Kaplan Math Fundamentals
Pattern with factors:
- If 2 numbers have the same factor, then the sum or difference of the two numbers will have the same factor. - include 1 and the number itself when counting the number of factors a # has For A number to be a factor of another (B) number - the prime factorization of the smaller A number needs to include factors of the bigger B number
Finding the sum of a quantity of consecutive integers (ex sum of 3 integers that sum=165)
1) divide the sum by the number of terms - 165/3 = 55 2) the next consecutive integers will be that number +/- 1 - 55-1 = 54 - 55 + 1 = 56 integers: 54, 55, 56
how to find a common multiple of two numbers?
1) do the prime factorization of each number 72: 3*3*2*2*2 64: 2*2*2*2*2*2 2) look at the number of times that each prime number occurs in the two sets - find the amount of times each factor appears in each number, choosing the one where it appears the most 72: 3² (most 3s between them) 64: 2⁶ (most 2s between them) 3) multiply each prime factor by the # of times it appears in the original number 72: 3² -> 64: 2⁶ LMC: 2⁶*3² = 576 OR 28=2 x 2 x 7 42= 2 x 3 x 7 LCM= 2 x 2 x 3 x 7 = 84
How to answer:: How to answer: the integer n is a multiple of both 18 and 30. How many different positive integers must n be a multiple of?
1) find LCM of both numbers 2) list factors of LCM 3) count the amount of unique factors that can make the LCM (this is the number of different potive integer factors)
How to solve: The remainder when the positive integer x is divided by 56 is 48. What is the remainder when x/4 is divided by 7 ?
1) find a value for X by adding dividing number and remainder 56+48= 104 2) Find what the second part of the problem is dividing by (in this case, combine (my multiplying denominators/dividers) x/4 is divided by 7 = x/4*7 > x/28 3) determine the remainder when it is dividing by 28 = find remainder after simplifying fraction
How can you estimate weighted average problems?
1) find the average/mean of the two values/averages - ex. march: 78 April: 72 -> average: 78+72/2= 75 2) the number with more weight (or more occurrences/values) will be closer to the average - ex. the weighted average will be closer to march(78) than april (72) because march is closer to the average
Finding the total reduction in a sequence?
1) find the sum of both lists when you pick numbers 2) subtract the two sums - the difference/result will be the amount the sum was reduced by OR: back solve by finding the sum of both lists with picking numbers then seing which solution would create the same differnece between the 1st and 2nd sequence
How to find the sum of a set of scores within a group of scores
1) find the sum of the set that you have the average for (sum = average * # of terms) 2) find the sum of the entire set by multiplying the gernal mean/average by the total number of scores 3) subtract the set of scores fro mteh whole group of scores to get the sum of the other set 4) to get the average of this sets sum, divide the sum by the number of scores in that set
Solving Multiple fixed arrangements - several pairs
1) find the ways the pairs can be ordered (factoral!) ex. 4 pairs: ab, cd, ef, gh - arranged: 4! = 24 2)if each pair can be aranged 2 ways (BA or AB) multiply the number by 2 for every pair - 24 x 2(the number of pairs) = 24 x (2)(2)(2)(2) = 384
How to find what an integer is divisible by based off another divider? ex. any numerb divisble by 84 is also divisible by what?
1) prime factor the dividing number - 84 = 2 × 2 × 3 × 7 2) the correct answer (also divisible) will have the same factors as the initial number - the other numbers will be divisible by those factors on their own or multiplied by any of the other factors. - the answers will be the product of any of these factors multipeld togetr (ex 3*7 = 21 and 84 is divisible by 21) - will be a factor of the original number numbers that have factors other than 2 3 and 7 will NOT be divisble/correct or too many of those factors (ex having three 2s) - 84 = 2 × 2 × 3 × 7, this number is divisible by (2²)(3)(7) - and when those numbers are multiplied (ex 3*7 = 21 and 84 is divisible by 21)
How to find standard deviation (formula)
1) subtract each term in lsit from the mean > then raise to the second power 2) add terms 3) divide by n/total 4) apply root √
How to find the Standard devation when you are given the mean and a score that is # SD's aboove the mean
1) subtract the known value from the mean ex. 78 is 3 SD's above the mean (42) -> 78-42= 36 2) devide hte difference between the value and the mean by the ampunt of SD's the numebr is from the mean ex. 36 / 3 = 12 (SD is 12
How to solve absolute value equations
1. Rewrite as two separate equations: when "= value" is positive or negative 2) Solve each equation separately If Comparing, its often D - BUT make sure that BOTH possible values smaller/greater than the other
how to solve: what is the sum of the integers from 10 through 50, inclusive?
