Quant Flashcards
(a-b+1)/((ab-b)-(1+a)(1-a))
(a-b+1)/((b(a-1)+(1+a)(1-a))
(a-b+1)/((-b(1-a)+(1+a)(1-a))
(a-b+1)/((1-a)(-b+1+a)
1/(1-a)
Given a/b and c/d
a/b+c/d=?
a/b-c/d=?
ad+cb/(db)
ad-cb/(bd)
Zero product property
If 2 things multiply to 0, at least one of them must be 0
Standard Deviation
measures “how far” a set of values are from the average of that set
High value= mean + x(sd)
Low value = mean - x(sd)
x–> the number of standard deviations from the mean
sqrt(2) sqrt(3) sqrt(5) sqrt(6) sqrt(7) sqrt(8)
=1.4 =1.7 =2.2 =2.4 =2.6 =2.8
|a-b| >=|a|-|b| if…
b does not equal 0 and |a-b|=|a|-|b|
a and b share the same signs and |a|>=|b|
only true of (+)(+) or (-)(-)
Compound interest
A=P(1+r/n)^(nt)
A=Future value of investment P=Initial value R=Interest rate per year n=Number of compounding periods per year t=time(#years)
Area of a regular hexagon
3sqrt(3)/2*s^2, where S is the the length of any of the sides
3sqrt(3)/2 = 2.6
When solving “at least” problems, first calculate the probabilities of the mutually exclusive scenarios, then add those probabilities to determine the final
In city Y, the probability that it will snow more than 5 inches during any given snowstorm in the month of January is 1/3. If there are 3 snowstorms, what is the probability that it snows more than 5 inches in at least 2 of the snowstorms
Scenario 1:
YYN
Arranged in 3!/2!= 3 ways
3(1/31/3*2/3)= 6/27
Scenario 2:
YYY
(1/31/31/3) = 1/27
6/27+1/27 = 7/27
Determining the number of primes in a factorial when base of a divisor is power of a prime
30!/4^n
Determine number of 2’s in 30!. Then create and simplify an inequality
30!/4^n-->30/2^2n 30/2 = 15 30/4=7 30/8= 3 30/16= 1 15+7+3+1=26
2n<=26
n<=13
Zero slope vs undefined slope
which is horizontal and which is vertical
Zero slope is horizontal
undefined slope is vertical
Exponential decay example:
If money in an account decreased by 50% each week, the amount of money by which the account decreased during the 3rd week is what fraction of the amount of money at the end of the first week
Initial Amt Amt Removed Amt Remaining
1st: x 1/2x 1/2x
2nd: 1/2x 1/2x1/2=1/4x 1/4x
3rd: 1/4x 1/4x1/2 = 1/8x 1/8x
(1/8x)/(1/2x) = 1/4x
For a function in the form of f(x)=kx^n+c where n is a positive even integer and K is nonzero:
if k>0, the range of f(x) is all real numbers >=C
if K<0, the range is all real numbers <=c
Trailing 0’s
Created by 5x2 pairs. Each pair in a number creates 1 trailing 0
Square roots and squares of fractions
(a/b)^2
sqrt(x/y)
(a/b)^2= a^2/b^2 sqrt(x/y)= sqrt(x)/sqrt(y)
x is jointly proportional to y and z
x = ykz
Leading 0’s
If x is an integer with K digits, then 1/x will have k-1 leading 0’s. If x is a perfect power of 10, there will be k-2 leading 0’s
2/7
.286
Slope intercept equation
y = mx+b
y= y coordinate for a point on the line
x = corresponding x coordinate for the point on the line
m - slope of the line
b = y intercept
If a number x has y prime factors, then x^n will have the same y factors
18 = 3^2*2 18^3 = 3^3 *2^2
If W is divisible by 6 and 9. W must be a multiple of which of the following:
4 12 18 24 36
18
We are given that W is divisible by 6 and 9. In other words, W is a multiple of both 6 and 9. To determine what must be a multiple of 6 and 9, we can determine the LCM(6,9)
If Z is divisible by both x and y, Z must also be divisible by the LCM of x and y
Linear Growth Example:
An investment grows by the same amount each year. The value of the portfolio after year 8 was 5/4 the value after year 5. If the portfolio began with $100, what is the amount the portfolio grew by each year?
