GMAT
45-45-90 triangle/square
A 45-45-90 is exactly half of a square. Two 45-45-90 triangles put together make up a square. So, if you are given the diagonal of a square, you can find the side by using the relation above
Circles: Area of a Circle
A = πr² --> All we must know is the radius Area of a Sector = πr² . (x°/360°) An inscribed angle is equal to half of the arch it intercepts. You can be asked this property by having the outer angle different then half the "central angle". If that is the case, the point of the "central angle"
Circles
A circle: a set of points in a plane, equidistant from a fixed center Any line segment that connects the center to a point in the circle is called radius. Any line segment that connects two points on a circle is called chord. Any chord that passes through the center is called diameter. Diameter = 2.radius
Circles: Arc Length
A portion or distance ON a circle is called ARC. The length of Arc AxB can be calculateded by: Length AxB = Circumference x Angle/360
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
X percent of Y percent of Z is decreased by Y percent. What is the result?
Answer: XYZ/10,000 - (Y/100)*(XYZ/10,000) = (100XYZ - XY²Z ) / 1,000,000
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.
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.
Parallel and perpendicular lines
Parallel lines have equal slopes. Perpendicular lines have negative reciprocal slopes!!! VERY IMPORTANT PROPERTY
Combinatorics formulas
Permutation formula: n! / (n - r)! Combination formula: n! / (n-r)! r!
Combinatorics 2
Slot method: 1) Set up the slots: __ __ __ __ __ 2) Fill the restricted: 5 __ __ __ 4 3) Fill the remaining: 5 . 8 . 7 . 6 . 4 = 6,720 The number of codes possible is 6,720
Formulas 2
Strange symbol formulas The symbol is irrelevant. Just follow each step of the procedure. Example: if x @ y = x² + y² - xy, what is 3 @ 4? Just replace: 3² + 4² - 3.4 = 25 - 12 = 13 Example: x¥ is defined as the product of all integers smaller than x but greater than zero. This equals (x-1)! Formulas with unspecified amounts Example: if the side of a cube is decreased by 2/3, by what percentage will the volume decrease? For those problems, pick numbers. Since 3³ = 27 and 1³ = 1, the percentage of decrease is 26/27, which is 96,3%
3 Dimensions: Surface Area
Surface Area = the SUM of the areas of all of the faces Rectangular Solid: 2(Base x Height) + 2 (Width x Height) + 2 (Base x Width) Cube: 6 x (Side)²
Cylinders
Surface Area: 2π(r²) + 2πrh 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.
Testing Inequality cases
xy>0 --- x and y are both positive or both negative xy<0 --- x and y have different signs x^2 - x < 0 ----- x^2<x so 0<x<1 (x is a proper fraction)
Slope-intercept Equation
y = mx + b m = slope b = y-intercept Horizontal and vertical lines Horizontal: y = some number Vertical: x = some number
Disguised Quadratics 2
36/b = b - 5 : This is a quadratic, as if you multiply both sides by b you get 36 = b² - 5b. The equation x³ + 2x² - 3x = 0 can be solved. Factoring will result on x(x² + 2x - 3) = 0, so 0 is one solution and the factored quadratic will have two roots. The equation has 3 solutions. If you have a quadratic equation equal to 0 and you can factor an x out of the expression, x=0 is one of the solutions. You are only allowed to divide an equation by a variable/expression if you know that this variable/expression does not equal 0.
45-45-90 triangle relationship between sides
45° - 45° - 90° Leg - leg - hypotenuse 1 : 1 : sq-rt 2
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.
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
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.
Rates and work - Advanced
Advanced rates and work problems involve either more complicated language to translate, more complicated algebra to solve or trickier Data Sufficiency logic to follow Be ready to break rate or work problems into natural stages. Likewise, combine workers or travelers into a single row when they work or travel together.
Word Translations
Algebraic Translation: Translating correctly A is half the size of B: A = 1/2B A is 5 less than B: A = B - 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 *** Be ready to use expressions such as (n - x), where x will be the number of units with a different price
If Cecil reads T pages per minute, how many hours does it take for her to read 500 pages?
