Kaplan Math Reference
How to FACTOR certain POLYNOMIALS
A polynomial is an expression consisting of the sum of two or more terms, where at least one of the terms is a variable. Learn to spot these classic polynomial equations: ab + ac = a(b + c) a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a-b)^2 a^2 - b^2 = (a - b)(a + b)
How to find the AREA of a TRAPEZOID
A trapezoid is a quadrilateral having only two parallel sides. You can always drop a perpendicular line or two to break the figure into a rectangle and a triangle or two triangles. Use the area formulas for those familiar shapes. Alternatively, you could apply the general formula for the area of a trapezoid: Area= (average of parallel sides)x(height)
How to MULTIPLY/ DIVIDE VALUES WITH EXPONENTS
Add/subtract the exponents
How to find the NEW AVERAGE when a number is added or deleted
Use the sum of the terms of the old average to help you find the new average EXAMPLE: Michael's average score after 4 tests is 80. If he scores 100 on the fifth test, what's his new average? Find the original sun from the original average: original sum= 4 x 80= 320 Add the fifth score to make the new sum: new sum= 320 +100= 420 Find the new average from the new sum: new average= 420/5= 84
How to calculate a simple PROBABILITY
Probability= number of desired outcomes/ number of total possible outcomes EXAMPLE: What is the probability of throwing a 5 on a fair six-sided die? There is one desired outcome- throwing a 5. There are 6 possible outcomes- one for each side of the die. Probability= 1/6
How to use the ORIGINAL AVERAGE and NEW AVERAGE to figure out WHAT WAS ADDED OR DELETED
Use the sums. Number added= (new sum) - (original sum) Number deleted= (original sum) - (new sum) EXAMPLE: the average of 5 numbers is 2. After one number is deleted, the new average is -3. What number was deleted? Find the original sum from the original average: original sum: 5x2=10 Find the new sum from the new average: new sum: 4x(-3)=-12 The difference between the original sum and the new sum is the answer number deleted: 10-(-12)=22
4.How to convert FRACTIONS TO DECIMALS and DECIMALS TO FRACTIONS
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the digits in the numerator and use the decimal name in the denominator.
How to plug a number into an ALGEBRAIC EXPRESSION
To evaluate an algebraic expression, choose numbers for the variables or use the numbers assigned to the variables. Evaluate 4np+1 when n=-4 and p=3 4(-4)(3)+1= -47
How to use actual numbers to determine a RATIO
To find a ratio, put the number associated with "of" on the top and the number associated with "to" on the bottom. Ratio= of/ to Ratios should always be reduced to lowest terms. Ratios can also be expressed in linear form, such as 5:3
How to find the SURFACE AREA of a RECTANGULAR SOLID
To find the surface area of a rectangular solid, you have to find the area of each face and add the areas together. Here's the formula: l=length w=width h=height surface area=2(lw)+2(wh)+2(lh)
How to solve for one variable IN TERMS OF ANOTHER
To find x "in terms of" y, isolate x on one side, leaving y as the only variable on the other
How to plot points on the NUMBER LINE
To plot the point 4.5 on the number line, start at 0, go right to 4.5, halfway between 4 and 5.
How to solve an INEQUALITY
Treat it much like an equation- adding, subtracting, multiplying, and dividing both sides by the same thing. Just remember to reserve the inequality sign if you multiply or divide by a negative quantity. EXAMPLE: Rewrite 7-3x>2 in its simplest form.
How to solve a DIGITS problem
Use a little logic- some trial and error. EXAMPLE: if A, B, C, and D represent distinct digits in the addition program below, what is the value of D? AB + BA = CDC Two 2-digit numbers will add up to at most something in the 100s, so C-1. B plus A in the units column gives a 1, and since A and B in the tens column don;t add up to C, it can't simply by that B + A= 1. It must be that B + A= 11, and a 1 gets carried. In fact, A and B can be any pair of digits that add up to 11 ( 3 and 8, 4 and 7, etc.), but it doesn't matter what they are; they always gives you the value for D, which is 2. 47+74= 121 83+38= 121
How to SOLVE a simple LINEAR EQUATION
Use algebra to isolate the variable. Do the same steps to both sides of the equation. 28=-3x-5 33=-3x -11=x
How to find the DIAGONAL of a RECTANGULAR SOLID
Use the Pythagorean theorem twice, unless you spot "special" triangles
How to find the THIRD ANGLE of a TRIANGLE, given the other 2 angles
Use the fact that the sum of the measures of the interior angles of a triangle always equals 180 degrees.