year 1: 100+x
year 2: 100+2x
…
year 8=100+8x
year 8=5/4(year 5)
100+8x= 5/4(100+5x)
x=14
“Percent of” means to multiple a given percent by a given value
5 percent of z = ?
400 percent of y =?
m percent of p =?
(5/100)z
(400/100)y
(m/100)*p
x^7 has the same sign as x
Numbers raised to odd powers reveal positvity/negativity
It’s impossible to determine the sign of numbers raised to even powers
Using trailing zeros to determine the number of digits in an integer
- PF number
- Count number of 5x2 pairs
- collect # unpaired 5 and 2 along with other non-0 pf’s and multiply together
- Add 5x2 pairs with total digits of number in step 3
Determining the number of primes in a factorial when base of a divisor is power of a prime
30!/4^n
Determine number of 2’s in 30!. Then create and simplify an inequality
30!/4^n-->30/2^2n 30/2 = 15 30/4=7 30/8= 3 30/16= 1 15+7+3+1=26
2n<=26
n<=13
Consecutive multiples of integers
Represented as x + (x+”multiple) + (x+”2*multiple)
Consecutive multiples of 5: x + (x+5) + (x+10) + (x+15)
Length of a diagonal of a rectangle
Sqrt(L^2+W^2)
When A and B are not mutually exclusive:
P(A or B)= P(A)+P(B) - P(A and B)
Exponential decay table
Initial Amt Amt Removed Amt Remaining
First reduction
Second reduction
3rd reduction
Number of seconds is one day
86,400
606024
Exponential Growth
amount of growth=(initial value)*(growth factor)^growth periods
15^2
225
Circle 3 part ratios
central angle/360 = arc length / circumference = area of sector/ area of circle
If 2 events are independent:
P(A and B)= P(A) * P(B)
Probability of a magician pulling a rabbit out of his hat on any attempt is 1/3. What is probability he will pull it out on the first but not the other 3.
Comparing Standard deviations without fully calculating standard deviation
- Determine the mean for each set
- For each individual mean, determine the absolute difference between the mean of the set and each data point in that set
- sum the differences obtained from each individual set
The set with the greater sum has the greater standard deviation
Pythagorean Theory
In any right triangle, C^2=A^2+B^2, where C is the length of the triangle’s hypotenuse.
17^2
289
Multiplication rules of even and odd
The product of an even number and any integer is always even
The product of an odd number and odd number will always be odd
sqrt(x+y)^2 = abs(x+y)
sqrt(x+5)^2= sqrt(36)
abs(x+5)=6
6/7
.857
(x+y)^2 other forms
(x+y)(x+y)
x^2+y^2 + 2xy
9^3
729
If we add or subtract the same amount to or from each term in a data set, the standard deviation does not change
That is, we can have data sets with the same standard deviation and different averages
Distance between 2 points formula
Sqr((x2-x1)^2+(y2-y1^2))
LCM of a set provides all unique prime factors. Thus, it provides all the unique prime factors of the product f all numbers in the set
pf of x = 2,3
pf of y = 3,5
pf of z = 2,5
xyz cannot have any prime factors other than 2,3,5
6^3=
7^3=
8^3=
9^3=
6^3= 216
7^3=343
8^3= 512
9^3=729
If we know y divides into x, LCM(x,y) is x and GCF(x,y) is y
LCM(100,25) = 100 GCF(100,25) = 25
Area of equilateral triangle
A=S^2(sqrt(3)/4
Properties of 0
Any number divided by 0 is undefined Square root of 0 is 0 0 raised to any positive integer is 0 Any number raised to 0 is 1 0 is even
0!=?
