Number Properties
Multiplying Binomials
FOIL: First, Outer, Inner, Last... Combine Like Terms
Using the Average to Find the Sum
Sum=(Average) x (Number of Terms) if the average of ten numbers is 50 then they add up to 10 x 50 or 500
adding and subtracting monomials
To combine like terms, keep the variable part unchanged while adding or subtractracting coefficients 2a +3a = (2+3)a =5a
greatest common factor (GCF)
To find the greatest common factor break down both numbers into their prime factorization and take all the prime factors they have in common 36=2x2x3x3 and 48=2x2x2x2x3 what they have in common is two 2s and one 3 so the gcf is 2x2x3= 12
prime factorization
a prime number is a postive integer that has exactly 2 postive integers factors 1 and itself the first eight prime numbers are 2 3 5 7 11 13 17 19 to find the prime factorization of a intager just keep breaking it up into factors until all the factors are prime to find the prime factorization of 36 for example you could begin breaking it into 4x9 36=4x9=2x2x3x3
rational/irrational
a rational number is a number that can be expressedas a ratio of two integers irratonal numbers are real numbers they have locations on the number line they just cant be expressed precisley as a fration or decimal for the purposes of the ACT the most important irrational numbers are root 2 root 3 and pie
Real/Imaginary
a real number is a number that has a location on the number line. on the ACT, imaginary numbers are numbers that involve the square root of a negative number. root -4 is an imaginary number.
multiples of 5 and 10
an integer is divisable by 5 if the last digit is 5 or a 0 an integer is divisable by 10 if the last digit is 0 the last digit of 665 is 5 so 665 is a multiple of 5 but not a multiple of 10
multiples of 3 and 9
an integer is divisable by three if the sum of its digits is divisible by 3 and intieger is divisable by 9 if the sum of its digits is divisible by 9 the sum of the digits in 957 is 21 which is divisable by 3 but not by 9 so 957 is divisible by 3 but not by 9
multiples of 2 and 4
an intiger is divisable by 2 if the last digit is even an integer is divisable by 4 if the last two digits form a multiple of 4 the last digit of 562 is 2 wich is even so 562 is a multiple of 2 the last two digits make 62 wich is not divisable by 4 so 562 is not a multiple of 4
average rate
average rate is not simply the average of the rates
part to part part to whole ratios
if the parts add up to the whole a part to part ratio can be turned into two part to whole ratios by putting each number in the orginal ratio over the sum of the numbers. if the ratio of males to females is 1 to 2 then the males-to people ratio is 1/1+2= 1/3 and the females to people ratio is 2/1+2 = 2/3 or 2/3 of all the people are female
Integer/Noninteger
integers are whole numbers; they include negative whole numbers and zero
undefined
on the ACT, undifined almost always means division by zero the expression a/bc is undefined if either b or c equals zero
comparing fractions
one way to compare fractions is to re express them with a common denominator 3/4=21/28 and 5/7 =20/28; 21/28 is greater than 20/28 so 3/4 is greater than 5/7 another way to compare fractions is to convert them both to decimals. 3/4 converts to .75 and 5/7 converts to approximately .714
probability
probability = favorable outcomes / total possible outcomes if you have 12 shirts in a drawer and 9 of them are white the prob of picking a white shirt at random is 9/12=3/4 this prob can also be expresed as .75 of 75%
Multiplying other Polynomials
product is found by multiplying each term in the first polynomial by each term in the second.
solving a propotion
proportion cross multiply
absolute value
the absolute value signs a lot like parentheses do what's inside of them first and then take the absolute value of the result dont take the absolute value of each piece between the bars before calculating in order to calculate [(-12)+5-(-4)] - [5+(-10)] first do whats inside the bars to get [-3]-[-5] which is 3-5 or 2
Factor/Multiple
the factors of intiger n are the postive integers that divide into n with no remainder the multiples of n are the integers that n divides into with no remainder six is a factor of 12 and 24 is a factor of 12 12 is both a factor and a multiple of itself
counting the posibilities
the fundamental counting principle if there are m ways one event can happen and n ways a second event can happen then there are m x n ways for the two events to happen for example with 5 shirts and 7 pairs of pants to chose from you can put together 5 x 7 =35 different outfits
identifyong the parts and the whole
the key to solving most fraction and percent story problems is to identify the part and the whole ussaly youll find the part associated with the verb is/are and the whole associated with the word of in the setnece half of the boys and blonds the whole is the boys and the part is the blonds
multiplying and dividing roots
the product of square roots is equal to the square root of the product root 3 x root 5 = root 3x5 =root 15 the quotient of square roots is equal to the square root of the quotient root 6 / root 3 =root 6/3 =root 2
remainders
the remainder is the whole number left over after division 487 is 2 more than 485 which is a multiple of 5 so when 487 is divided by 5 the remainder will be 2
Adding subtracting signed numbers
to add a postive and a negative first ignore the signs and find the postive diffrence between the number parts. then attach the sign of the original number to the larger number part. for example to add 23 and -34 first we ignore the minus sign and find the postive diffrence between 23 and 34 thats 11 then we attatch the sign of the number with the larger number part in this case its the minus sign from the -34 so 23+(-34)=-11 make subtraction situations simpler by turning them into addition for example think of -17-(-21)as -17+(+21) to add or subtract a string of positives and negatives, first turn everything into addition then combine the postives and negtives so that the string is reduced to the sum of a single postive number and a single negative number
adding subtracting fractions
to add or subtract fractions first find a common denominator and then add or subtract the numerators
adding and subtracting polynomials
to add or subtract polynomials combine like terms (3x^2 +5x-7) - (x^2 +12) =(3x^2-x^2) + 5x + (-7-12) = 2x^2 +5x-19
converting fractions to decimals
to convert a fraction to a decimal divide the bottom into the top to convert 5/8 divide 8 into 5 yielding .625
