fractions
finding the prime factorization of large numbers close to squares
2491 2491 is 9 away from the square of 2500 2500-9=difference of squares (50+3)(50-3)=(53)(47)=prime factorization
A dry mixture consists of 3 cups of flour for every 2 cups sugar. How much sugar is in 4 cups of the mixture?
2x+3x=5x 5x=4 X=4/5 2(4/5)= 8/5 cup sugar
surface area of a cylinder
2πrh+2πr²
arc length
sector angle/ 360 degrees (2 pie r)
to find x and y intercepts
set x=0 to find y intercept set y=0 to find x intercept
when to use substitution or elimination for systems of equations
substitute if one coefficient is equal to +/- 1 linear elimination if all coefficients are greater than one to avoid fractions
to find the number of even factors of positive integer N
subtract odd factors from total factors; no direct way to find the even
combinations
order does not matter nCr
the number of integers from x to y inclusive is
y-x +1
probability: is it possible for two events to happen at the same time?
yes: NOT mutually exclusive P(A OR B)= P(A)+P(B)-P(A AND B) no: mutually exclusive P(A OR B)= P(A) + P(B)
circumference of a circle
πd
volume of a cylinder
π×r^2×h
1/9
0.11111
sum of arithmetic series when the first number is 1
n*(n+1)/2
volume of cylinder
V=πr²h
notwithstanding
despite
surface area of a cylinder
(2πr^2)+ 2πrh
if simplifying (48 x 75) / 3, you can only divide one numerator by the denominator
(48x75)/3= 16x75 or 48x25
(3x 10^20)(8x 10^30)
(8x3)(10^20 x 10^30) 24(10^50) (2.4 ^1 ) (10^50)= 2.4x 10^51
How to find the Least Common Multiple
(P X Q) / GCF = LCM
square of sum
(a+b)^2 = a^2 + 2ab + b^2
square of difference
(a-b)^2 = a^2 - 2ab + b^2
look for consecutive integers in variable form IF t is an integer
(t^2-2t)(t-1)= t(t-2)(t-1)
factor x^8 - 9y^2
(x^4+3y)(x^4 - 3y)
express |x + 7| < or equal to 3 as an ORDINARY inequality
-4 < or equal to x < or equal to 10
to simplify multiplication, you can move the decimals in opposite directions
.0003 x 40,000 3 x 4 = 12
to simplify division, you move decimals in the same direction
0.0045 / 0.09 45/900
1/8
0.125
1/7
0.143
1/6
0.1667
1/5
0.2
1/3
0.3333
3/8
0.375
2/5
0.4
3/5
0.6
5/8
0.625
2/3
0.66667
7/9
0.77777
4/5
0.8
5/6
0.8333
7/8
0.875
30 60 90
1 : square root of 3, 2
the multiplier for a P% increase
1+(P% in decimal form)
multiplier for a P% decrease
1-(P% as a decimal)
to find the number of factors of positive integer N
1. find prime factorization of N and write in exponent form 2. create a list of exponents of the prime factors, do not forget the 1's 3. add one to every number on the list, creating a new list 4. find the product of the new list; that product = factors of N
1/600
1/6 x 1/100 = 0.16666 x 10^-2= 0.0016666
factorial short cuts
10!-8!/7! 10!/8!= 10x9x8 - 8!/7!= 8 = 720-8=712 cancel out any factorial that are repeated in the bottom
you can rationalize a number with a radical in the denominator by multiplying the top and bottom by the radical
12/sq rt3= 12sq rt 3/ 3
we have 12 different major rides. We can choose any of the 3 of them. So this is simply going to be, 12 choose 3.
12C3 or 12C9 12 x 11 x 10/3!= 220
2SD
13.5 % (27% on each side of the mean)
adding to numerator and denominator
150/203 -> 147/200 ; 150/203 is bigger than 147/200; by subtracting 3 you are moving farther from one
sum of angles in a traingle
180
3 sd
2.5 % (5% on each side of the mean)
1SD
2/3 of data
factoring out a factorial
20!= 19x18x17x16!
