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Gravity
Terms in this set (129)
Percent
part = percent X whole
Percent Increase
Amount of Increase/Original Whole X 100%
Percent Decrease
amount of decrease/original whole X 100%
common multiple of 2 numbers
prime factor both numbers; multiply each factor the greatest number of times it appears in factorization.
Average
Average = Sum of terms/number of terms
Average to find the sum
Sum = (Average) X (Number of Terms
Average of Consecutive numbers
Add the two numbers and divide by 2 (ave. of integers from 13 to 77 = 13 + 77/2)
Count consecutive numbers (inclusive)
B-A +1 (how many integers from 73-419 = 419-73 + 1
Sum of Consecutive numbers
Sum = (Average) X (Number of Terms) (so must find average and the number of terms and add them together)
Median
Middle number
Mode
number that appears most often
Range
Subtract the largest and smallest number in the list of numbers
Ratio
Ratio= Of/To - 20 oranges to 12 apples = 20/12 = 5/3
Ratio to get actual number
ratio of boys to girls is 3/4. If 135 boys, then how many girls = 3/4 = 135/g
Rate
identify quantities and units to be compared. Keep units straight. If question gives you a rate in hours, but wants an answer in minutes, then convert the hours in the problem to minutes to solve.
Simple probability
Probability = # of desired outcomes/# of total possible outcomes
Area of a Trapezoid
Area = Average of parallel sides X Height OR A=1/2 (b1 + b2) h
Circumference of a circle
C=2(pi)r Or (Pi)D
Area of a circle
Area = Pi)r2 (pi R squared)
Slope of a Line
Slope = Rise/Run = change in Y/Change in X (EX) slope of line with (1, 2) and (4, -5) = -5-2/4-1 = -7/3
Quadratic Equation
ax2 + bx2 + c = 0 (2= squared
Quadratic Formula
x= -b =/- square room of b2 - 4ac/2a
distance formula
d= rt
Pythagorean theorem
for right triangles to find sides; a2 + b2 = c2
Isosceles Right Triangle
1 to 1 to square root of 2
special right triangle - half of equilateral triangle
has angles 30 60 90/ ratio of 1 - square root of 3, 2
Area of a Triangle
A=1/2 bh
Area of a Parallelogram
A= bh
Length of an Arc
Arc ABC/Dpi = Angle in degrees/360
surface Area
A=2(lw+lh+wh)
Volume of a Right Circular Cylinder
V=pir2h (pi r squared x h)
Revenue Formula
Profit = Revenue - Cost
Solve for both or neither
Group1+group2+ neither - both= total
Volume of cylinder
V = pi r2 h
Surface area of sphere
SA=4pi r2
Volume of a sphere
4/3pi r2
Volume of a right circular cylinder
Surface Area of Right Circular Cylinder
Pi or π
Ratio of circumference to the diameter
Area of a Circle Equation
πr^2
Circumference Equation
2πr
Surface Area of a Cylinder Equation
2πr^2 · 2πrh
Area of a Triangle Equation
1/2(base·height)
Area of a Hexagon Equation when side length is known
(3√3·s^2)/2
Arithmetic Mean (Same as Average)
Using arithmetic to figure out the average of data = add all / by number of data set
Pythagorean Theorem
Finding the hypotenuse of a triangle
a^2 + b^2 = c^2
Determining slope direction
Coefficient of x (neg. or pos.) shows the direction of the slope. Negative slopes down left to right, while positive slopes up left to right.
Quadratic Equation
Ax^2 + Bx + C = 0
[-b^2±√(b^2 - 4ac)]/2a
Graphing Quadratic Equations
· Factor equation
· Set each factor = 0
· Solve for x
· xV = (x intersect 1 + x intersect 2) / 2
· To find yV, plug in xV into original equation
· To find y intercept, make x=0 into original equation and solve for y
Quadratic Inequalities
· Remember a quadratic formula = a parabola
· You're looking for the two x axis intersections and the direction of the remaining
· When you divide both sides by a negative you must switch the direction of the inequality sign
Solving for # of variables
Determine number of unique values per variable and multiply them by each other
Multiplying Fractions, 5/6 x 2/3
· Multiply numerators and denominators separately.
5 x 2 / 6 x 3 = 10/18 = 5/9
· Denominators do not have to be the same.