Answer: 500/60T
X is what percent of y, in terms of x and y?
Answer: x = w/100 . y -> x = wy/100 -> 100x/y = w
Advanced factoring and distributing
Distributed Form Factored Form x² + x x(x + 1) X^5 - x³ X³(x² - 1) = x³(x + 1)(x - 1) 6^5 - 6³ 6³(6² - 1) = 35.6³ 4^8 + 4^9 + 4^10 48 + (1 + 4 + 4²) = 21.48 p³ - p p(p² - 1) = p(p - 1)(p + 1) a^b + a^b+1 ab(1 + a) m^n - m^n-1 mn(1 - m-1) = mn-1 (m-1) 5^5 - 5^4 55(1 - 1/5) = 54(5-1) xw + yw + zx + zy -- (w + z) (x + y) = w(x + y) + z(x + y) Be able to recognize any of the above and quickly manipulate it in both directions.
Combining inequalities
First, make all the inequality signs face the same direction. Them, add them up. Example: Is a + 2b < c + 2d? (1) a < c (2) d > b a < c + b < d + b < d = a + 2b < c + 2d
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
Productivity ratio
For productivity ratios, you should invert the time. Example: If Machine A works for 80 minutes to produce X and has a productivity ratio of 4:5 to Machine B, how long does machine B take to produce X? 5/4 = x/1/80 5.(1/80) = 4x 4x = 1/16 x = 1/64, so machine B takes 64 minutes to produce X.
Combinatorics
Fundamental counting principle: if you must make a number of separate decisions, then multiply the number of ways to make each individual decision to find the number of ways to make all decisions together. Example: If you have 4 types of bread, 3 types of cheese and 2 types of ham and wish to make a sandwich, you can make it in 4.3.2 = 24 different ways. For problems where some choices are restricted and/or affect other choices, choose most restricted options first (slot method): Example: you must insert a 5-digit lock code, but the first and last numbers need to be odd, and no repetition is allowed. How many codes are possible?
GMAT Word Translation specific
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
Symmetry
GMAT has difficult symmetry problem, such as: For which of the following f(x) = f(1/x) ? For those problems, you can either pick each alternative and substitute for the 2 variables (here, x and 1/x) or pick numbers. Normally, picking numbers is easier. The same is valid for problems like: for which of the following f(x-y) does not equal f(x) - f(y)? For which of the following f(a(b+c)) = f(ab) + f(ac)?
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.
Trapezoid GMAT specfiic
GMAT may require you to divide some shapes. Notice, for example, that a trapezoid can be cut into 2 right triangles and 1 rectangle.
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
Multiple ratios: Make a common Term
If C:A = 3:2, and C:L = 5:4, what is A:L? C : A : L C : A : L 3 : 2 : -> Multiply by 5 -> 15 : 10 : 5 : : 4 ->Multiply by 3 -> 15 : : 12 Once you have converted C to make both ratios equal (15), you can combine the ratios. Combined Ratio C:A:L = 15:10:12, and ratio A:L = 10:12 or 5:6 The combined ratio will normally be asked as "the least number" of a population. The least number is the sum of common minimum denominator. In the situation above, the least number is 15 + 12 + 10 = 37.
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
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. 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.
Multiplying or dividing two equations:
If xy² = 96 and 1/xy = 1/24, what is y? We can easily solve by multiplying both equations: xy² . 1/xy = 96 . 1/24 : y = -4 If you see similar/inverse equations, dividing/multiplying them may be a good idea to cancel variables.
Square-Rooting inequalities
If x² < 4, then |x| < 2. This means x > -2 and x < 2. So, -2 < x < 2. You can only square-root an inequality if you know x is not negative
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.
Multiple arrangements
In a group with 12 seniors and 11 juniors, you need to pick 3 seniors and 2 juniors. In how many different ways can you do that? For these problems, you have to calculate the arrangements separately and then multiply both. Answer: 220 . 55 = 12,100
Population problems
In such problems, normally a population increases by a common factor every time period. You can build a population table to avoid computation errors.