1
Units digits of powers: 2= 3= 4= 5= 6= 7= 8= 9-
2= 2,4,8,6 3= 3,9,7,1 4= 4,6 5= 5 6= 6 7= 7,9,3,1 8= 8,4,2,6 9= 9,1
sqrt(a)/sqrt(b) = sqrt(a/b)
sqrt(54)/sqrt(6) = sqrt(9)
X is a dividend of y
Means x/y
|a-b| >=|a|-|b| if…
b does not equal 0 and |a-b|=|a|-|b|
a and b share the same signs and |a|>=|b|
only true of (+)(+) or (-)(-)
“What percent of” problems
100 is what percent of 50
Can translate “what percent” into x/100, and then set up an equation to solve for x
or we can use (a/b) * 100
100/50*100
When an event has more than one possible outcome, each possible outcome must be considered when calculating the probability that the event will occur. Thus, to determine the actual probability:
The prob that it rains in town X on any given day is 40 percent. What is the prob that it will rain in town X on exactly 3 days in a certain 4 day period
(number of outcomes producing the event) * (probability of one outcome)
RRRN can be arranged in 4!/3!=4 ways.
4(2/5 * 2/5 * 2/5 * 3/5)
1/7
.143
18^2
324
If the product of 2 integers is 1…
Both are either 1 or -1
Adding/subtracting a constant to numerator and denominator of a fraction
(a+c)/(b+c)
Adding constant to both will always make the fraction larger –>3/5–> 3+2/5+2 –> 5/7 > 3/5
Subtracting will always make the fraction smaller
Multiplying will always keep the fraction the same size
3/5 –> 32/52 = 6/10
Formula for division
x/y = Q + r/y
y=xq+r
If 2 events are complementary:
P(A)+P(Not A) =1
eg. Probability that it will rain in town x is 25 percent. what is probability it does not rain in town x tomorrow
Area of a trapezoid
((Base 1 + Base 2)*Height)/2
When (x,y) is reflected over the line y=x
its image is (y,x)
2,3) becomes (3,2
When (x,y) is reflected over the line y=b
its image becomes (x, 2b-y)
(5,-2) becomes (5,4) when reflected over the line y=1
Determine the number of primes in a factorial
Y!/x
Divide 1 by powers of x (x^1, x^2, x^3…) until quotient = 0
Add quotients
21!/3^n –> 21/3= 7, 21/9=2
7+2=9
The sum of the interior angles of any hexagon is 720. Any one interior angle of a regular hexagon measures 120
A regular hexagon can be divided into 6 equilateral triangles
(A or B)=
Numbers of members in either set
A car dealership has 50 red cars and 35 convertibles. If there are 65 cars that are either red or convertible, how many are both red and convertible
(A or B)= #A + #B -#(A and B)
= 50 + 35 - 65
Geometric sequence is one in which the ratio between every pair of two consecutive terms is the same.
an=a1*r^(n-1)
Where an is the nth term, a1 is the first, and r is the common ratio
Sum of integers from 101 to 202, inclusive
(101+202)/2 = 151.5 – average
Number of terms = 202-101+1 = 102
151.5*102=15453
Base 1/6 fractions 1/6= 2/6= 3/6= 4/6= 5/6=
1/6=.167 2/6= .333 3/6=.5 4/6=.667 5/6= .833
Catch up and pass problems
The faster objects distance is equal to the slower objects distance plus any difference in starting points and any distance by which the faster object must pass the slower
a fast way to calculate catch up and pass is:
time= Change in distance/Change in rate
Dont attemplt this on catch up problems
Simple Interest
simple interest= principal x rate x time
When an investment grows via simple interest, the interest earned in any period on the principal investment is not factored into future amounts of interest earned
When A and B are mutually exclusive, P(A) or P(B):
P(A or B)=P(A)+P(B)
What is the remainder of (12 * 13 * 17)/5?