converting an improper fraction to a mixed number
to convert a improper fraction to a mixed number divide the denominator into the numerator to get a whole number quotient with a remainder. the quotent becomes the whole number part of the mixed number and the remainder becomes the new numerator with the same denominator. for example to convert 108/5 first divide 5 into 108 wich yields 21 with a remainder of 3 therefore 108/5=21 3/5
converting a mixed number to an improper fraction
to convert a mixed number to an improper fraction multiply the whole number part by the denominator then add the numerator the result is the new numerator. to convert 7 1/3 first multiply 7 by 3 then add 1 to get the new numerator of 22 put that over the same denominator 3 to get 22/3
counting consecutive integers
to count consecutive intigers subtract the smallest from the largest and add 1 to count the integers 13 through 31 subtract 31-13=18 then add 18+1=19
relitive primes
to detemine whether two integers are realitive primes break them both down to their prime factorization for example 35=5x7 and 54=2x3x3x3 they have no prime factors in common so 35 and 54 are relative primes
combined precent increase and decrease
to determine the combined effect of multiple precents increase and or decrease start with 100 and see what happens example a price went up 10% one year and the new price went up 20% the next year what was the combined precent increase set up first year 100+(10% of 100)=110 second year 110+(20% of 110) =132 thats a combined 32% increase
dividing fractions
to divide fractions invert the second one and multiply
evaluating an expression
to evaluate an algabreic exprssion plug in the given values for the unknowns and calculate according to pemdas to find the value of x^2 + 5x -6 when x= -2 plug in -2 for x (-2)^2 +5(-2)-6 = 4-10-6= -12
Finding the Missing Number
to find a missing number when youre given the advrage use the sum if the average of four numbers is 7 then the sum of those four numbers 4 x 7 or 28 supose that three of the numbers are 3,5,and 8 these numbers add up to 16 of that 28 which leaves 12 for the fourth number
repeating decimal
to find a particular digit in a repeating decimal note the number of digits in the cluster that repeats. if there are two digits in that cluster then every second digit is the same if there are three digits in that cluster then every third digit is the same and so on. for example the decimal eguivalent of 1/27 is .037037037..... there are three digits in the repeating cluster so every third digit is the same 7. to find the 50th digit look for the multiple of 3 just less than 50 thats 48 tha48th digit is 7 and with the 49th digit the patern repeats with 0 the 50th digit is 3.
Setting up a Ratio
to find a ratio put the number associated with the word of on top and the quanity associated with the word to on the bottom and reduce the ratio of 20 oranges to 12 apples is 20/12 which reduces to 5/3
average formula
to find the average of a set of numbers add them up and divide by the number of numbers average= sum of the terms/ number of terms to find the average of the five numbers 12,15,23,40,and 40 first add them 12+15+23+40+40=130 then divide the sum by 5 130/5=26
average of evenly spaced numbers
to find the avrage of evenly spaced numbers just average the smalest number and the largest number. the average of all the integers from 13 through 77 is the same as the average of 13 and 77 13+77 /2 = 90/2 = 45
LCM (Least Common Multiple)
to find the least common multiple check out the multiples of the larger number until you find one thats also a multiple of the smaller to find the LCM of 12 and 15 begin by taking the multiples of 15 15 is not divisible by 12 30 is not nor is 45 but the next multiple of 15 60 is divisible by 12 so its the least common multiple
finding the original whole
to find the orginal whole before a precent increase or decrease set up an equation think of a 15% increase over x as 1.15x set up 1.15x=59,346
reciprocal
to find the reciprocal of a fraction switch the numerator and the denominator the reciprocal of 3/7 is 7/3 the reciprocal of 5 is 1/5 the product of recipocals is 1
percent increase/decrease
to increase a number by a precent add the precent to 100% convert to a decimial and multiply to increase 40 by 25% add 25% to 100% convert 125% to 1.25 and multiply by 40. 1.25 x 40 = 50
multiplying/dividing signed numbers
to multiply and or divide postives and negatives treat the number parts as usal and attach a negative sign if there were originally an odd number of negitives to multiply -2,-3,and -5 first multiply the number parts 2x3x5=30 then go back and note thet there were three-and odd number negatives so the product is negative (-2)x(-3)x(-5)=-30
multiplying fractions
to multiply fractions multiply the numerators and multiply the denominators
Multiplying monomials
to multiply monomials multiply the coeficents and the variables seperatly
multiplying and dividing powers
to multiply powers with the same base add the exponets y^3 x y^4 = y^3+4 = Y^7 to divide powers you just subtract insted of adding
Even/Odd
to predict wehter a summ diffrence or product will be even or odd just take simple numbers like 1 and 2 and see what happens there are rules "odd times even is even" for example but theres no need to memorize them what happens with one set of numbers generally happens with all similar sets
raising powers to powers
to raise a power to an exponent, multiply the exponents
reducing fractions
to reduce a fraction to lowest terms factor out and cancle all factors the numerator and denominator have in common
Simplifying Square Roots
to simplify a square root factor out the perfect squares under the radical unsquare them and put the result in front
rate
to solve a rate problem use the units to keep things straight
PEMDAS
when preforming multiple operations remember PEMDAS, which means parentheses first then exponets then multiplication and division (left to right)then addition and subtraction (left to right)
precent formula
wheter you need to find the part the whole or the percent use the same formula part=percent x whole
adding and subtracting roots
you can add or subtract radical expressions only if the part under the radical is the same
common multiple
you can allways get a common multiple of two numbers by multiplying them but unless the two numbers are relative primes the product will not be the least common multiple for example to find a common multiple for 12 and 15 you could just multiply 12x 15= 180