Geometry
1 is the multiple of all numbers/integers
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Evenly spaced consequtive ..
.A= Middle number if set has odd numbers A= average two middle numbers if set has even numbers or A= 1st+last term/2 Don't have to get all numbers of even spaced consecutive integers, only find 1st and last...
Convert nickle, dime, and quarters to dollar
1 N= 0.05 dollar 1 Dime= 0.10 dollar 1 quarter= 0.25 dollar
two consecutive odd integers
x+(x+2)
30-60-90 triangle, half of equilateral triangle (60-60-60)
x: x v3 (LONG LEG): 2x (HT)
If two lines in a plan do not intersect, they are parallel no pair of numbers (x,y) will satisfy both equations. Another possibility: both equations are the same. Then, infinitely many points will solve both equations. To test whether a point lies on a line, just test it by plugging the numbers into the equation.
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Inequalities and Extreme Values
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Testing Inequality cases It is important to know the following
xy>0= x,y :+ve or both -ve xy<0: x & y different signs x2-x<0 : x2<x, 0<x<1
Parallel lines cut by a transversal: Sometimes parallel lines cut by a transversal appear when a rectangle, parallelogram, rhombus or trapezoid is cut in half by a diagonal
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Surface Area OF SOLID = the SUM of the areas of all of the faces
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sd
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Slope-intercept Equation
y = mx + b m = slope b = y-intercept Horizontal and vertical lines Horizontal: y = some number Vertical: x = some number
The slope is defined as
"rise over run": (Y1 - Y2) / (X1 - X2) In a line, any two other points have different "rise" and different "run", but the slope will always be the same. A line with positive slope rises upward from left to right. A line with negative slope falls downward from left to right. A horizontal line has zero slope, and a vertical line has undefined slope. The x-axis has zero slope, and the y-axis has undefined slope.
Distance between two points
(1) Draw a right triangle connecting points (2) Find the lengths of the 2 legs of the triangle (3) Use pythagorean theorem Example: what is the distance between (1,3) and (7,-5)? First leg = 6 (7-1); second leg = 8 (3- (-5)). Pythagorean: x² = 6² + 8² --- x = 10
Step by Step: from 2 points to a line
(1) Find the slope with rise over run (2) Plug in the Slope (3) Solve for b by plugging the coordinates of one point (4) Write the equation in the form y = mx + b Example: Point 1 = (5, -2) - Point 2 = (3,4) (1) Rise over run: (-2 - 4) / (5 - 3) = -3 (2) y = -3x + b (3) 4 = -3.3 + b --- b = 13 (4) y = -3x + 13
Example: x > 8, x < 17, x + 5 < 19 (1) Solve any inequality that needs to be solved: x + 5 < 19 = x < 14
(2) Make all the inequality symbols point in the same direction: 8 < x x < 17 x < 14 (3) Eliminate the less limiting inequalities: x < 14 is more limiting than x < 17, so ignore x < 17 The final answer is: 8 < x < 14 Manipulating compound inequalities If you perform an operation on a compound inequality, be sure you do it on every term. Combining inequalities First, make all the inequality signs face the same direction. Them, add them up.
Combining Inequalities
(2) Make all the inequality symbols point in the same direction: 8 < x x < 17 x < 14 (3) Eliminate the less limiting inequalities: x < 14 is more limiting than x < 17, so ignore x < 17 The final answer is: 8 < x < 14 Manipulating compound inequalities If you perform an operation on a compound inequality, be sure you do it on every term. Combining inequalities First, make all the inequality signs face the same direction. Them, add them up.
The area of an equilateral triangle of side S is equa
(S² . (st-rt 3)) / 4. This is because an equilateral triangle can be cut in 2 30-60-90 triangles, and the proportion of the height will be S . (sq-rt 3) / 2.
Polygons and Interior Angles: The sum of the interior angles of a polygon depends on the number of sides (n) the polygon has:
(n - 2) x 180 = Sum of Interior angles of a polygon In the figure above, (f + g + h + i + j) = 540°, because the polygon has 5 sides Every polygon can be cut in x triangles, and the sum of each triangle's interior angles is 180°. This is an alternative method to find the sum of a polygon's interior angles
FOIL: First terms, Outer terms, Inner terms, Last terms
(x + 7) (x - 3): F: x.x + O: x.-3 + I: 7.x + L: 7.-3 The result is: x² - 3x + 7x - 21 = x² + 4x - 21 tips For the equation (x² + x - 12) / (x - 2) = 0, you cannot multiply both sides for x - 2, and also, x - 2 cannot be zero. So, the solution will be the solution of the equation in the numerator: x = -4 and x = 3. Be careful when the denominator has x!!..need to knw abt this more???