12/5 - R=2
13/5 - R=3
17/5 - R=2
232=12. Remove excess. Remainder= 2
Y divides into x evenly
Means x/y
Division rules for even/odd
even/odd = even odd/odd= odd
the sum of the first n terms of an arithmetic sequence:
Sn=n/2(a1+an)
work(object 1)+work(object 2)= Work(Total)
Work(Total) usually =1, because 1 job is being completed
Catch up and pass problems
The faster objects distance is equal to the slower objects distance plus any difference in starting points and any distance by which the faster object must pass the slower
a fast way to calculate catch up and pass is:
time= Change in distance/Change in rate
Dont attempt this on catch up problems
Surface area of a cube:
6s^2
5/7
.714
Finding the number of factors in a number
x = pf^e1 x pf^e2…
factors = (e1+1)(e2+2)(e3+3)
240= 2^453
factors=(4+1)(1+1)(1+1)
factors= (5)(2)(2)
factors = 20
Price per item equation
Price per item = Total cost of identical purchases/(number of identical purchases)
Change in worker problems
Proportion method
x workers/(combined rate of x workers) = y workers/(combined rate of y workers)
Volume of a cylinder
v=pi* r^2*h
If LCM (x,y) = p and GCF(x,y)=q, then xy=pq
m=24—> 2^33
n=30 –>23*5
LCM(m,n) =2^335=120
GCG(m,n) = 2*3 = 6
mn= 24*30 = 720 LCM(x,y)*GCF(x,y) = 720
Diagonal of a cube:
s*sqrt(3)
“percent less problems”
1 percent less than z=
60 percent less than z=
98 percent less than z
In percent less than problems, x% less than y is equivalent to (100-x)% of y
(99/100)z (99 percent of z)
(40/100)z ) 40 percent of z
(2/100)z (2 percent of z
X is inversely proportional to cube of y and directly proportional to square root of z
x=k*sqrt(z)/y^3
Area of parallogram
base x height
Median
When a set is numerically ordered, the median is the value in the middle of the arranged set
for even numbered sets, the median is the average of the 2 middle
(n+1)/2 gives the spot of the median for an odd numbered set
Subtracting whole number and fraction
A-b/c
(ac-b)/c
or (A-1)(c/c)-b/c
6-2/3 = 5(3/3)-2/3 = 5 1/3
Y is a divisor of x
Means x/y
Difference of 2 squares
X^2-Y^2 = (x+y)(x-y) 4x^2-100= (2x+10)(2x-10) 1/36x^2-25 = (1/6x-5)(1/6x+5) 2^12-1 = (2^6-1)(2^6+1)
If the value of 1 variable (x) increases while that of a second variable (y) decreases and they are inversely proportional, they are related by what equation?
y=k/x
To find the number of terms between 2 numbers in a set of consecutive integers, excluding both endpoints:
subtract the first number from the last number and then subtract 1
Last number - first number - 1
If 2 absolute values are equal, they are either equal or opposites
|r+s|=|m+n| if r+s=m+n or r+s=-(m+n)
15q/17 –> Q must be multiple of 17 in order to be an integer
14rst/18=?
rst must be multiple of 9 to be an integer
Number of minutes in 1 day
1440
60*24
16^2
256
Surface area of cube and rectangle
Cube = 6 s^2 rectangle = 2(LW) + 2(LH) + 2(WH)
Functions domain vs range
for domain- look at graph from left to right
for range - look at graph from top to bottom
The slopes of 2 perpendicular lines multiply to -1 and are negative reciprocals of each other. Negative reciprocals multiply to -1
10 * -1/10 = -1
Bow tie method to compare fraction size
Given a/b & c/d,
a/b > c/d if ad>bc
Weighted average
sum of weighted items/total number of weighted items
OR
(data point 1)(frequency of data point 1) + (data point 2)(frequency of data point 2)/total frequency of data points
Dry mixture problems
Create a table with component 1, component 2, and final mixture as the rows.