1. find the mean of the two numbers given. (+ then ÷ by 2) 2. determine the number of terms B-A+1 3. find the sum= average x number of terms average=(10+50)/2 =30 number of terms= 50-10+1 = 41 sum= 30 x 41 =1230
how to do long division
1. how many times does x go into y? (if not possible, add decimal and *100) 2.multiply x*y to see if anything is left (subtract result from y) (if so, carry down z) 3. how many times does x go into z? 4. continue, then multply for left over/remainders (continue till no remidners or pattern)
How to manually calculate Square Root / reduce roots
1. look at factors of the number - are any of the factors perfect squares? 2. change the # into choosen factors inside √ 3. take any perfect squares out of √ leaving remaining factor inside
Relationship betewen n³ and 1/n³ when n= negative fractoin (-1/2)
1/n³ is the recorpical of n³ (reverse the fraction) - find value of n³ = -1/2³ = -1/8 - Raise the denominator/n value up -> 1/n³ -> 1/(-1/2)³ -> 1/ (-1/8) = -8 (a fraction divided by a fraction > mulitply by recrpirocal)
prime numbers 1-49
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 (1 is NOT a prime number) (0 is not prime - NIETHER)
How do you recognize the multiples of 2
2: last digit is even
How do you recognize the multiples of 3
3: sum of digits is a multiple of 3
How do you recognize the multiples of 4
4: last two digits are a multiple of 4
prime numbers 50 - 100
53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103
How do you recognize the multiples of 5
5: last digit is 5 or 0
How do you recognize the multiples of 6
6: sum of digits is a multiple of 3, and last digit is even
How do you recognize the multiples of 9
9: sum of digits is a multiple of 9
How to solve: Integer A is dividable by 12 and 98 . Find numbers that A will be divisible by
A is a multiple of 12 and 18 - so find the LCM and find which numbers will divide into the LCM solving: - find the LCM of the two numbers. the answer can be divided into the LCM if its correct OR back solving: - prime factor answers and find the #s with factors not included in the original #s prime factors
Perfect cubes: definition and examples
A perfect cube - multiplying the same integer three times
Perfect Squares: definition and examples
A perfect square is a positive integer that is obtained by multiplying an integer by itself. (ex. 5x5=25)
how many integers are there from 73 through 419, inclusive?
B-A+1 = (419-73)+1 = 347
Counting methods: Combination
Choosing a group/item from a larger population/group - ORDER does NOT matter ex. picking people to form a group Formula: n! k!(n - k)! (or use double slots)
If you know that something is 4/5ths of another value, how do you reverse find? ex. the area of 112 square feet is 4/5 of the area of the whole floor, what is the floors area?
Formula: value = (fraction/percentage) * (unknown) ex. 112 = 4/5*(floor) - Solve for variable (floor) - mulitply value by inverse of fraction: 112 * 5/4 = 140
How to find the sum of consecutive integers
Formuula: sum = # of integers x average of integers 1) find the number of terms (or what terms are included) 2) find the mean/average: add the largest and smallest term then divide by 2 3) multiply the average by the number of terms
Fractions and Decimal conversion
Fraction to decimal - divide numerator by dominator using long division Decimal to fraction - move the decimal over right till number is a whole integer. count # places decimal was moved right then put the same # of 0s after a 1 to calculate denominator
How to know the amount of zeros to put behind a number multiplied by 10⁻ⁿ - negative numbers
If negative, you are moving the decimal up n (exponent value) places or, you are putting the number - (already decimal, 3.3) 1 zero less than the exponent - (not a decimal , 33) 2 zeros less than exponent 2.1×10⁻⁶ = .0000021 (behind 5 zeros)
Dividing Fractions
Invert (reciprocal) the second fraction and multiply (1/2 divided by 3/4 = 1/2 * 4/3)
if the product of two integers is odd, what else is true?
