Math equations Flashcards
definitions
Sum: addition
Difference: subtraction
Product: multiplication
Quotient: division
Integers
positive and negative whole numbers, and zero (…, –2, –1, 0, 1, 2, …)
zero as an integer
Zero is an even integer, neither positive nor negative
Digits
Digits: the integers from 0 through 9 (ten digits total)
ABCD.EFGH:
A = thousands
B = hundreds
C = tens
D = ones = units;
E = tenths
F = hundredths
G = thousandths
H = ten-thousandths
Prime
numbers with exactly two factors (so 1 is not prime)
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97 are prime
Prime numbers are numbers that have only 2 factors: 1 and themselves.
By contrast, numbers with more than 2 factors are call composite numbers.
“Can the positive integer p be expressed as the product of two integers, each of which is greater than 1”
is the same as asking “is p NOT prime”
Through or inclusive: including the endpoints
Through or inclusive: including the endpoints
(to count consecutive integers through: last number – first number + 1)
including endpoints- what is the sum of the multiples of 3 between 1 and 99 or 2 3 adn 99, does not include 3 adn 99, first 3 last 96, how many multiples of 3 are there from 3 to 99 inclusive, then includes both 3 adn 99 so then 99-3 plus 1 divided by 2 but rounding up
Between: excluding the endpoints
Between: excluding the endpoints
(to count consecutive integers between: last number – first number – 1)
Factors or divisors
when listing factors/divisors, always list them in pairs beginning with 1 and the number itself
ex: 24 = 1 × 24, 2 × 12, 3 × 8, 4 × 6
add 1 to each exponent of prime factorization, then multiply all of those numbers = total # of factors
ex: 144 = 24
2^4 X 3^2 → (4+1)(2+1) = 15 → 144 has 15 total factors
lets take a number 12 numbers are made up of prime factors every number has its unique set of prime factors what makes a number distinct, and in fact as we have discussed think about numbers not as numbers but as composites of their dna. governs every aspect of the that number, governs the entire identity of that number, it is also the case that numbers have a set of factor pairs factor pairs of 12, 1, 12, 2, 6, 3,4 - if you want to know how many factors that number has you can caluclate it using its prime factorization is using process on sheet so 12 2^2 x 3, circle exponents 2 and 1 we add 1 to each exponent then multiply the result 2+1=3 and 1+1= 2, 3x2 6 12 has 6 factors, 1, 12, 2, 6, 3, 4 and thats what it means
Multiples
Multiples: the multiples of 5 are 5, 10, 15, 20, 25, …
Distinct
Distinct: different
Reciprocal
the reciprocal of x is 1/x
the reciprocal of a/b is b/a
Mean
average ([Sum] Total = Average × Number of Numbers)
Median
Median: middle number or the average of the two middle numbers, when numbers are arranged in order (equivalent to the 50th percentile)
Mode
Mode: the most commonly occurring number
Range
Range: greatest number – smallest number
Standard deviation
a measure of how spread out a set of numbers is
- adding a constant to every number in a set does not change the SD
- multiplying by a constant will change the SD by the absolute value of that same constant
-if all the numbers in a set are the same, the SD will be 0 - in a normal distribution, within 1 SD = 68%; within 2 SDs = 95%; within 3 SDs = 99.7%
** question was if X, Y, Z haev same sd as 10, 15, 20? z-x=10 and 2. z-y=5. yes C Because the standard deviation is unaffected when the same number is added to each member of a data set, the standard deviation of the data set consisting of the numbers z – 10, z – 5, and z is the same as the standard deviation of the data set consisting of the numbers 10, 15, and 20;***
Divisibility
3: if the sum of the digits is divisible by 3, the number is also divisible by 3
9: if the sum of the digits is divisible by 9, the number is also divisible by 9
4: if the last two digits as a number are divisible by 4, the number is also divisible by 4
Divisibility 2
8: if the last three digits as a number are divisible by 8, the number is also divisible by 8
6: must be divisible by 2 and 3
15: must be divisible by 3 and 5
- Any integer divided by any power of 2 or 5 (or combination of those two numbers, such as 10 or 20) will be a terminating decimal; anything else in the denominator has the potential to make a repeating decimal
Fractions 1
Fractions 2
Cross multiplying
Odd/Even Rules
Counting
Equally spaced Lists 1
Equally spaced Lists 2
Number Tricks
7 × 11 × 13 = 1,001
Difference between consecutive perfect squares is the set of consecutive odd integers
–>Differences between (0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …) are (1, 3, 5, 7, …)
Dividing by 5: double the number and divide by 10
Multiplying by 5: halve the number and multiply by 10
Unit Patterns
Perfect Squares
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, …..