· If a denominator from one fraction and the numerator of the other fraction share a common factor, simplify before multiplying
Dividing Fractions
· Same as multiplying the reciprocal
· e.g. 3/5 ÷ 1/2 = 3/5 x 2/1
· Simplify as directed
· e.g. "Simplify to a mixed number" 6/5 = 1 1/5
Adding / Subtracting Fractions
· Least common multiple to make denominators the same
· Add together numerators, while keeping the denominators the same
· Simplify if needed
Simplifying exponential equations
·When comparing expressions to find which is greater find a common root
· e.g. 64^5 & 16^8 = (4^3)^5 & (4^2)^8 or 4^15 & 4^16
Simplify sqrt72
· sqrt9 x sqrt8 = 3sqrt8
· When simplifying an answer, look for perfect squares
30º, 60º, 90º Triangle
...
Ratios
3 : 5 = 3/5
Multiplying ratios is the same as multiplying fractions
Determining Slopes with 2 Sets of Coordinates
(y2 - y1) / (x2 - x1)
Improper fractions to mixed numbers
e.g. 18/7 = 7√18 = 2 R4 or 2 4/7
Mixed number to improper fractions
e.g. 2 4/7
denominator is given, 7
numerator = denominator x integer + original numerator
7 x 2 + 4 = 18
18/7
Combinations
A permutation where you DON'T care about the order
nCr = (n!/(n-r)!)/r!
Permutations
When you DO care about the order.
For instance, if there are 5 people, how many different ways can they sit in 3 chairs. When a person sits down, it removes the possibility of sitting in any other chair.
nPk = n!/(n-k)!
Probability
P is probability, m is # of favorable ways, n is total # of ways
P = m / n
when combining probability of two mutual exclusive events
Pa x Pb
How many more times is 17% than 3%
17/3 or 5 2/3
how many more times is m than n?
x = m/n
Integer divisible by 3
Sum of the digits. If that digit is a 3,6 or 9, the number is a multiple of 3
e.g.
314159265
3 + 1 + 4 + 1 + 5 + 9 + 2 + 6 + 5 = 36
Then, 3 + 6 = 9
Integer divisible by 4
The last 2 digits are a multiple of 4. (e.g. 144 .... 44 is a multiple of 4, so 144 must also be a multiple of 4.)
Integer divisible by 6
Sum if the digits are divisible by 3 and the last digit is even
e.g.
1,458: 1 + 4 + 5 + 8 = 18
Integer divisible by 8
Examine the last three digits
e.g.
34152: Examine divisibility of just 152: 19 × 8
Integer divisible by 9
Sum the digits. If the result is divisible by 9, then the original number is divisible by 9.
e.g.
2,880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9
a^x / a^y
a^x-y
(a^x)(a^y)
a^xy
a^0
1
(ab)^x
(a^x)(b^x)
y^2 = x
y = ±√x
e.g.
y^2 = 4
so, y = ±√4 = ±2
x^2-2x+1 = 0, solve for x
· Factor: (x-1)(x-1) = 0
(x-1)^2 = 0
· Square root both sides
x-1 = 0
· Solve for x
x = 1
(a+b=c) + (d=d)
a+b+d = c+d
(b > c) + (d > e)
b+d > c+e
Intercepted Arch (Circles)
Angle is 1/2 of intercepted arch
90º, 45º, 45º Triangle - Side Measurements
or
legs = √2/2
hypotenuse = 1
60 60 60 Triangle - side measurements
All sides are equal
Irregular Triangle
a + b > c
a + c > b
b + c > a
x^-n
1/x^n
a/(b/c)
invert denominator, then multiply
a/1 x c/b = ac/b
odd x odd
odd always
odd^odd
odd always
even^odd
even always
10^5
100,000
10^5 means 5 zeros
107075.6 = 1.070756 x
10^5
the n exponent = n decimal places
(a/b)^x
a^x/b^x
(a^x)(b^x)
ab^x
(a^-x)(b^y)
(b^y)/(a^x)
(2a^m)(1/3a^-n)
= (2/3a^m)(a^-n)
= 2a^m / 3a^n
value of x for:
x + 4y = 7 & x - 4y = 8
· Add Equations
· No need for a common factor since 4y & -4y cancel each other out
2x = 15
x = 15/2
find value for x & y for:
x - 2y = 2 & 2x +y = 4
· Add Equations
· Find common factor for x to cancel it out
(2x + y = 4) + -2(x - 2y = 2)
· Solve for y
5y = 0 so, y = 0
· Replace y value into either one of the original equations and solve for x
x - 2(0) = 2 so, x = 2
inscribed angle whose triangle base = diameter
...