Properties of WA
- Fall between numbers just like normal A - Will be more toward high weight. - FOR DS: Don't need to have exact weights of each number to solve, RATIO is enough to solve ex: A mixture of 10% LEAN bEEF and 4% super lean beef contains twice as much lean beef than S.L, what is the % of fat in mixture? 10%*2 +4%*1/2+1= 8% (2;1 is the ratio= weights)
Finding copies of sell or revenue by expanses.
- REVENUE(number of items) = ALL kind of variable cost(number of items)+fixed cost. - 1 dollar = 100 cents, - converting cents to dollar, 40 cents= 0.40, 8 cents= 0.08 dollar.
FOR PERCENTAGE INCREASE. TWO WAYS: FIND 150% GREATER THAN 200. 15% GREATER THAN 100 170% GREATER THAN 322 25% LESS THAN 100
1) 100+150% = 250% (250%) (200)= 420 2) (200) (2.5) = 420 1. (115%)(100) 2. (1.15) (100) = 115...HOW 1.15 CAME? (1+ 15/100) = 115/100= 1.15 1 (270%) (322) 2 1.7(322) 1 (75%) 100 2( 0.75) 100
Probability of at least one event happening between A and B ...veno diagram overlapping with each other
1. P(A OR B) = P(a) +P(b)- P(A * B) 2. 1- P(A not happening) P(B not..) 3. P(A)P(notB) + P(B)P(not a) + P(A)P(B)
Rectangular Solid: AREA Volume
2(Base x Height) + 2 (Width x Height) + 2 (Base x Width) V: Length x Width x Height
Cylinders Surface Area
2π(r²) + 2πrh
common right triangles
3-4-5, 9-12-15, 6-8-10,12-16-20 5-12-13, 10-24-26 8-15-17
5C0
5!/0! (5-0)!= 1
Cube Volume of a Cube
: 6 x (Side)² V: Side³ GMAT TRICK: How many Books, each with a volume of 100 in³, can be packed into a crate with a volume of 5,000 in³? You cannot answer to this question without knowing the exact dimensions of each book. Remember: if you are fitting 3 dimensional objects into other 3-dimensional objects, knowing the respective volumes is not enough
) Area of a PARALLELOGRAM
: Base x Height
sequence problem
: If each number in a sequence is 3 more than the previous and 6th number is 32, what is the 100th? We know that we have 94 jumps of 3 between the 6th and the 100th, so the answer is: 94.3 + 32
) Area of a TRAPEZOID
= ((Base1 + Base 2) x Height) / 2 Notice that the height is also a line perpendicular to the two bases, inside the trapezoid Sometimes, you will have to draw a right triangle. If you know both base 1 and 2, you know the base leg. If you know the outer leg, you have 2 dimensions and can find the third, which is the height, via pythagorean theorem!!!
Area of a RHOMBUS
= (Diagonal 1 x Diagonal 2) / 2
Probability of impossible event
= 0
Mutual exclusive events ..ex Probability of happening a and b together ..p(a) or p(b) ex: If Jessica rolls a die, what is the probability of getting at least a "3"?
= 0 = p(a) +p(b) solu: There are 4 outcomes that satisfy our condition (at least 3): {3, 4, 5, 6}. The probability of each outcome is 1/6. The probability of getting at least a "3" is: P = 1/6+1/6+1/6+1/6= 4/6=2/3
Independent event example: If there is a 20% chance of rain, what is the probability that it will rain on the first day but not on the second?
= 0.2 *0.8 = 0.16
Probability of certain event, ex: getting tail or head in coin
= 1
The length of Arc AxB can be calculateded by
= Circumference . Angle/360
Circles: Area of a Circle Area of a Sector
= πr² All we must know is the radius = πr² . (x°/360°)
Volume: V
= πr²h To find either the surface area or the volume, you only need the radius and the height. GMAT TRICK: Two cylinders can have the same volume but fit a different larger object. Different combinations of radius and heights can produce the same volume but very different cylinders.
Quadratics
A quadratics function graph (y = ax² + bx + c) is always a parabola. If a>0, the parabola opens upward. If a<0, the parabola opens downward. If |a| is large, the curve is narrow. If not, the curve is wide. b² - 4ac is the discriminant. If b² - 4ac >0, the function has 2 roots. If b² - 4ac = 0, the function has 1 root. If b² - 4ac < 0, the function produces no roots.