Perfect cubes
…..–8, –1, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000…
Power of 2
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, …
Square root of 2
1.41
Square root of 3
1.73
Square root of 5
2.24
Arithmetic- Percents!
Percent Change
Ratios
Distance, Rate and Time
Weighted Averages
Sets
Venn
part of Sets
Union
part of Sets
Intersection
part of sets
Work/Rate Questions
Proportions
Inequalities
Solving Equations
Special Factoring
Absolute Values
Radicals
Exponent Rules 1
Exponent Rules 2
Quadratics
Interest equations
Compound interest equation
Simple interest equation
Rule of 72 (for Interest)
Combinations/Permutations/Factorials
Factorials
Permutations and Combinations
Circular arrangements, multiply, divide
Probability
Arithmetic Progressions
Geometric Progressions
Revenue/Profit
Points and Lines
Geometry
Slope
- bc perpendicular lines have negative reciprocal slopes, so if one of them was a slope of 2, then the other would be negative 1 over 2 so -1/2,
x intercept
y intercept
Midpoint (for a line)
Distance (for a line)
Reflections
for a line/geometry
Functional Shifts (for a line/geometry)
Angles (geometry)
quadtrilaterals
interior angles add up to 360 degrees!!
pentagon
interior angles add up to 540 degrees
hexagon
interior angles add up to 720
Square, Rectangle
Area, perimeter, diagonal
Parallelogram, Rhombus, Trapezoid
Area, perimeter, diagonal
Polygons
what are 2-D shapes?
Area of a circle
circumference of a circle
diameter of a circle
Chord (for a circle)
Square in a circle
Central angle and inscribed angle for a circle
Arc length for circle
Area of sector for circle
Equation of a circle and Equation of a circle centered at the origin
area of a triangle
Isosceles
Equilateral triangle and area of an equilateral triangle
triangles 2
triangles 3
similar triangles, exterior angles….
Special Right triangles
Pythagorean triples summary
Pythagorean triples 1
3,……
5,…..
6, …..
7,…..
3, 4, 5
5, 12, 13
6, 8,10
7, 24, 25
Pythagorean triples 2
8, …….
9, ……
10, ……
12,…..
8, 15, 17
9, 12, 15
10, 24, 26
12, 16, 20
Pythagorean triples 3
14,…..
15,…..
15,…..
16,…..
14, 48, 50
15, 20, 25** (more important to know)
15, 36, 39
16, 30, 34
3-D Shapes geometry
Cube
volume, surface area, diagonal
Rectangular solid
volume, surface area, diagonal
Cylinder
Sphere and cone
permutations vs combinations
arrangement have order !!! =permutations, each of which has a particular position in the order! that is a permutation!!! so that you do slots do not use the 5 c 3 use slots 5 x4 x3
group is a disordered collection it is simply a certain number of things that has no order, has no difference of position that is a combination ex with marbles 5 chose 3
memorize for data sufficiency
data sufficiency- if 1 says x=4, 2 will say x= 4 or 7, but 2 will not say x is 7, so they are committed to the same answer if you get 1. and 2. say x=4 they want one answer** you will not get two different answers
so data sufficiency has to be one single answer, so long as you have a single answer to the question but it needs to be a single answer
has to be one single answer!!!
what is a dividend
a dividend is a simple percentage- the result of a simple percentage, if your stock had a 5% dividend means 5% of the stock will be sent to you compound interest environment interest stays in account account grows exponetially bc inc money, in simple interest it is the same amount bf you are getting a check amount of money staying the same
Q 18. 18. positive digit= ……
- positive digit= 0-9 tells us n is a number from 0-9
if they say the unit digit of another then 14 or 4 has a unit digit of 4, but a positive digit of 4 can’t be 14, need a 10s digit in order to have a unit digit, they say square, triangle and star represent a positive digit a set of positive digits are the numbers 0-9
Q22 complementary angles …..
Q22 complementary angles around a line add up to 180, two parallel lines and draw transversal the lines in the same positions on that transversal will be equal so those x have to b equal these xs are equal adn ys are equal if have parallel lines where they could be
parallelogram the angles right next to each other have to be 180, opposite angles are equal