x + y > z, then y > z - x
Always
-4 < -x, then +4 > +x
Always
7/12 + 3/5
· Cross multiply & add both products together
7x5 + 12+3 = 71
· Multiply both denominators
12x5 = 60
· So, 71/60, then simplify to a mixed number
1 11/60
or
· Find a multiple that makes the denominators equal
60 is the lowest common denominator
· Multiply each fraction by their respective multiple
4(7/12) + 12(3/5) = 28/60 + 36/60
· Add the two numerators
64/60 or 16/15 or 1 1/15
what is c if,
200 = (a+b+c)/2 & 80 = (a+b)/3
· Get rid of the denominator in each equation by multiplying both sides by the denominator value
· Now subtract both equations by each other
(400 = a+b+c) - (240 = a+b)
· Therefore,
160 = c
Solve without multiplying,
(140 x 15) / (5 x 7)
· Find common factors in the numerator and denominator
(20 x 15) / (5 x 1)
· Repeat for remaining numerator and denominator
(20 x 3) / (1 x 1)
· Therefore,
60/1 or just 60
Solve for x,
x^2 + 2xy + y^2 = 25, with x + y > 0 & x-y=1
· factor the equation
(x+y)(x+y) = 25 or, (x+y)^2 = 25
· Square root both sides
x + y = ±5
· Since it is stated that "x + y > 0" we know that 5 can only be positive
· add the remaining equations together
(x+y=5) + (x-y=1)
· Because the y value cancels it's self out, there's no need to multiply by a common factor
2x = 6, x = 3
if a/b = 1/4, where a is a pos. integer, which of the following is possible for the value a^2/b?:
1/4, 1/2, 1
All
· cross multiply
4a=b
· substitute b in equation
a^2/4a
· square root both nominator and denominator
a/4
· looking at the options, plug in values to see if they equal any of the possible answers
If a lamp decreases to $80, from $100, what is the decrease in price?
= (actual decrease/Original amount) x100%
= 20/100x100% = 20%
If a>b then
-a<-b
1ⁿ
1
25^(1/2) or sqrt. 25 =
5 or -5
0^0
undefined
(6sqrt3) x (2sqrt5) =
(6 x 2)(sqrt3 x sqrt5) = 12sqrt15
(12sqrt15) / (2sqrt5) =
(12/2) x (sqrt15 / sqrt5) = 6sqrt3
Formula to calculate arc length?
Arc length = (n/360)(2πr) where n is the number of degrees.
Finding the vertex of a quadratic equation
-b/2a
Vertex
· The minimum or maximum of a parabola
· Equal distance from the two points intersecting the x-axis.
7!
7x6x5x4x3x2x1
How many different ways can 5 people sit in 3 chairs?
· Written,
5P3
· Equation for solving
5!/(5-3)!
· Because 5x4x3x2x1/2x1, the "2x1" cancel out from both the numerator and denominator, you only solve for:
5 x 4 x 3 = 60 different possibilities
(x+y)/xy =
1/x + 1/y x,y≠0
(x-y)/xy=
1/y - 1/x x,y≠0
x^2 - y^2 =
(x + y)(x - y)
xy + xz =
x(y+z)
xy - xz =
x(y-z)
x^2 + 2xy + y^2 =
(x+y)(x+y) = (x+y)^2
x^2 - 2xy + y^2 =
(x-y)(x-y) = (x-y)^2
7 divided by ∅
Null
∅ divided by 7
∅
a/∅
Null
Ratio of boys to girls is 2:3. Of 40 students, how many are girls?
· Boy/Girls = 2/3 or 5 students total
· There are 8 sets of 5 students in a classroom of 40 students
· 8 sets x 3 girls/set = 24
Ratio of ages of Anna and Emma is 3:5 and of Emma and Nicolas is 3:5. What is the ratio of Anna to Nicholas' ages?
9 : 25
For similar triangles, the ratio of their corresponding sides is 2:3. What is the ratio of their areas?
4:9. The ratio of the areas of two similar triangles equals the square of the ratio of the corresponding sides.
Simplify the expression [(b^2 - c^2) / (b - c)]
(b + c)
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