Quadratic equations
A quick way to work with quadratics is to factor them. If you have the equation ax² + bx + c, when a=1: Find two integers whose product equals c and whose sum equals b Rewrite the equation in the form (x + k)(x + w), where k and w are those two numbers which resulted in the product of c and in the sum of b.
Circles: Inscribed Triangles
A triangle is said to be inscribed in a circle if all of the vertices are points on the circle. Main property: if one of the sides is the diameter, the triangle IS a right triangle. Conversely, any inscribed right triangle has the diameter as one of its sides. A right triangle can be opposed to a semicircle. If you need to calculate that arc, it is 180° Take care: A triangle inscribed in a semicircle doesn't have the same properties as a properly inscribed triangle
Inequalities and Extreme Values
ADDITION; 8+ LT2 = LT10 Substraction: 8-LT2= GT6 Multi..: 8*LT2= LT16 -7*LT2= GT14 Divison: 8/LT2= GT4 Mutiply 2 extreme values: LT8*LT2= LT16 (if both are +ve) Optimization problems On problems involving minimization or maximization, focus on the largest and smallest possible values of each variables
Perimeter of a Sector
ARC + 2 Radius NOTE:To find either the Arc Length or the perimeter of a sector, all you need is the radius plus the angle of the sector
Combo Problems Example: What is 2/y/4/x? (1) (x + y)/y = 3 (2) x + y = 12 2/y/4/x = 2/y . x/4 = 2x/4y = 1/2 . x/y (we have isolated x/y) Working on (1): (x + y)/y = 3 = x + y = 3y, x = 2y, 2 = x/y Now it is easy to notice that, if x/y = 2, 1/2 . x/y = 1. Equation 2 is insufficient, as its not possible to isolate x/y. Answer: A
Again: the key to solve some combos is to try to find similarities between 2 equations, instead of trying to solve them. Isolating terms instead of working with single variable is essential.
The intercepts of a line
An intercept is a point where the line intersects a coordinate axis X-Intercept = (x,0) Y-Intercept = (0,y) To find x-intercepts, plug in 0 for y. To find y-intercepts, plug in 0 for x
Example of AVERAGE A N S Table... Same earned 2000 commission on a big sale, raising his average commission by 100. If same's new average commas is 900, how many sales has he made?
Average Number = Sum Old total 800 n = 800n This sale 2000 1 = 2000 nEW TOTAL 900 n+1 = 900(n+1) Equation: 800n+2000= 900(n+1) n= 11 Total n+1 = 12 sales
DO"S - Do think about inequalities as ranges on a number line - Do treat inequalities like equations when adding or subtracting terms -Do add inequalities together when necessary -Do use extreme values to solve range problems -Do set terms with even exponents equal to zero
DONT'S - Don't forget to flip the inequality sign if you multiply or divide by a negative number - Don't multiply or divide an inequality by a variable unless you know the variable's sign - Don't forget to perform operations on all expr. - Don't subtract one inequality from another - Many pos/neg problems are disguised as ineq
If asked small sitting arrangements then just manually write how many possible: don't use any formula.. ex: Three people are to be seated on a bench. How many different sitting arrangements are possible if Erik must sit next to Joe?
E J x, x E J, J E x, x J E So 4 possible ways
GMAT can also hide positive constraints. Sides of a square, number of votes, etc will always be positive. When you have a positive constraint, you can:
Eliminate negative solutions from a quadratic function Multiply or divide an inequality by a variable Cross-multiply inequalities: x/y < y/x x² < y² Change an inequality sign for reciprocals: x<y; 1/x > 1/y Multiply inequalities (but not divide them) Square/unsquare an inequality: x < y ; x² < y². x < y ; √x < √y
Exponential Equations
Even exponents are dangerous, because they hide the sign of the base. For any x, √x² = |x|. The equation x² = 25 is the same as |x| = 5. x² = 0 has only one solution. X² = -9 has no solutions, as squaring cannot produce a negative number. If an equation has even and odd exponents, treat it as dangerous. It probably has more than 1 solution. Same base or same exponent: Try always to have the same base or the same exponent on both sides of an equation. This will allow you to eliminate the bases or exponents and have a single linear equation. This rule does not apply when the base is 0, 1 or -1. The outcome of raising those bases to power is not unique (0 = 0³ = 0²³).