How to handle NEGATIVE POWERS
A number raised to the exponent -x is the reciprocal of that number raised to the exponent x
How to find the AREA of a PARALLELOGRAM
Area= (base)(height)
How to find the CIRCUMFERENCE of a CIRCLE
Circumference= 2[pie]r, where r is the radius Circumference= [pie]d, where d is the diameter
How to predict whether a sum, difference, or product will be EVEN or ODD
Don't bother memorizing the rules. Just take simple numbers such as 2 for even numbers and 3 for odd numbers and see what happens. Example: If m is even and n is odd, is the product nm off or even? say m=2 and n=3 2x3=6, which is even, so mn is even
How to work with EQUILATERAL TRIANGLES
Equilateral triangles have three equal sides and three 60 degree angles. If a GRE question tells you that a triangle is equilateral, you can bet that you'll need to use that information to find the length of a side or the measure of an angle.
2.How to add, subtract, multiply, and divide FRACTIONS
Find a common denominator before adding or subtracting fractions To multiply fractions, multiply the numerators first and then multiply the denominators. Simplify if necessary. You can also reduce before multiplying numerators and denominators. This keeps the products small. To divide by a fraction, multiply by its reciprocal. To write the reciprocal of a fraction, flip the numerator and denominator.
How to find the HYPOTENUSE or a LEG of a RIGHT TRIANGLE
For all right triangles, the Pythagorean theorem is a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse
How to find a WEIGHTED AVERAGE
Give each term the appropriate "weight" EXAMPLE: The girls' average score is 30. The boys' average score is 24. If there are twice as many boys as girls, what is the overall average? weighted average= (1x30) + (2x24)/ 3 = 78/3 = 26 IHINT: Don't just average the averages
How to find the PERIMETER of a RECTANGLE
Perimeter= 2(length + width)
How to find the SURFACE AREA of a CYLINDER
Surface area= 2pie(r^2)+2pierh
How to COUNT CONSECUTIVE NUMBERS
The number of integers from A to B inclusive is B-A+1 Example: How many integers are there from 73 through 419, inclusive? Set up: 419-73+1= 347
How to find an ANGLE formed by INTERSECTING LINES
Vertical angles are equal. Angels along a line add up to 180 degrees.
How to find the VOLUME of a CYLINDER
Volume= area of the base x height= pie(r^2)h
4.How to add, subtract, multiply, and divide POSITIVE AND NEGATIVE NUMBERS
When addends (the numbers being added) have the same sign add their absolute values; the sum has the same sign as the addends. But when the addends have different signs, subtract the absolute values; the sum has the sign of the greater absolute value. 3+9=12, but -3+(-9)=-12 In multiplication and division, when the signs are the same, the product/ quotient is positive. When the signs are different, the product/ quotient is negative. 6x7=42 and -6x-7=42 -6x7=42 and 6x-7=42
How to use PEMDAS
When you're given a complex arithmetic expression, it's important to know the order of operations. Just remember PEMDAS--> PARENTHESES EXPONENTS MULTIPLICATION DIVISION ADDITION SUBTRACTION *going from left to right*
1.How to add, subtract, multiply, and divide WHOLE NUMBERS
You can check addition with subtraction. 17+5=22 22-5=17 You can check multiplication with division. 5×28=140 140÷5=28
How to handle LINEAR EQUATIONS
You may also encounter linear equations on the GRE. A linear equation is often expressed in the form y=mx+b where m= the slope of the line =rise/ run b= the y-intercept (the point where the line crosses the y-axis)
How to handle GRAPHS of FUNCTIONS
You may see a problem that involves a function graphed onto the xy-coordinate plane, often called a rectangular coordinate system on the GRE. When graphing a function, the output, f(x), becomes the y-coordinate. For example, in the previous example, f(x)=x^2-1, you've already determined 2 points, (1,0) and (0,-1). If you were to keep plugging them in numbers to determine more points and the plotted those points on the xy-coordinate plane, you would come up with something like this: This curved line is called a parabola. In the event that you should see a parabola on the GRE, you will most likely be asked to choose which equation the parabola is describing. These questions can be surprisingly easy to answer. Pick out obvious points on the graph, such as (1,0) and (0,-1) above, plug these values into the answer choices, and eliminate answer choices that don't work with those values until only one answer choice is left.
How to handle EXPONENTS with a base of ZERO and bases with an EXPONENT of ZERO
Zero raised to any nonzero exponent equals zero. Any nonzero number raised to the exponent 0 equals 1. The lone exception is 0 raised to the 0 power, which is undefined.
