n-k)! Example 1: Five runners in a race. You want to see how many ways a gold, silver and bronze medal could be awarded to these five runners. 5P3 = 5! / (5-3)! = 5x4x3x2x1 / (2! = 2x1) = 5x4x3 = 60.
Probability "and" rule (conditional probabilities - when A and B are not independent)
P(A and B) = P(A) * P(B|A) P(A and B) = P(B) * P(A|B)
Word Problems: Addition
Plus, added to, sum, combined, and, more than, total
Probability
Probability = ( # of successes / total # of outcomes)
Basic Permutations (for lists)
Say you want to determine the number of possible orders, or permutations of a set of all the objects. When the first object is assigned (n) there are (n-1) possibilities, etc. So n(n-1)(n-2)... This product is written as n! and pronounced n factorial Example 1: You are arranging 4 trophies on a shelf. How many distinct ways are there to arrange the trophies? 4(4-1)(3-1)(2-1) = 4x3x2x1 = 24 possible arrangements
Empty Set
Since circle A and circle C do not overlap at all, A∩C is called the empty set
Probability "complement" rule (short cut!)
The "complement" of A = NOT A P(A) = 1 - P(A) *guaranteed to work when you see the words "at least" or "at least one" in a question
Solving Equations
There are difference cases depending on the equations: 1. There can be an equation with no solution such as taking the square root of a negative number is impossible. Ex: x² = -5 2. There can be a case of two solutions to an equation where x = 3 or 7 for example. 3. There can be a case of only one solution to an equation when we get a difference of squares like x²-6x+9, when factored looks like (x-3)(x-3). 4. There can be a case of an infinite number of solutions where there are two equations with no intersection. For example: x+y=2 and 2x+2y = 7. Solve for y on both sides and you get: y=7/2 - x y = 2-x Both lines never intersect and so there is no single solution.
Probability "and" rule (not mutually exclusive)
To find the probability that both A and B will occur, we must indicate that the probability of A occurs independently of the probability of B. P(A and B) = P(A) * P(B) "and" means to MULTIPLY For example: If we want to choose 2 managers from a group of 5 managers to attend a conference in Las Vegas, we must find the P(A) and then the P(B). So, P(A) = 2/5 and the P(B) = 1/4. So the probability of both manager A and B being selected = (2/5) x (1/4) = 1/10
Inequality Rules
Treat the same way you treat a regular equation, except that when you Multiply or Divide by a negative number, you must switch the inequality sign. Rules: 1. You can add inequalities with the same directions, but you cannot subtract. Ex: if 15>8 and 10>2 then 15+10>8+2 2. You can subtract inequalities with opposite directions but you cannot add them. Ex: if 15>8 and 10<12 and you subtract them: 15>8 -10<12 ---------- 5>-4 Note that the solution inequality sign, in this case > is the same inequality sign that we subtracted from in the beginning. Since we took 10<12 FROM 15>8, we get an equality with a > sign. Multiplication and Division of Inequalities Rule: -You can never multiply or divide an inequality from another. Why? Because you could have positive or negative numbers. Ex: -10<2 and -8<3. If you multiply -10(-8) you get +80 and 2(3) = 6. So 80 < 6 is not correct.
Rules
We never want to divide by a 0 We never want to have a square root of a negative #
Word Problems: Variable (x, n etc.)
What, how much, how many, a number
Age Questions with Time
When a word problem deals with ages, you simply sub the variable in for the name of the person and create an equation. However, when a word problem has to do with changes in time, you want to add that time to each variable. Example: Right now, Steve's age is 1/2 of Tom's age. In eight years, twice Tom's age will be 5 more than 3 times Steve's age. How old is Tom right now? S=1/2T or can also be T=2S Make sure to use S+8 and T+8 in the equation: 2(T+8) = 3(S+8) +5
Multiplication Principle
When two events occur sequentially, and the first does not influence the second, the multiplication principle states that the number of possible outcomes is m x n, where: m= number of possible outcomes for the first event n= number of possible outcomes from the second event Similarly, the probability of two independent events occurring is: P(A∩B) = P(A) x P(B) Example 1. : How many possible outcomes are there if a six-sided die is rolled three times? first roll = 6 possible outcomes second roll = 6 possible outcomes third roll = 6 possible outcomes so 6x6x6 = 216 possible outcomes Example 2: Two cards are chosen sequentially from a deck of 52 cards. The first card is not returned to the deck. What are the number of possible outcomes? 52 x 51 = 2652 possible outcomes
Linear Function
Y=mx+b m= slope = (change in y / change in x) b= y intercept = plug in (x,y) point and the slope to find b
Combinations (for groups)
You have a set of n objects, you want to select some number (k) of them, but the order does not matter. The number of possible groups chosen is called the number of "combinations of n objects taken k at a time": nCk = n!/[k!(n-k)!] Example 1: A choir director randomly selects 3 of his 6 members to form a group. How many possible groups of 3 members are there? 6C3 = 6!/[3!(6-3)!] = 6x5x4x3x2x1/ 3!(3!) = 6x5x4x3x2x1 / 3x2x1x3x2x1 = 6x5x4 / 3x2x1 = 20.
Difference of Squares
a^2 - b^2 = (a+b)(a-b) Example: Simplify the expression: (b^2 - c^2) / b-c? Solution:(b^2 - c^2) is a perfect square, so it equals (b-c)(b+c). Divide (b-c)(b+c) by (b-c) and you get (b+c).
Distance Formula
d = rt d = √[( x₂ - x₁)² + (y₂ - y₁)²]
Functions: Input and Output
f(x) = 2x-1. x = input, solution to the function = output
Absolute Value Function
f(x) = |x| Because the absolute value function of x represents a distance from the origin, its output is always a positive number or zero. Ex: Find all the possible integers for x that range from 1-100 that satisfies this equation: |x-30| >20. This of this absolute value on a number line. You start with positive 30. Then move right and left 20 spaces. Your possible values for x on the left side of the line must be 1-9 (cannot include the number 10) and all the possible values for x on the right side of the line are 51-100 (cannot include 50). So 9 values on the left, plus 50 values on the right = 59.
Word Problems: Equals
is, was, will, be, has costs, adds up to, is the same as...
The Binomial Situation
p = probability of success on one trial n = number of trials r = number of successes p^r = successes happening r times P = (nCr) * (p^r) * [(1-p)^n-r]
Stranger Operators
plug and chug
Word Problems: Multiplication
times, of, product of, twice, double, half, triple...
Absolute Value Equations Extraneous Solution = No Solution to the Equation
|x| = 5 so x = +5 or -5 |2x+3| = 5 so 2x+3 = +5 OR 2x+3 = -5 Remember: |-3| = 3 but |9| ≠ -9 Why? Because an absolute value cannot equal a negative number! What happens with both roots to an equation do not work? It is called an Extraneous solution or No Solution to the equation.