Perpendicular Bisectors The perpendicular has the negative reciprocal slope of the line segment it bisects. To find the equation of a perpendicular bisector: (1) Find the slope of the line segment (2) Find the slope of the perpendicular bisector (reciprocal) (3) Find the midpoint of AB (4) Find b for the bisector, by plugging the midpoint of AB
Example: Find the perpendicular bisector of line with points (2,2) and (0,-2) Slope = 2 - (-2) / 2 - 0 = 2; Slope of the bisector = -1/2 Midpoint = (1,0) Plugging: 0 = -1/2 . 1 + b ---- b = ½ Equation of the perpendicular bisector: y = -1/2x + 1/2
Compound Functions If f(x) = x³ + √x and g(x) = 4x - 3, what is f(g(3))?
First, solve g(3): 4.3 - 3 = 9. Now, plug-in 9 on f(x): 9³ + √9 = 729 + 3 If f(x) = x³ + 1 and g(x) = 2x, for what value f(g(x)) = g(f(x))? f(2x) = g(x³ + 1) = (2x)³ + 1 = 2(x³ + 1) = 8x³ + 1 = 2x³ + 2 x³ = 1/6
When asked question where need to find numbers then utilize this method for all. Don't forget to satisfy condition which is asked, like not to use same number etc. ex: How many 3-digit numbers satisfy the following conditions: The first digit is different from zero and the other digits are all different from each other?
For 1st number - 9 choice (except 0, 1 to 9) for 2nd number - 9 choice ( eliminate 1st number) 3rd number- 8 choices ( eliminate both numbers) 9*9*8= 648
Proportional and inversely proportional functions
For direct proportionality, set up ratios for the "before case" and the "after case" and equal them (regra de 3). The proportionality is defined by y = kx, where k is the proportionality constant. For inverse proportionality, set up products for the "before" and "after" cases and equal them. The inverse prop. formula is y = k/x Example: amount of current and resistance are inversely proportional. If current was 4 amperes and resistance is cut to one-third the original, what will be the current? x1y1 = x2y2 - 3.4 = 1.x x = 12
COIN Probability: Whats the Probability of getting 3 heads and 1 tail if coin is tossed for 4 times.
HHHT: 1. first write probability of each one: 1/2*1/2*1/2*1/2 = 1/8 2. Total numbers of head and tail / Numbers repeating(H) & Number of times T repeating.= 4!/ 3!* 1! 3. Multiply everything= 1/8* 4!/ 3!= 1/2
whats the probability of getting at least 2 heads and 1 tail if tossed 3 times
HHT + HHH (1/2)3 3!/2! + (1/2)3 3!/3!
Using extreme value: LT and GT
If 2h + 4 < 8 and g + 3h = 15, what is the possible range for g? Since h<2, let's say h = LT2 (Less Than 2). g + 3.LT2 = 15 g = 15 - LT6 g = Greater than 9 g > 9
Sequences and patterns
If Sn = 3n, what is the units digit for S65? There is a pattern: 3¹ = 3, 3² = 9, 3³ = 27, 34 = 81, 35 = 243. So, the units digit for 365 will be 3 again (64 is a multiple of 4)
The intersection of 2 lines
If lines intersect, both equations at the intersect point are true. That is, the ordered pair (x,y) solves both equations At what point lines y = 4x + 10 and 2x + 3y = 26 intersect? To quickly solve this, replace y = 4x + 10 in the second equation. x = 4, y = 6
Similar Triangles Triangles are defined as similar if all their corresponding angles are equal and their corresponding sides are in proportion.
If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a²:b²
Of all quadrilaterals with a given perimeter, the square has the largest area. Conversely, of all the quadrilaterals with a given area, the square is the one with the smaller perimeter
If you are given 2 sides of a triangle or a parallelogram, you can maximize the area by placing those two sides perpendicular
examples of sequence problems/exponent
If you are given that the first two terms of a sequence are 20 and 200, you know that k = 200/20 = 10. So, replacing: S = x.10n. Now, you can find x: 20 = x.10¹, so x = 2, and the sequence is 2.10n
Inequalities
If you multiply or divide an inequality by a negative number, you have to flip the inequality sign. You cannot multiply or divide an inequality by a variable, unless you know the sign of the variable. The reason is you wouldn't be able to know whether you have to change the inequality sign. Combining Inequalities
Sequence Formulas
Linear sequences are also called arithmetic sequences. In those, the difference between two terms is constant. S = kn + x Exponential sequences: S = x(kn)
MEDIAN..MEDIAN IS DIFFERENT THAN MEAN/AVERAGE...ITS NOT AT ALL SAME
M= Middle number if set is odd in ascending or descending order ( don't has to be evenly spaced like in average ) M= average of two middle terms if set is even ..MEDIAN WONT BE IN THE SET. If values are unknown then we can find median depends on the set. EX: x,2,5,11,11,12,33 = MEDIAN WILL BE 11 NO MATTER x .. x,2,5,11,12,12,33= MEDIAN can be 11 or between 11 and 12, or 12..varies..