Pythagorean triple
3, 4, 5 5 12 13 8 15 17
1 SD
34 % (68% on each side of the mean)
rationalizing with conjugates to get rid of radical in the denominator
4/ sq rt 5 + 1 4(sq rt 5 - 1)/ (sq rt 5 + 1 )(sq rt 5 -1) = 4(sqrt 5-1)/ (sqrt5^2)-(1^2)
use prime factorization to figure out if a number is a factor of a bigger number
4680= 2^3 x 3^2 x 5 x 13 25 5x5=NO 45 5 x 3^2= YES 65 5 x 13 = YES 85 5 x 17 =NO 120 12 x 10 = 2^3 x 3 x5= YES 180 15 x 12 = 5 x 3^2 x 2^2 = YES
what is the smallest POSITIVE INTEGER that when divided by 12 has a remainder of 5?
5 if dividend is less than the divisor it goes into it zero times and you are just left with the remaining 5
6C1
6
how many arrangements can 7 books be in if there are 4 different novels and 3 identical dictionaries?
7! / 3! divide by 3! since there are three dictionary repetitions
how many multiples of 8 are there from 200 to 640: inclusive counting
8 goes into 200 25 times 8 goes into 640 80 times 80-25= 55 + 1 = 56
combination short cuts: 20C3
=20 x 19 x 18 / 3! = 20 x 19 x 18 / 6 = 20 x 19 x 3 =
For average rate problems: use RTD or RTW chart with smart numbers for the distance
A car travels from town a to town b at an avg speed of 40 miles per hour and returns immediately along the same rate at an average speed of 50 miles per hour. What was the average speed in miles per hour for the round trip? R T D 40 t. 100 50. T. 100 Avg velocity= 200/ (100/40)(100/50)= 44.44
octagon area
A= 2(1 + square root of 2) a^2; a=side length
compound interest
A=P(1+r/100n)^nt
compound interest annually
A=Pr^y
the area of an equilateral triangle
A=s^2√3/4
Area of paralellogram
B x h
List/dataset
Can repeat numbers
Set
Cannot repeat numbers
on a ferry there are 50 cars and 10 trucks. the cars have an average mass of 1200 kg and the trucks have an average mass of 3000 kg. what is the average mass of all 60 vehicles on the ferry?
DO NOT ADD AVERAGES add sums!!!! avg(number of item)=sum of averages car sum= 50 (1200) = 60000 truck sum= 10 (3000) = 30000 add sums-> 60+30=90000 total average= 90/60=3/2=1.5= 1500 kg
Diagonal of a cube
Diagonal = side√3 D=s√3
Parallelogram: opposites sides are equal and opposite angles are equal
Diagonal will divide the parallelogram into two equal triangles
how to find the greatest common factor
Find the prime factorization of all numbers. Find the common prime factors from all numbers. Only use numbers that match from all prime factorizations. Multiply the common prime factors to find the GCF.
when counting problem contains restrictions, always do the most restrictive calculation first
In a certain summer school program, there are five periods in the day. Each student takes English, Math, History, Science, and Science Lab. In how many different orders can a student's schedule be arranged given that science lab must immediately follow the science class. _ _ _ _ _ science=4 science lab=1 english=3 math=2 history=1 24 ways
The double matrix method can enormously simplify problems in which each member is placed into two different kinds of categories. The entries in the row sum to the row totals in the rightmost column. The entries in the columns sum to the column totals in the bottom row. And the grand total, in the lower right-hand box, is everyone:
In a company of 300 employees, 120 are females. A total of 200 employees have advanced degrees and the rest have college degree only. If 80 employees are males with college degrees only, how many are females with advanced degrees?
probability formula:
P(A or B)= P(A) + P(B) - P(A and B)
difference of 2 squares formula
P^2-Q^2= (P+Q)(P-Q)
A regular polygon with all sides equal and all angles equal will maximize area for a given perimeter and minimize perimeter for a given area
Squares
time formula
distance/speed
divisible by 6
divisible by 2 and 3
sum of angles in a polygon formula
Sum of angles in n-gon=180×(n−2)
permutations:
order does matter n!
