Math Flashcards
sqrt(5)*sqrt(8)
sqrt (5*8)
Logb(M) - Logb(N)
Logb(M/N)
Log(10)-Log(5)= Log(2)
Logb(M^z)
zLogb(M)
Log(10^3)= 3Log(10)
Formula for combining options (where must pick one option per number of options)
Ie. there are 3 types of shells, 4 types of meat, 3 types of cheese, and 5 types of salsa. How many distinct tacos can be ordered assuming that any order contains exactly one of each of the above choices?
Total combinations = # of options x # of options etc…
Total combinations of tacos = 3 x 4 x 3 x 5 = 180
x^(a/b)
b-root(x^a)
Isosceles triangle
Two sides are equal
The reflection of y=2x+5
x=2y+5
Height of an equilateral triangle
(1/2)(length of sides)(√3)
Value of deposit using simple interest
V=P(1+(rt/100))
Assuming r is in decimal form (eg. 10% int= .10)
V= P (1+rt)
Points on graph: P P' P'' P'''
P: quadrant I (4,2)
P’: quadrant IV (4, -2)
P’’: quadrant II (-4, 2)
P’’’: quadrant III (-4, -2)
Formula for finding arc,angle, or area of sector when it is positioned at center of a circle
(sector angle/circle angle)=(sector arc/circle circumference)=(sector area/circle area)
Q3
Median of values above Q2 (median of data set)
Probability for all events happening together (“AND” question)
Multiply probabilities together
Rhombus
- A parallelogram with equal sides
- Area = bxh
- If all the angles are right angles then it is a square
- The rhombus with the largest possible area is a square
Sum in geometric sequence
(a[1-r^n])/[1-r]
Logb(M)+Logb(N)
Logb(MN)
Log(10)+Log(5)= Log(50)
Slopes of perpendicular lines
Inverse negatives of each other
Line A = 8
Line B = -1/8
Third side rule of triangles
Any side of a triangle must be greater than the difference of the other two sides, and less than their sum
Trapezoid
- Quadrilateral with two sides parallel
- Isosceles trapezoid: base angles are equal, so non-parallel sides are equal
- Area of trapezoid: [(b1+b2)/2)] x height, where b1 and b2 are the lengths of the parallel sides
How to find vertex of parabola
Set equation = 0 to find x- intercepts
The x value of the vertex will be halfway between those intercepts.
Plug in x-value to find y-value of the vertex
Solving inequalities
Same as regular equations, except when both sides of the inequality are multiplied or divided by a negative number, the direction is reversed
5^2 x 2^2
10^2
Q1
Median of values below Q2 (median of data set)
What does it mean if the product of two numbers is odd?
Both numbers must be odd
Slope
Rise over run
Change in y/change in x
Sum in arithmetic sequence
Average of the first and last term multiplied by n
What is the sum of the first 20 terms? ([A20+A1]/2)*20
In an arithmetic sequence, the sum of the first and last term/2 = median = mean
a^m/a^n
a^(m-n)
What is the approx. percentile?
1) -2 SD
2) -1 SD
3) 0 SD
4) 1 SD
5) 2 SD
6) 3 SD
1) 2%
2) 16%
3) 50%
4) 84%
5) 98%
6) 99%
Formula for combinations where order doesn’t matter
1) If picking select group: the number of possible pairings of 2 colors that can be selected from 5 possible options
2) If in subgroups, the order doesn’t matter: in how many ways can the word “MISSISSIPPI be rearranged?”
1) Everything!/(Picked!Notpicked!)
5!/(2!3!)
2) Everything!(Group1!Group2!Etc…)
For each individual subgroup (aka letter), the order does not matter
M=1 , I=4, S=4, P=2
11!/(1!4!4!2!)
Parallelogram
- Quadrilateral with opposite sides parallel (and opposite angles equal)
- Area = bxh
- A parallelogram with equal sides = rhombus
- A parallelogram whose angles are all right angles = rectangle
Value of deposit compounded annually
V= P(1+ r/100)^t
Quadratic formula
To find x-intercepts in equation ax^2+bx+c
x= {-b+/- root(b^2-4ac)}/2a
Does Q1 or Q3 include the mean?
No. Median of the lower half or the upper half of the data (excluding the mean)
Finding equation of circle
1) Circle with radius 3 and center at (6, -5)
2) Circle with radius 10 and center at origin
1) (x-6)^2+(y+5)^2 = 9
2) x^2 + y^2 = 100
Establishing proportional trinagles
1) Side of one triangle is parallel to the corresponding side of another
2) All three angles are the same within the two triangles
What happens to SD and range when
1) multiply all values in data set by x
2) divide all values in data set by x
3) Add x to all values in data set
4) Subtract x to all values in data set
**Check to see in study guides where SD changes due to addition and subtraction
1) SD= multiply by x, range= multiply by x
2) SD= divide by x, range= divide by x
3) SD= unchanged, range= unchanged
4) SD= unchanged, range= unchanged
Value of deposit compounded n times per year
V= P(1+r/100n)^nt
What are properties of 1 and 0?
Neither of them are prime numbers
0 is an even number
0 is an integer
a^2 x b^2
(ab)^2
Formula for combinations where order matters
1) If everything included: five students are lining up. How many different lines are possible?
2) Where only a select group from total is chosen: 10 student class is to choose a prez, VP, and secretary from the group. If no person can occupy more than one post, in how many ways can this be accomplished?
1) Everything!
5!= 5x4x3x2x1
2) Everything!/Not chosen!
10!/7!= 10x9x8
Another way of thinking about this is, how many options for first position (10), how many options for second position (9), how many options for third position (8)= 10x9x8
sqrt(5)/sqrt(8)
sqrt (5/8)
What do you need in order to say where a score falls (in what percentile?)