Means that x*y = odd or that its: odd x odd - the sum or difference of two odd numbers is even (add/subtract odds = even) - Any odd integer squared is odd - Any odd number multiplied by another odd number = odd number, (or solve by picking Numbers like 1 and 3) (0 is even)
Multiplying Fractions
Multiply the numerators and put the product over the product of the denominators
Units: how to individually quantify numbers in the unit, tens, and hundreds positions
Multpply the number by the digit value it is in ex: 236 - unit: 1x or the value of the number itself (1x6 = 6) - tens: 10x (3x10 = 30) - hundren: 100x (2x100=200)
What are the factors of a number
Numbers that can be multiplied to make a specific number - OR factors = divisors: the positive integers by which a number (multiple) is evenly divisible by ex a factor of 28, - 1x28, 2x14, 4x7 so there are 6 factors: {1, 2, 4, 7, 14, 28}.
the rules for adding odd and even integers:
ODD + EVEN = ODD ODD + ODD = EVEN EVEN + EVEN = EVEN
the rules for multiplying odd and even integers:
ODD × EVEN = EVEN ODD × ODD = ODD EVEN × EVEN = EVEN
how to find the range
RANGE is the postiive difference between the largest and smallest terms in a set of numbers Range = the greatest value - the least value
What is the sum vs difference vs prodct vs quotent?
Result of: Sum - addation + diffference - subtraction - product - mulitplcation x quotent - division /
Ratios - finding the quantity of a item by reversing the other proportions ex. yellow marbles: 1/5 the number of purple marbles purple marbles: 1/8th the number of green marbles. Green = #?
Reverse formula: y= p(1/5) -> p=5y Pick numbers, starting with yellow = 1 - p=5(1) = p=5 Then plug 5 into other formula - p = g(1/8) -> g =8p -> 8*5 = 40
adding square roots
Separate roots: √2+√ 2 = 2√ 2 Inside the same root: √ 2+ 2 = √ 4 = 2
what is the average of a list of numbers starting with 13 and ending with 77?
The average of evenly spaced number is simply the average of the smallest number and the largest number. (or the mean of both numbers added together) EX the average of all the integers from 13 to 77: - Add 13 + 77 = 90 - Divide by 2 = 90/20 = 45 average of list of numbers = 45
Divisibility - relationship between factors and multiples
The concepts of multiples and factors are tied together by the idea of divisibility. A number is said to be evenly divisible by another number if the result of the division is an integer with no remainder. - 52 / 4 = 13, which is an integer. A number that is evenly divisible by a second number is also a multiple of the second number. - 52 is evenly divisible by 4, and 52 is a multiple of 4.
How to determine if products are even or odd based on multiples:
The product of any number of odd integers is always odd the product of an odd integer and an even integer is even. for the product of a group of integers to be even, at least one of the integers must be even. Cube of an odd number is always even.
How do you calculate a value if you are given what percentage it is of another number? AKA: if January's wages ($20.60) are 90% of April's wages, what were April's wages?
To QUICKLY SOLVE, divide (the resulting value)/the percentage 20.60 / .90 = 22.89 written as an equation, it would be (the resulting value of the %) = percentage% * (the original value) 20.60 = .90x
patterns as you increase/decrease expoents
UP: as you go up (2^2 to 2^3), you are multiplying the previous exponent/value by the original # again (ex. 2^4 is the same as 2*8 (or 2^3) = 16). DOWN: as you reverse, you divide the next/higher expoent/value by the original # again (ex.2^4 is the same as 32/2 (or 2^5/2) = 16)
The place values/names of a number
Unit, tens, hundreds, thousandths, ect. Each place value represents a power of 10, where the rightmost digit is the units place (10^0), the next digit to the left is the tens place (10^1), the next is the hundreds place (10^2),x
Counting methods: How to solve Permeation
When arranging only some items (subset) of a group, use Formula: n! (n - k)! 1) plug in numbers 2) write out factorials 3) cancel reduant factorals 4) Caculate Or use slots (or just calculate n!) 1) create slots for each position your trying to fill 2) write the # of possibilities available in each slot/choice 3) multiply (basically, n!)
Absolute value equations in comparative questions
With comparative questions, find the possible value of x that is true in both equations its often D - BUT make sure that BOTH possible values smaller/greater than the other
What does five consecutive multiples of 6 means? (question type)
all of the numbers in the set will be equally separated by 6 ex. 6, 12, 18, 24, 30, 36....
How to know if an integer is even or odd from formulas
as long as an integer can be divided by 2, it is an even number. k and R are integers. - If k=2r+1, then k is an odd number. (K can't be divided by 2 evenly with this formula) - If k=2r, then k is an even number.