Units and total are the columns
Dividing Decimals
- Move decimal of divisor(#after division sign) until it becomes whole number
- Move decimal of dividend (#before division sign) to the right same number of places as divisor
- Divide by long division. Keep decimal in same location as new dividend
|a+b| <= |a|+|b| if…
|a+b|=|a|+|b| and a&b share the same sign
The equation for a vertical line
the equation for a horizontal line
vertical line - x =a, where a is the x-intercept of that line
horizontal line - y=b, where b is the y intercept of that line
Bookend method to find average
For an odd numbered set spaced evenly, add the first and last number and divide by 2.
For even numbered set spaced evenly, average of the terms is the average of the 2 middle terms
A y=-x reflection causes the original x and y coordinates to switch positions and causes the signs of the coordinates to flip
(2,10)–?
(-10,-2)
Prime numbers under 100
25 total
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
If a number x has y prime factors, then x^n will have the same y factors
18 = 3^2*2 18^3 = 3^3 *2^2
Addition/Subtraction rules for even and odd numbers
If both numbers are even or odd, the result will be even
14^2
196
Balance point method to find average
In an evenly spaced set with an odd number of terms, the average of the terms in the set is the exact middle term of the set when the terms are in numerical order
In an evenly spaced set with an even number of terms, the average of the terms in the set is the average of the two middle terms of the set
Splitting the cost
If 10 people split a bill, the cost per person is d/10. If 2 people leave or cant pay, the new cost is now d/8. There is a d/8-d/10=d/40 difference.
When the ratio between 2 variables is constant, they are related by what equation?
y=kx
where K is a positive constant. When x increases y increases and when x decreases y decreases.
Interior angle of a polygon
sum of interior angles= (n-2)*180
You can add and subtract only like radicals
5sqrt(4)+4sqrt(4) = 9sqrt(4)
Range
Highest in a set - lowest in a set
When (x,y) is reflected over the point (a, b)
its image becomes (2a-x, 2b-y)
5, -2) becomes (-1, 4
Fractional parts of the whole example:
A pilot is required to fly a certain number of hours in one day. In the morning, she flew 1/4 of the required hours plus 2 more. In the afternoon, she flew 2/5 of the remaining hours. In the evening, she flew 4 more. How many hours did she fly?
h=Total number of hours
AM: 1/4h +2
X is a multiple of y
Means x/y
8^3
512
The rate at which an object is performing a task =?
work/time
Profit equation
profit=total revenue - (fixed costs + variable costs)
Divisibility Rule
Div by 4- if last 2 digits are divisible by 4
Div by 6- even number whose digits sum to a multiple of 3
Div by 8- Even, last 3 digits divisible by 8
Div by 11- Sum of odd placed digits minus sum of even place digits is div by 11
Div by 12- Div by both 3 and 4
sqrt(a)*sqrt(b) = sqrt(ab)
sqrt(5) * sqrt(7) = sqrt(35)
A y=-x reflection causes the original x and y coordinates to switch positions and causes the signs of the coordinates to flip
(2,10)–?
(-10,-2)
Finding the LCM
- Prime factorize each number
- Take highest exponent for any repeated PF
- Take remaining PF’s
- Multiply remaining
Perfect square ends
0 1 4 5 6 9
Mixed fraction A B/C
(AC+B)/B
|a-b| >=|a|-|b| if…
b does not equal 0 and |a-b|=|a|-|b|
a and b share the same signs and |a|>=|b|
only true of (+)(+) or (-)(-)
When (x,y) is reflected over the line x=a
its image becomes (2a-x,y)
(5,-2) becomes (-1,-2) when reflected over the line x =2
Word problems with divisibility
W= Original Price N= New/Sale Price
Example: Sale price of 16% of original Price N= (16/100)W N=(4/25)W W=(25N/4) N must be a multiple of 4
Probability of dependent events:
P(A and B)= P(A) * P*B|A), where P(B|A) is the probability that B will occur after A has already occurred
x is divisible by y
Means x/y
Y is a factor of x
Means x/y
Any integer that has prime factors of at least one 2 and at least one 5 must end with at least one zero
What is units digit of positive integer x?