SD and mean
x^(-a)
1/(x^a)
Pythagoreum equation
leg^2+leg^2=H^2
- If true, then the triangle must be a right triangle
- If the sum of the squared legs is smaller than the square of the hypotenuse than the angle opposite the H is greater than 90
- If the sum of the squared legs is greater than the H^2, then the angle opposite the H is smaller than 90
Probability for “OR” question:
1) If both probabilities are independent (can happen at the same time)
2) If both probabilities can’t occur at the same time
1) 1-P(event never happening)
2) Add probabilities together
30-60-90 rule
If triangle’s angles are 30-60-90 then the length of its sides are:
30: x
60: x√3
90: 2x
Formula for finding arc,angle, or area of sector when it is positioned at the border of the circle (aka inscribed)
The central angle is 2x the inscribed angle; solve ratios from there
45-45-90 rule
If triangle’s angles are 45-45-90, then the length of its sides are:
45: x
90: x√2
Formula for sum of shape’s interior angles
180(n-2), where n is the number of sides
Logb(N)
Eg. Log2(6)
Logx(N)/Logx(B)
Log2(6)= Log(6)/Log(2)
If Number= base^power,
eg. 15=10^x
Logbase(number)=power
Log10(15)=x
a^n * a^m
a^(n+m)
If 15:x and x:100 are equivalent, what is x?
15/x = x/100
solve x from there
Formula for overlapping sets, ie. students studying languages– 14 french, 12 Spanish, 10 Latin, 5 two languages
What about 14 french, 12 spanish, 10 latin, and 3 all three languages?
Total = set + set - overlapping set
Total students = 14 + 12 + 10 - 5
If studying all three languages, subtract 3-1 = 2 for each student
For example, if there were 3 students studying all three the total would be = 14 + 12 + 10 - (2)(3)
Tangent line to a circle
Perpendicular to the radius of the circle
(a^m)^n
a^mn
How to find n for
A) odd numbers from x to y (inclusive)
B) even numbers from x to y (inclusive)
C) consecutive numbers from x to y (inclusive)
A) (y-x)/2 + 1 ex. what is n of odd integers from 1 to 199? (199-1)/2 + 1 = 100 B) (y-x)/2 + 1 ex. what is n of even integers from 2 to 50? (50-2) / 2 + 1 = 25 C) y-x + 1 ex. what is n of numbers from 4 to 50? (50-4) + 1 = 47
Factor out 2^99 - 2 ^96
2^96(2^3 - 1)
Value in geometric sequence
a*r^(n-1)
How can we know if the triangle that is inscribed in a circle is a right triangle?
-Right triangle if and only if the hypotenuse of the triangle is equal to the diameter of the circle (aka right triangle inscribed in a circle must be inscribed within a semi circle)
Value in arithmetic sequence
v = a + d (n-1)
d= difference between each number a= first number
Group A’s average score is 90 and group B’s average score is 85. There are more than twice as many students in group A than group B. What is the mean?
Weighted mean.
2(90)+1(85)/(2+1)= 88.333
Since there are MORE than twice as many students in group A than group B, and group A’s average is higher than group B’s average then the mean is greater than 88.3333.
Two similar triangles. The ratio of their sides is 1:9. What is the ratio of their area?
1:81
Two coal carts, A and B, started simultaneously from opposite ends of a 400 yard track. Cart A traveled at a constant rate of 40 ft/s. Cart B traveled at a constant rate of 56 ft/s. After how many seconds of travel did the two carts collide? (1 yard=3ft)
Distance= 400 yd= 1200 ft
When doing a problem where objects are moving towards each other, add the rates together.
Rate of A and B= 40+56= 96 ft/s
Seconds until carts collide: 1200/96= 12.5 seconds
In a probability experiment, G and H are independent events. The probability that G will occur is (1/2) and the probability that H will occur is (1/2).
What is the probability that either G will occur or H will occur, but not both?
What is the probability that either G will occur, H will occur, or both?
Since independent events can occur at the same time the probability is:
(1/2)+(1/2)-(1/4)= 3/4
This is the probability of G, H, or both.
Therefore the probability of G, H, but not both is
(3/4)-(1/4)= 2/4= (1/2)
For a certain probability experiment, the probability that event A will occur is 1/2 and the probability that event B will occur is 1/3. What possible values could be the probability that the event AUB (that is, the event A or B, or both) will occur.
We do not know the relationship between the two events, so we can only find the minimum possible value and the maximum value for the event.
IF OVERLAPPING COMPLETELY
Minimum possible value: the larger value, eg. 1/2.
(A can’t be a subset of B since the probability of A occurring is greater than the probability of B occurring; if B is a subset of A, then the least probability is just the probability of A– this is if they intersect completely)
IF NOT OVERLAPPING AT ALL
Maximum possible value: if A and B do not intersect at all (aka if they are mutually exclusive and cannot happen at the same time), P(A)+P(B)= 1/2 + 1/3 = 5/6.
Therefore, the probability is between 1/2 and 5/6, inclusive.
How many zeros does 10^5 have?
5
The ratio of Kim’s time to paint a house to Jane’s time to paint a house is 3:5. If Kim and Jane work together at their respective constant rates, they can paint a house in 10 hours. How many hours does it take Kim to paint the house alone?
- If it takes Kim and Jane 10 hours to paint a house then their combined rate is 1/10= .1 house/hr
- ***so K+J= .1
- If K:J time is 3:5 then K:J rate is 5:3 (less time means faster speed
- ***in other words, Kim is 5/3 times faster than Jane
- ***so K=5/3 J or J= 3/5K
- Solving from here, you can find that K= .0625, aka .0625 house/hr
- Dividing 1 by .0625, you can find that it takes Kim 16 hours to paint the house alone