Testing Inequality cases
Is bd < 0? (1) bc < 0 (2) cd > 0 b c d bc<0 cd>0 bd? + - - YES YES - - + + YES YES - bd must be < 0 , as b and d will have different signs Whenever you need to proceed with a positive/negative analysis, building a table like the one above may be a good idea.
Rates and work - Advanced Example
Liam is pulled over for speeding just as he is arriving at work. He explains that he could not afford to be late today, and has arrived at work only 5 minutes before he is to start. The officer explains that if he had driven 5mph slower for his whole commute, he would have arrived on time. If his commute is 30 miles, how fast was he actually driving? Since the slower trip takes 4 minutes more, 30/r = 30/(r+5) + 1/15 -> 30/r = (r + 455)/(15(r+5)) -> 450r + 2,250 = r² + 455r -> r² + 5r - 2,250 = 0 -> r = 45
Optimization problems
Maximizing linear functions: since linear functions either rise or fall continuously, the max/min points occur at boundaries: at smallest or largest possible x as given in the problem. Maximizing quadratics: The key is to make the squared expression equal to 0. Whatever value of x makes the squared expression equal 0 is the value of x that maximizes/minimizes the function
Rates and Work - Translations
Normally, you will have to convert the rate. This is the only difference to the RTD problems. Example: Oscar can perform one surgery in 1.5 hours. You have to convert it to: Oscar can perform 2/3 surgeries per hour. This is the actual rate. If two or more agents are performing simultaneous work, add the work rates. (Typical problems are: "Machine A, Machine B"). Exepction: one agent undoes the other's work, like a pump putting water into a tank and another drawing water out. Rate will be a - b. Example: Larry can wash a car in 1 hour, Moe can wash it in 2 hours, and Curly washes it in 4 hours. How long does it take for them to wash 1 car? Answer: Rate is 1/1 + 1/2 + 1/4 = 7/4 cars/ hour. To find the rate for 1 car, just take the reciprocal: 4/7 hours to wash 1 car (34 minutes)
Maximum Area of a quadrilateral
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
Optimization problems
On problems involving minimization or maximization, focus on the largest and smallest possible values of each variables. 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.
Inequalities and Extreme Values
Operation Example Procedure Addition 8+LT2 LT10 Subtraction 8-LT2* LT6 Multiplication a) 8xLT2 LT16 b)-7xLT2* GT14 Division 8/LT2 GT4 Multiply 2 LT8XLT2 LT16 only if extreme values both are positive Take care with the ones with *
VIC Problems (Variable in answer choice)
Picking number to solve VICs: (1) 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 Common Translations: "y percent less than z" = z - yz/100
Formulas
Plug-in Formulas GMAT may give you a strange formula and ask you for the result by giving you the values of all variables. These problems are just a matter of careful computation.
Word Translations
Positive Constraints (cont.) Multiply inequalities (but not divide them) Square/unsquare an inequality: x < y ; x² < y². x < y ; √x < √y
Equations for Exponential Growth or Decay
Problems like: How long does it take to double the population? Formula: Population = S.2t/I, where S is the starting value, t is the time and I is the Interval, or the amount of time given for the quantity to double. If the quantity triples: Population = S.3t/I. If the quantity is cut by half: Population = S.(0.5)t/I Example: If a population of rabbits double every 7 months and the starting population is 100, what will be the population in 21 months? Answer: Population = 100 . 221/7, = 100.2³ = 800
Rates and Work
Rate x Time = Distance or Rate x Time = Work RTD Chart: R x T = D, or R = D/T, or any other equation Always express rates as "distance over time". If it takes 4 seconds for an elevator to go up one floor, the rate is 0.25 floors / second. Translations: Train A is travelling at twice the speed of train B: A = 2r, B = r. Wendy walks 1 mph slower than Maurice: W = r-1, M = r Car A and car B are driving toward each other: Car a = r1, Car b = r2 - Shrinking distance = r1 + r2 (if they are driving away, this is growing distance) Car A is chasing car B and catching up: a < b, and shrinking distance = a - b
Isosceles Triangles and the 45-45-90 triangle
The 45-45-90 triangle is a special triangle with 2 equal sides and a relation between each side. If you are given one dimension on a 45-45-90 triangle, you can find the others.