1. 7, 15, 24, and n are all factors of x.
Since integer x has factors 15 and 24, it has at least one prime factor of 5 and at least one prime factor of 2. Thus, integer x will always have a units digit of 0
Counting consecutive multiples in a set
(Highest # Divisible by the given number - Lowest # divisible by the given number)/given number + 1
How many multiples of 2 are between 51 and 99 inclusive?
(98-52)/2+1 = 24
19^2
361
Base 8 fractions 1/8 2/8 3/8 4/8 5/8 6/8 7/8
1/8=.125 2/8=.25 3/8=.375 4/8=.5 5/8=.625 6/8=.75 7/8=.875
When liquid is filling a container:
time=volume of container/rate
When setting up equation for rate problem with actual speed vs hypothetical speed:
time of actual= time of hypothetical + additional time
Median strategy for even numbered set
If a set of numbers has n terms and if N is even, the median is the average of the values at n/2 and n+2/2, when the numbers are in numerical order
Average Rate Questions
Average Rate = Total Distance/Total Time
or = (D1+D2)/(T1+T2)
Remember that T=D/R
if x/y is an integer, then x/(any factor of y) is an integer
100/20
100/(1,2,4,5,10,20) all work
At the Vet, there are 5 cats and 3 dogs. If 5 of these animals are to be randomly selected, what is the probability that 3 cats and 2 dogs will be selected
Method 1:
P(CCCDD)- arranged in 5!/3!2! = 10 ways
10(5/84/73/63/52/4) = 30/56
X-intercept
x= - b/m
Slope
Rise over run
y2-y1)/(x2-x1
Another way to express -1
(x-y)/(y-x)=-1
Arithmetic sequence
Sequence in which the difference between every pair of two consecutive terms is the same
1,4,7,10,13 …
an=a1+(n-1)d
an is the nth term in the sequence, a1 is the first term, d is the common difference
When setting up equation for rate problem with actual speed vs hypothetical speed:
time of actual= time of hypothetical + additional time
(x-y)^2 other forms
(x-y)(x-y)
x^2+y^2- 2xy
Inequality word problems
Up to= More than= At least= Exceeds= No more= At most= As few as=
<= less than or equal to > greater than >= greater than or equal to >greater than < less than <= less than or equal to >= greater than or equal to
4/7
.571
Midpoint formula
midpoint (x,y) = (x1+x2)/2, (y1+y2)/2
Properties of 1
1 is a factor of all numbers, all #'s are multiples of 1 1 raised to any power is 1 1 is odd 1 is only number with 1 factor 1 is not prime
To find the longest line that can be drawn within a rectangular solid, use the extended pythagorean theorem:
Solid is a cube?
d^2=L^2+W^2+H^2
d=s*sqrt(3)
Linear growth formula
F=KN+P F=Final value after growth P=Initial value K=Growth periods N=Number of growth periods
An initial value grows by the same amount each growth period
7^3
343
3 Circle venn diagrams
Total # Unique Members= #A Only + #B only + #C Only + #Double Overlaps + #Triple Overlap + #Neither
6^3
216
Another way to express -1
(x-y)/(y-x)=-1
Wet mixture problems
Create a table with solution 1, solution 2. and final mixture as the rows.
Concentration, quantity, and total are the columns
Multiples
X is a multiple of y only if x/y is an integer.