How many positive integers less than 100 have a remainder of 2 when divided by 13?
To solve this problem, start listing numbers that have a remainder 2 when divided by 13. If you're having trouble thinking of one, start with 15. 15 is 2 more than 13, so when you divide 15 by 13, you get a remainder of 2. Now that we have 15, just keep adding 13: 15, 28, 41, 54, 67, 80, 93
Isosceles triangle
Two equal angles and sides
in a certain company, 70% of employees are marketers who make an average of 40,000; 20% are programmers who make an average of 80,000; and 10% are managers, who make an average of 120,000. what is the average salary of all employees at this company?
average total= .7(40) + .2(80) + 0.1(120) = 56 56,000
formula for the nth term in a series of consecutive integers
a + multiple(n-1)
|A|<19
a<19 a>-19
work equation
a=rt a=amount of energy or work r=work rate t=time
we can add inequalities that are in the same direction
a>b c>d a+c > b+d
we can subtract BIG minus SMALL of inequalities that are in the same directions
a>b c>d a-d > b-c
If Michael can shovel all the snow off a standard driveway in 12 minutes, and Eamon can shovel all the snow off a standard driveway in 36 minutes, then working together, how many minutes would it take for them both to shovel all the snow off a standard driveway?
add the rates 1/12 + 1/36 = 1/9 they can shovel the driveway together in 9 minutes
if given sixth number and asks for 100th number
always find difference between 100 and 6 and then mulitply by multiple given
for problems with multiple workers/machines create rates for each and add the rates to get the combined time
andrea and bob take 3 hours to clean a car andrea takes 4 hours to clean a car how long does it take bob to clean the car? ab= 1/3 a=1/4 1/3-1/4=1/12 it takes bob 12 hours to clean a car
difference of two squares
a² - b² = (a + b)(a - b)
Area of trapezoid
b1 + b2 /2 * h
what is the probability that the sum of two dice rolls will roll anything but an 8?
calculate the prob that the sum WILL yield a sum of 8 and then subtract from 1 5 combos of 2 dice= a sum of 8 2 + 6, 3 + 5, 4 + 4, 5 + 3, 6 + 2. P(sum=8)= 5/36 36 comes from 6x6 for how many combs you get when rolling a die twice 1- 5/36= 31/36
generalized OR rule: MUTUALLY EXCLUSIVE P(A OR B)= P(A) + P(B) - (P(A AND B))
cannot land on both heads and tails simultaneously -drawing a heart and spade from the same deck -cannot occur at the same time
remainder formula-use if must find remainder
d/s = Q + r/s d=dividend s=divisor q=quotient r=remainder
The diagonal of a square
d=s√2
divisible by 5
end in 0 or 5
divisble by 2
end in an even number
to find the number of odd factors of positive integer N
get rid of factors of two and follow the same procedure
shrinking and expanding gaps
going in opposite directions ADD going in the same direction SUBTRACT think about directions with expanding vs shrinking solving D=RT for the gap itself can enormously simplify such problems
30 60 90 triangle
hypotenuse: leg opposite 60 degree angle = 2: √3
an inscribed angle is equal to half of the arc it intercepts
if central angle is 60 degrees than inscribed angle of the same intercepted arc is 30 degrees
the complement of at least one is none: P(at least one)= 1- P(zero successes)
if i roll a six-sided die eight times, what is the probability i will roll a six AT LEAST once? find the probability you will not roll a six at all P(will not roll a six)= 5/6 roll 8 times= P= (5/6)^8 1-P(5/6^8)= P(will roll a six at least once)
fundamental counting principle:
if k ways n ways i ways= k x n x i = total ways (order does not matter)
for finding the hypotenuse of a triangle with large sides: if the sides given are larger, the sides of the triangle given are relatively large numbers, divide down by the greatest common factor
if legs are 45 and 24 GCF=3 legs are now 8 and 15: hypotenuse is 17 3x17=51 51 is the hypotenuse of the larger triangle
divisible by 4
if the last two digits form a number that is divisible by 4
if the median of list b is exactly 4 higher than the median of list a, what is the value of x?