Combo Problems If GMAT asks you for x + y instead of only x or y, never try to solve for the isolated variables. It will almost always be much easier to isolate the combo and get the answer. Look for similarities in the numerator and denominator. You can cancel variables or similar numbers. This will save you a lot of time.
Mismatch problems GMAT may induce you to think one equation has no solution by giving you 3 variables and 2 equations. You must try to solve each of these problems, specially in data sufficiency. Sometimes, you can solve a problem for one variable but not for the others. If there are any non-linear terms in an equation, there will usually be two or more solutions. Double check each one anyway.
ALGEBRA: VIC Problems Picking number to solve VICs:
Never pick 0 or 1 (2) Make sure all the numbers you pick are different (3) Pick Small numbers (4) Try to pick prime numbers (5) Don't pick numbers that appear as a coefficient in several answer choices Direct Algebra Break the problem down into manageable parts Write every step, as it may be difficult to verify the answer
If you see question with: How many ways can arrange??? ex:there are 4 kinds of beds, 3 kinds of closets, 2 kinds of shelves and 7 kinds of chairs. In how many ways can a person decorate his room if he wants to buy in the workshop one shelf, one bed and one of the following: a chair or a closet
ONLY COMBINATION: 4*2*(7+3)= 80 ways....
If the probability of raining on any given day in Atlanta is 40 percent, what is the probability of raining on exactly 2 days in a 7-day period?
P = C^7_2*0.4^2*0.6^5 or as explained earlier in perfect score: (slide 18) (0.4)2 *(0.6)6 * 7!/2! *5!
Jane bought twice as many apples as bananas: A = 2B
P is X percent of Q: P = xq/100 OR P/Q = x/100 Pay $10 for the first 2 cds and 7 for additional CD: T = 10.2 + 7.(n-2 *** 1 year ago, Larry was 4 times older than John: L - 1 = 4J - 4 ***
Probability of both events happening
P(A)*P(B)
ODDS IN FAVOR
P(A)/ P(not A)
ODDS AGAINST
P(NOT A)/P(A)
Independet events Probability ..ex, P(a and b) ex: Q:There is a coin and a die. After one flip and one toss, what is the probability of getting heads and a "4"?
P(a) * P(b) both are independent events, not connected with each other. iTS DIFFERENT than mutually exclusive events. In independent events, they happen together but independently. Solu: 1/2 * 1/6= 1/12
Probability of exactly 1 event happening
P(a)P(B not ) +P(B)P(A not)
Parallel and perpendicular lines
Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes!!! VERY IMPORTANT PROPERTY
Formulas and Functions - Advanced
Recursive formulas for sequences A recursive formula looks like this: An = An-1 + 9 To solve a recursive sequence, you need to be given the recursive rule and also the value of one of the items. 1) Linear sequence: S1 = k + x x is the difference between two terms, and k is the value of the sequence for S0. 2) Exponential sequence: Sn = Sn-1k You can find k by having 2 terms of the sequence. Exponential growth The formula for exponential growth is: Y(t) = Y0.kt - y0 is the quantity when t = 0.
AVERAGE=MEAN=Equally shared among = per capita income
SUM/n S= A.n similar to D= RT (use RTD table while solving complex Average problems) To find average don't have to know all numbers of set ..but need total sum. In conseque stes: 1st+2nd/2 or sum of set= average * number of terms
For more complicated problems, like |x + 2| < 5, a good method is to shift the entire graph down by 2. The center point will then change from 0 to 2
Standard formula: when |x + b| <c, center point is -b, and "less than" symbol tells us x is less than c units away from -b
Linear Growth or decay
They are defined by the function y = mx + b. m is the constant rate at which the quantity grows. b is the quantity at time zero. Example: a baby weighs 9 pounds at birth and gains 1.2 pounds per month. The function of his weight is: w = 1.2t + 9 (t = nº of months
Absolute Value Equations
Three steps to solve absolute value equations: (1) Isolate the abs expression: 12 + |w - 4| = 30 |w - 4| = 18 (2) Once you have a |x| = a equation, you know that |x| = +/- a. So, remove the brackets and test both cases: w - 4 = 18, w = 22; w - 4 = -18, w = -22. (3) Check in the original equations if both solutions are valid. It is very important to check both values. Some data sufficiency problems may seem insufficient as you would have 2 answers. Although, when you check both, you may find that one solution is invalid, so the alternative is sufficient.