Only get clean divisions when x/y (multiple/factor) is an integer
Multiplying/Dividing complex fractions (multiplying num. and den. by the LCD)
(1/3)/(1/5)= 15(1/3)/15(1/5)= (15/3)/(15/5)
= 5/3
(1+3/(b+2))/(1+7/(b-2))
= multiply both top and bottom by (b+2)(b-2)
Determining the # primes in a factorial when divisor is not prime
40!/6^n
Break into pf’s. Use larger pf of x and apply divisibility shortcut
40!/6^n=50/2^n*2^n 40/3 = 13 40/9=4 40/27=1 13+4+1= 18
One object travels faster than another object. Which is “r” and which is “r + difference in speeds”
When one object is travelling faster than another object, consider letting the slower object’s speed be some variable “r” and the faster object’s speed be “r+ difference in speeds”
Surface area of a cylinder
c=2pir^2+2pir*h
Percent greater than problems
25 percent greater than z
450 percent greater than z
Final value = initial value (1+percent greater than/100)
(125/100)z (125 percent of z)
(550/100)z (550 percent of z)
Finding GCF
- Prime factorize each number
- Identify repeating pf’s and take the smallest exponent. If no repeating PF’s, GCF = 1
- Multiply numbers from step 3
To find the number of terms in a set of consecutive integers that includes only one of its endpoints but not both:
Subtract the 1st number from the last number
last number-first number
Fractional parts of the whole
Must sum to the whole
Fractional part 1 + fractional part 2 + fractional part n +any remaining = original number
Surface area of a rectangular solid
=2(LW) + 2(LH) + 2(WH)
Change in worker problems
Defining the rate of 1 worker method
find rate of all workers (work/time), then divide that rate by the number of workers to find the rate of 1 worker.
Dividing decimal example (10.36/2.8)
Move decimals over to make it 103.6/28
3/7
.429
Leading 0’s
If x is an integer with K digits, then 1/x will have k-1 leading 0’s. If x is a perfect power of 10, there will be k-2 leading 0’s
2 Objects leave at different times. Which is represented as “T’ and which is “T+difference in time”
When two objects leave at different times and converge at a constant rate, the travel time of the object that leaves later can be represented by some variable t; the travel time of the object that leaves earlier can be represented by “t + the difference between their departure times”
Probability that some number of items must be selected:
ways some number of items can be selected/# ways that all items can be selected
When (x,y) is reflected over the line y=-x
its image becomes (-y, -x)
-1,4) becomes (-4, 1
Equations with fractions efficient approach
Eliminate fractions by multiplying each term by the LCM of denominators 3/x + 3/5= 3 (3*5*x)/x + (3*5*x)/5 = 3*5*x 15+3x = 15x 15=12x 15/12 = x
Bow tie method to compare fraction size
Given a/b & c/d,
a/b > c/d if ad>bc
x is directly proportional to the square of y
x = k * y^2
When a question asks for the probability that “at least 1” outcome will occur, consider using complementary events to simplify the problem
Cannot be used on non-complementary events
On any given day, the probability that a certain student is abset from class is 1/5. In a span of 4 days, what is probablity that the student will be abset at least 1 day
P(At least 1 outcome) = 1 - P(none of the outcomes occur)
1-(4/54/54/5*4/5) = 369/625
There are 10 marbles in a jar, consisting of only red and blue. If the prob of choosing 2 red marbles, one after the other is 2/15, how many red marbles are in the jar
R/10 * (R-1)/9= 2/15
Age problems
- Define variable for present day
- Represent each age in the future or past
- Organize into matrix
- Use matrix and problem stem to solve
Vertical line test
If a graph is the graph of a function, then any vertical line can only intersect the graph at exactly one point or no points
A vertical line itself is not a function
Probability that some number of items must not be selected
ways some number of items must not be selected/#ways that all items can be selected
One object is relatively faster than another
When the first object is “x” times as fast as the second object, let the rate of the second object be “r” and the rate of the first object be xr. When the first object is “x” percent as fast as the second object, let the rate of the second object “r” and the rate of the first object “(x/100)*r”