What is the average formula (and its variations)
average = sum of terms / number of terms sum = average * number of terms
how to find a common factor of two numbers?
break both numbers down to their prime factors to see which they have in common. then multiply the shared prime factors to find all common factors.
Factors vs multiples
factor - positive integer that divides evenly into another number (# you divide by) (small numebrs that go into bigger numebrs, LFC/GFC) multiple - integer that is the result of mulitplying two other integers (result of mulitplicaiton) (unlimited mulitples) If two numbers have a common factor, then they also have common multiples. ex. the common factors of 12 and 18 = 1, 2, 3, 6. Therefore, the common multiples of 12 and 18 are 12, 24, 36, etc. Multiple divided by Factor = Integer.
Prime factors of Factorials
factors of n! (ex 15!) - the factors are just the numbers that make the 15! (numbers before 15 > 15*14*13*12...*3*2*1) any number bigger / not included in the factorial will not be factors - if another n=number has a prime factor that is bigger than the numbers included in the factorial, it is not a factor
How to combine different ratios that share one common value/item
find the common value/item in each ratio - make that value the same in each ratio by multiplying the whole ratio by a value now you have the ratios in terms of each other/equal in their total - you can compare the values to the total or find a new ratio
Finding unknown values with Ratios
formula: quantiy of a = ratio of a to total * total if you know total quanity, use alegra to solve: ratio = x/total you need to know at least one of the real values in order to use the ratio to solve
How to solvle for divisibitu probelms with the ansers in fractios/varaibles ex. x/16
if a integer x is divisible by 24 and 50 - then the dividing numbers are factors and you need to find LCM 1) find LCM 2) plug in LCM for x/variable 3) test answers with LCM - it will be divisible if pluging in the LCM gives you a whole integer . Since the choices are fractions, the numerator must be a multiple of the denominator in order for the fraction to be an integer
How do you infer the greatest increase in a set of numbers
if all sets/items are seeing the same amount of increase, the greatest percentage increase will be in the set/item that had the lowest value/least quantity originally
Multiplying Decimals
ignor decimal points and multiply the two numbers, mupl;y the biggest # by each digit of smaller number add these numbers then count the digits to the right of the decimal points in the original numbers and place the decimal so there are the same number of digits to the right of the decimal
How do you recognize the multiples of 10
last digit is 0
standard deviation
measure of how much scores vary around the mean score/how spread apart a set of numebrs. the bigger the SD, the more spread apart are the #s
How to know the amount of zeros to put behind a number multiplied by 10ⁿ - positive numbers
move the decmial right n (exponent value) places (if already a decimal (3.3), you add 1 less zero than the value of the exponent) 2.1×10⁶ = 2,100,000 (practically, you are moving the decimal 6 times to the right)
ading and subtracting fractions
must have a common dominator. find LFC of denominators then multiply that factor to the numebrator and then add.
Fraction in the exponent rules
numerator is the expoent inside radical demonnator is before the radical X^a/b = ^b√x^a
the rules for raising odd and even integers to an exponent?
odd number - will awys be an odd numebr even number - always produce an even number Multiplying like #s (exe, oxo) will produce the same type of number
prime integer
only dividible by 1 and itself must be greater than 1 and positive
Types of prime factors (asked for on test)
prime factors - all listed factors (2 * 2 * 2 * 2 * 4*7) distinct prime factors - non repetitious list of all factors (2*4*7) the number of prime factors - the quantity of factors total (6)
How to find the probability of rolling something with dice when roled multiple times
probability of rolling an individual #: 1/6 probability you will roll a 3 once during 3 rolls of a dice: - multiply the probability of getting 3 (1/6) by the probability of getting any other value during the other rolls 1/6 * 5/6 * 5/6 = 5/18 then, multiply this probability by the amount of rolls you will do in total 5/18 * 3 rolls = 5/6
Getting an non perfect square expoent into another form (ex. 42⁸)
put the term into its two factors that are the same as waht your are tying to compare 42: 6*7 - raise the factor pair to the expoent value: 42: 6⁸*7⁸ then, elimate factors and compare
Radical / square root rules
rasing a radical to ^2 cancels out radical radical basically means a^1/2 the result of a square root can be positive or negative
How do you recognize the multiples of 12
sum of digits is a multiple of 3, and last two digits are a multiple of 4
variables squared on comparative probelms
the GRE wants you to remember that any number squared could be negative or posiiitve when you see a² on the GRE, the value you get for a could be +/- - often, answer is D because of the negative or possitive difference/multiple possible values
Absolute value: |2x-2|=4
the distance from zero that a number is on the number line, without considering direction. The absolute value of a number will never be shown as negative (but you cacualte it twice as is the = # is negative or positive)
What do you know if --- If an integer is the product of prime numbers
the integer was made by multiplying prime numbers (or prime factors) For an integer to be a multiple of another number, that number must be the product of multiplying two prime number
Standard Deviation: percentiles
the mean is the 50th percentile (50% chance of happening) The increase in percentiles that are closer to the mean will be bigger than the increase in percentiles that are furthur from the mean (less are distrubuted furhtur away from teh mean) this means that percentile 1 - percentile 2 is less than the mean - percentile 1
how to find the x or y-coordinate of the midpoint of a segment
the mid point of a segment is the middle of a line between point A and B - or the average of the x or y coordinates Formula: (x₁ + x₂)/2 (y₁ + y₂)/2
Multiples:
the product of a integer and another integer - the result of multiplying two other factors multiples are even spaced conservative numbers (separated by what they are multiplied by (ex multiple of 8, separated by 8) An integer divisible by another integer without a remainder is a multiple of that integer. Example: 12 is a multiple of 3, since 12 is divisible by 3.