Diagonal of a square: d = s . (Sq-rt 2) Main Diagonal of a Cube: d = s . (sq-rt 3)
To find the diagonal of a rectangle, you must know either both sides or the length of one side and the proportion from this to the other side To find the diagonal of a rectangular solid, if you know the 3 dimensions, you can use pythagorean theorem twice: First, use pythagorean theorem with the length and the width to find the diagonal of the bottom face. Then, use pythagorean theorem again to find the main diagonal. The sides for this second pythagorean will be: the height, the bottom face diagonal and the main diagonal
Disguised Quadratics
When you find an equation similar to 3w² = 6w, don't forget that dividing both sides by 3w will cause you to miss one solution! If you factor, you get w(3w - 6) = 0, so w = 0 is also a solution You are only allowed to divide an equation by a variable/expression if you know that this variable/expression does not equal 0.
ISOSCELE TRIANGLE- 45-45-90 Degree
X-X- x V2 (Hypotenus) Half of the square.
SPECIAL RULE FOR INEQUALITIES
You can always add inequalities. However, you can only multiply inequalities if both sides or both inequalities are positive. We can never subtract or divide inequalities!!
MEAN and median correlation
You can not make any relation between two if you don't know exact numbers of set. ex: Set A, Set B, and Set C each contain only positive integers. If Set A is composed entirely of all the members of Set B plus all the members of Set C, is the median of Set B greater than the median of Set A? (1) The mean of Set A is greater than the median of Set B. (2) The median of Set A is greater than the median of Set C. ANS: E ..bcs mean median does not say anything about each other here.
Squaring inequalities
You cannot square both sides unless you know the signs of both sides. If both sides are negative, the inequality sign will flip when you square. If both sides are positive, the sign will remain. If one side of the inequality is positive and the other is negative, you cannot square!
Optimization problems If x ≥ 4 + (z + 1)², what is the minimum value for x?
You have to recognize that a square value is minimized when set to zero.
Exterior angles of a triangle
a + b + c = 180; (a,b,c= angles of triangle) c + d = 180 d = a + b (D= Exterior angle of the triangle)
Disguised forms of common factored expressions:
a² - 1 = (a + 1)(a - 1) a² + b² = 9 + 2 ab = a² - 2ab + b² = 9 = (a - b)² = 9 = a - b = +/- 3 (x² + 4x + 4) / (x² - 4) = (x + 2)(x + 2)/(x + 2)(x - 2) = (x + 2)/(x - 2) Attention: you can only simplify the equation above if you know that x ≠ 2. Otherwise, the equation is undefined
SD
dispersion of set of numbers around the mean.
SD
distance of number from set's average. - Its spread of numbers in the set. - Mean can be same for two different sets but SD will be different. - If SD IS small- numbers-clutters- are mean, viceversa - IF Each data point is increased by 7 the SD??? = will be same. - IF Each DP is increased by factor of 7 - multiply each DP by 7= SD will increase bcd numbers will be increased 7 times
The number of possible arrangements of n different items...
n!
combination formula- unordered subgroups - when order does not matter
nCk= n!/ k! (n-k)! - How many are in total group= n - How many are we picking= k - does order matter????
Permutation- ordered subgroups- when order matters
nPk= n!/ (n-k)!
probability
ways in Favor/ total ways... 4 balls, 2 green , 2 red ..whats the probability of having red balls: 2/4= 1/2= 50% Probability is always between 0 and 1
Weighted averages
weighted A will be closer to heavy weight ex: 4/5(20) + 1/5(30)= 22 which is closer to 20. - all weights will sum up to 1 (4/5+1/5)= 1 - If weights does not sum up to 1 then divide whole by sum of weights. ex 2(20) +3(30)/2+3 = 26 weight(datapoint) +w (DP)/ Sum of weights. WEIGHTS CAN BE: frequency, ratio, percentage, fractions.