Exponential growth
The formula for exponential growth is: Y(t) = Y0.kt - y0 is the quantity when t = 0.
Eliminating roots
The most effective way to eliminate roots is to square both sides. Don't forget: whenever you do that, you need to check both solutions in the original equation
Perimeter of a Sector
The perimeter of a sector is ARC + 2 Radius To find either the Arc Length or the perimeter of a sector, all you need is the radius plus the angle of the sector
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
The Slope of a Line 1
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.
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
Coordinate plan: Difficult question What are the coordinates for the point on line AB that is 3 times as far from A as from B, and that is between points A and B, knowing that A = (-5,6) and B (-2,0)?
The total x-axis distance from one point to the other is 3 (-5 - -2) The total y-axis distance from one point to other is 6 (6 - 0) Consider that the distance between A and B is 4x, and we need a point which is 3x distant from A and x distance from B. Therefore, 3x = 4 --- x = 0.75. The point is located on x-axis -2.75 4x = 6, x = 1.5. The point is located on y-axis 1.5 The point is (-2.75, 1.5)
The Slope of a Line 2
There are 4 types of slopes: positive slope, negative slope, zero slope and undefined slope. 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.
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)
Ratios: The unknown multiplier
This technique can be used for complicated problems, such as: the ratio of men to women is 3:4. There are 56 people in the class. How many are men? Introducing the unknown multiplier: 3x + 4x = 56. x = 8, so we have 24 men. You can use this technique for 3 ratios as well.
Equilateral Triangles and the 30-60-90 Triangle
This triangle, when cut in half, yields two equal 30-60-90 triangles. The long leg will be the height, the hypotenuse is the equilateral triangle's side and the short leg is half of the equilateral triangle's half.
The 3 special factors
Those appear very often on GMAT. Recognize them instantly! 1) x² - y² = (x + y)(x - y) 2) x² + 2xy + y² = (x + y)² = (x + y)(x + y) 3) x² - 2xy + y² = (x - y)² = (x - y)(x - y)
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.
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
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).
Word Translations Hidden Constraints
When the object of the problem has a physical restriction of being divided, such as votes, cards, pencils, fruits, you must realize the problem has told you the variable is an integer. Example: Jessica bought x erasers for 0.23 each and y pencils for 0.11 each. She has spent 1.70. What number of pencils did she buy? 23x + 11y = 170 -- 11y = 170 - 23x -- y = (170 - 23x)/11 Since the number of pencils is an integer, y = (170 - 23x)/11 must be an integer as well. Test the values for x to find the answer.
Arrangements with constraints
When the problem presents you a constraint, such like "Jan won't sit next to Marcia", you should first calculate the arrangement without considering the constraint and then using a method for the constraint. Example 1: Greg, Mary, Pete, Jan, Bob and Cindy are to sit in 6 adjacent seats, but Mary and Jan won't sit next to each other. How many combinations? Without the constraint, you have 6!, which is 720. Now, use the glue method: consider that an and Marcia are one person. The combination would then be 5!, which is 120. You have to multiply this by 2, as Marcia could sit next to Jan OR Jan could sit next to Marcia. So, the number of total restrictions is 120 . 2 = 240, and the number of combs is 720 - 240 = 480
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!
Triangles and Area, revisited
You can designate any side of a triangle as the base. So, you have 3 ways to calculate the area of a triangle, depending on which side is considered the base. Obviously, those 3 calculations will all yield the same result In right triangles, if you choose one of the legs as the base, the other leg will be the height. If you choose the hypotenuse as the base, you will have to find the height. The area of an equilateral triangle of side S is equal to (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.