Properties of Ratios
the proportioanl relationship between quantities/numbers values of a ratio represent the smallest possible quanitty of each item (but you can have any multiple of each ratio) the sum of the ratio needs to be a factor of the total quantity
How to find the probability of a normally distributed variable being wihtin a certain range of values
the range that is closer to the mean is most likeley to occur - within 1 SD: 68% values - more than 1 SD from mean (up or down): 32%
expoent rules
to use exponent rules, base #s must be the same multiplation = add expoent = a^b + a^c = a^b+c division = subtract exponetns = a^b / a^c = a^b-c (exponet)expoent = multipy expoents = (a^b)^c = a^b*c negative expoent = fraction = a^-b = 1/a^b
how to count the number of possibilities
use multiplication to find the numbers of possibilities.
How to solve: The integer x is divisible by both 8 and 14. x=?
use prime factorization to determine divisibility 1) find the smallest number x can be > find the Lowest comon multiple of 8 and 14 by prime factorization 2) find the prime factors that are ost common in the two numbers 8: 2*2*2 14: 2*7 multiply: 2*2*2*7 = 56 3) find the values that would make a whole numebr (integer) if X is 56
Absolute value and inequalities: greater than sign >
when X is greater than somthing, you have an OR statment x>3 or x<-3 looking at numbers that are separated on a number line
Absolute value and inequalities: Less than sign <
when X is less than somthing, you have an AND statement x<3 and x>-3 and statement you are looking for numbers between the two numbers
percent change/difference questions - when both percentages are based of the same total Ex. what percent greater than the sales of brand E laptops were the sales of brand D laptops?
when both percentages represent parts of teh same total/quantatiy, you can direclty compare the percentages 1) subtract the %s 2) place the difference in the percentages over the smaller quantity/what your comparing it against 3) turn the resulting fraction into a percentage
Probabilities - Compound Events - And vs OR in questions
when there are multiple ways to get something (by doing x and y OR just y) - MULPTLY the probabilities that are dependent (x AND y must happen) - Use for two step processes (choosing blue then also choosing another blue) - ADD the separate probabilities (getting x OR y) - you add the various ways to get the desired outcome after calculating the probabilities separately
Absolute value problems with inequality
when you have an absolute value problem with an inequality -> - the inequality sign will flip for the negative version of the problem
inequalities and quantitative comparison questions
when you multiply or divide a variable across an inequality, you need to account for that variable being positive or negative (changing inequality) DO NOT multiply or divide negative numbers across both quantities in all QC question - you are flipping the relationship between the quantities
Integers
whole numbers and their opposites/negatives. 0 is the only non neg non positive integer (0 is even)
Counting methods: How to solve Combination with slots (double)
write the # of possibilites for each slots divided by the # of slots 1) write the # of possibilities available in each slot/choice 2) divide/write under the number of slots 3) multiply fractions (eliminate common factors)
percent formulas
x% = x/100 x% of y = (x/100)*y percent = part/whole * 100% final as percent of oiginal = final/original*100% percent change = difference/orriginal*100%
(x^m)^n equals what?
x^(m*n)
(x^m)(x^n) equals what?
x^(m+n) addation
anything to the power of 0 =
x^0 = 1