list a = (4,7,10,18) list b = (4,7,10,18,25,x) list a median=8.5 list b median=12.5 numbers in list b 4, 7, 10 are below the median, so x must be above the median 12.5= (10+x)/2 x=15
Q1 and Q3
median of the lower list median of the upper list
1/20
multiply both by 5 to get 5/100 and 0.05
rebuilding remainder formula
multiply both sides by S d = sQ + r
probability combination formula
n! / [r!(n-r)!]
sum of arithmetic series
n(A1+An)/2 n=number of terms in the series A1=first term in the series An=last term in the series you can solve for n through the inclusive formula : y-x +1
sum of sequences of evenly spaced numbers
n/2 (a1 + an) what is the sum of all multiples of 20 from 160 to 840? 8 x 20 = 160 42 x 20 = 840 42-8=34 +1 =35 (35/2)(160+840)= 17500
for finding the percent increase or decrease when given starting values
new= multiplier*old multiplier=new price /old price change ratio back to a percent if greater than 1=increase_> x-1=percent increase if less than one=decrease-> 1-x=percent decrease
a committee of three will be selected from 8 people including alice and bob what is the probability that the committee must include alice but not bob?
no restrictions = 8c3 =8x 7 x6/ 3!= 56 restrictions: must have a; cannot have b a- - - - - - - --p(cdefgh)= 6c2 6x5/2= 15 15/56
express 5< x< 17 as an ABSOLUTE VALUE inequality
take 5 + 17 /2 =11 notice that 5 and 17 are both 6 away from 11 |x-11|<6
six children will sit in a row of six chairs, but jackie and marilyn cannot be seated by eachother. how many arrangements are possible?
take total arrangements irregardless of order= 6!=720 take total arrangements if jackie and marilyn CAN sit by eachother jmxxx . mjxxx xjmxx . xmjxx xxjmx . xxmjx xxxjm . xxxmj = 10 arrangements plus the four empty seats= 4! 10(4!)= 240 720-240= 480 arrangements
to check if larger number is prime
test if divisible by prime number less than 10: 2, 3, 5, 7
a circle in the xy plane has a center of (6,3) and a radius of 5. find the two x-intercepts
the hypotenuse of two triangles formed between center point and x-axis=5 vertical leg=3 since y coordinate of center point=3 5^2-3^2= 4^2 two x-intercepts are 6+4 and 6-4
a parabola with a vertex (2,1) has a y-intercept of 9. what is the x-coordinate of the other point on the parabola with a y-coordinate of 9?
the line of symmetry of this parabola is x=2 since one x-intercept is 2,9 the other intercept is 4,9 the x-intercept is 4
divisible by 3
the sum of all the digits must be divisible by three
divisible by 9
the sum of digits is divisible by 9.
if events are INDEPENDENt: this is an AND-MULTIPLY P(A)XP(B)
tossing a head on a coin and rolling a six on a die -the probability of one does not affect the probability of the other
1 is not a prime number
true
2 is a prime number
true
inscribed angle
vertex on the circle itself rather than on the center of the circle
inclusive counting
we use inclusive counting when both endpoints, the starting value and ending value are included in what we are counting: perform ordinary subtraction of endpoints and then add one to the difference to account for the included starting value
to determine the probability that event X AND Y will both occur, MULTIPLY the two probabilities together
what is the probability that a fair coin flipped twice will land on heads both times? 1/2 x 1/2 = 1/4
to determine the probability that event X OR event Y will occur, ADD the two probabilities together: make sure events are mutually exclusive
what is the probability that a fair die rolled once will land on either 4 or 5?
if problem contains multiple success possibilities-with wording like at least and at most- find probability of failures; 1-x
what is the probability that on three rolls of a single roll die, AT LEAST ONE of the rolls will be a six? failure=what is the probability that NONE of the rolls will yield a six? 5/6 x 5/6 x 5/6= 125/216 1- (125/216) = 91/216
|x +3|=5
x+3=5 x+3= -5
30 60 90
x, x(SR3), 2x
45 45 90
x, x, x√2
any even power of x is the square of another power
x^10= x^5 * x^5 = (x^5)^2