Combinations x Permutations
You have to be ready to see the difference between the 2 statements below: If seven people are going to sit in 3 seats (with 4 left standing), how many different seating arrangements can we have? If three of seven standby passengers are to be selected for a flight, how many different combinations can we have? In the first problem, order matters. So, it is a permutation, and the formula is: n! / (n - r)!, where r! is the number of chosen items from a pool. For the second problem, all it matters is if the passenger is flying or not. Since the order of chosen passengers doesn't matter, the problem is a combination, and the formula is: n! / (n-r)! r!
Sequence Formulas
You must be given the rule in order to find a number in a sequence. Having two terms of the sequence is not enough. Linear sequences are also called arithmetic sequences. In those, the difference between two terms is constant. S = kn + x Exponential sequences: S = x(kn)
Inequalities and absolute value
You should interpret some inequality problems with absolute value expressions as a range on the number line. 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
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
Area of a TRAPEZOID
((Base1 + Base 2) x Height) / 2
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.
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
Area of a RHOMBUS
(Diagonal 1 x Diagonal 2) / 2
Common Right Triangles
3-4-5 and its key multiples: 6-8-10, 9-12-15, 12-16-20 5-12-13 and its key multiples: 10-24-26 8-15-17
30-60-90 Triangle relationship between sides
30° - 60° - 90° Short leg - long leg - hypotenuse 1 : sq-rt 3 : 2
Area of a PARALLELOGRAM
Base x Height
Circumference
C = 2πr Some GMAT problems will require you to calculate π. In those, use 3 or 22/7. A full revolution of a spinning wheel means a point on the edge of a wheel travels one circumference. If a wheel spins at 3 revolutions per second and its diameter is 10 units, a point on the edge will travel at a rate of 30π/s (Circumference = 10 π, and the point travel 3 circumferences per second).
Complex Absolute value equations
Complex Absolute value equations: If the equation contains only one variable, use algebra. If it contains 2 variables, go conceptual (positive/negative analysis). Example: |x - 2| = |2x - 3| You have to test 2 scenarios: the first with both equations positive, and the second with one of them negative. After finding the solutions, you have to check all the answers.
Diagonals of other polygons
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.
Rates and Work - Population problems
Example: the population of a bacteria triples every 10 minutes. If the population was 100 20 minutes ago, when will it reach 24,000? This population table can be very useful to avoid mistakes like doing the right computation but forgetting that the p population right now is 900 and not 100, which was 20 I minutes ago. Population problems are just an exponential sequence since the bacterias grow by a constant factor. If something grows by a constant amount, you have a linear sequence.
Combining Inequalities
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
Distributing factored equations
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
Sequence Formulas 2
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 Other sequence problems: 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 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)
Quadrants
If you are required to find out which quadrant a given line passes two and you have the equation, set x and y to zero to find the two intercepts and draw the line. This is the fastest way
Factoring / Distributing
If you encounter a quadratic equation, factoring may help you. If you encounter a factored equation, try distributing it. Never forget that (x + k) (x - m) = 0 means x = -k and x = m. Zero in the denominator: Undefined 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!!
Manipulating compound inequalities
If you perform an operation on a compound inequality, be sure you do it on every term.
Ratios
The relationship ratios express is division. A 3:4 rate is the same as a decimal 0.75. The order of the ratio has to be respected always.
Inequalities - Advanced
You cannot multiply or divide an inequality by a variable unless you know the sign of the variable!! Reciprocals of inequalities: If we do not know the sign of both x and y, we cannot take reciprocals If x<y, then: 1/x > 1/y when both x and y are positive 1/x > 1/y when both x and y are negative 1/x < 1/y when x is negative and y is positive If ab < 0 and a>b, which is true? (1) a>0 (2) b > 0 (3) 1/a > 1/b A good way to solve those problems is with a positive/negative analysis. We will see that 1 and 3 are true a b ab + - - - - +
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!