How to find the AREA of a CIRCLE
area= [pie]r^2 where r is the radius
How to find the SLOPE of a LINE
slope= rise/run= change in y/ change in x EXAMPLE: what is the slope of the line that contains the points (1,2) and (4,-5)? Slope= -5-2/4-1=-7/3
How to find the VOLUME of a RECTANGULAR SOLID
volume= lengthxwidthxheight
How to find a COMMON FACTOR of 2 numbers
Break both numbers down to their prime factors to see which they have in common. Then multiply the prime factors to find all common factors. Example: What factors greater than 1 do 135 and 225 have in common? First find the prime factors of 135 and 225; 135= 3x3x3x5, and 225= 3x3x5x5. The numbers share 3x3x5 in common. Thus, aside from 3 and 5, the remaining common factors can be found by multiplying 3, 3, and 5 in every possible combination: 3x3= 9, 3x5= 15, 3x3x5= 25. Therefore the common factors of 135 and 225 are 3, 5, 9, 15, and 45.
How to find an AVERAGE RATE
Convert to totals Average A per B= Total A/ Total B EXAMPLE: if the first 500 pages have an average of 150 words per page, and the remaining 100 pages have an average of 450 words per page, what is the average number of words per page for the entire 600 pages? Total pages= 500+100=600 Total words= (500x150) + (100x450) =75,000 +45,000 =120,000 Average words per page= 120,000/600= 200 To find an average speed, you also convert to totals average speed= total distance/ total time
How to add and subtract LINE SEGMENTS
.___A____.___B___._____C______ If AB = 6 and BC = 8, then AC = 6 + 8 = 14 If AC = 14 and BC = 8, then AB = 14 - 8 = 6
How to recognize MULTIPLES OF 2, 3, 4, 5, 6, 9, 10, AND 12
2: last digit is even 3: sum of digits is a multiple of 3 4: last 2 digits are a multiple of 4 5: last digit is 5 or 0 6: sum of digits is a multiple of 3, and last digit is even 9: sum of digits is a multiple of 9 10: last digit is 0 12: sum of digits is a multiple of 3, and last 2 digits are a multiple of 4
How to find the AREA of a RECTANGLE
Area= (length)(width)
How to find the AREA of a SQUARE
Area= (side)^2
How to find the AREA of a TRIANGLE
Area= 1/2(base)(height) Base and height must be perpendicular to each other. Height is measured by drawing a perpendicular line segment from the base- which can be any side of the triangle- to the angle opposite of the base
How to find the AVERAGE or ARITHMETIC MEAN
Average= sum of terms/ number of terms
How to solve PERMUTATION problem
Factorials are useful for solving questions about permutations (i.e. the number of ways to arrange the elements sequentially). For instance, to figure out how many ways there are to arrange 7 items along a shelf, you would multiply the number of possibilities for the first position times the number of possibilities remaining for the second position, and on on- in other words: 7x6x5x4x3x2x1, or 71 If you're asked to find the number of ways to arrange a smaller group that's being drawn from a larger group, you can either apply logic, or you can use the permutation formula: nPk= n!/(n-k)! where n = (the number in the larger group) and k = (the number you're arranging) EXAMPLE: five runners run in a race. The runners who come in first, second, and third place will win gold, silver, and bronze medals, respectively. How many possible outcomes for gold, silver, and bronze medal winners are there? Any of the 5 runners could come in first place, leaving 4 runners who could come in second place, leaving 3 runners who could come in third place, for a totaly of 5 x 4 x 3 = 60 possible outcomes for gold, silver, and bronze medals winners. Or, using this formula; 5!/(5-3)! = 5!/2! = 5x4x3x2x1/2x1
How to handle FRACTIONAL POWERS
Fractional exponents relate to the roots. For instance, x^(1/2) = square root of x
How to use the PERCENT INCREASE/ DECREASE FORMULAS
Identify the original whole and the amount of increase/ decrease. PERCENT INCREASE= AMOUNT OF INCREASE/ ORIGINAL WHOLE X 100% PERCENT DECREASE= AMOUNT OF DECREASE/ ORIGINAL WHOLE X 100% Example: the price goes up from $80 to $100. What is the percent increase? percent increase= 20/80 x 100% 25%
How to use PERCENT FORMULA
Identify the part, the percent, and the whole Part=PercentxWhole Find the part. EXAMPLE: What is 12 percent of 25? Part= 12/100 x 25= 300/100= 3 EXAMPLE: 45 is what percent of 9? 45= percent/100 x 9 4,500 = percent x 9 500 = percent
How to use actual numbers to determine a RATE
Identify the quantities and the units to be compared. Keep the units straight. Example: Anders types 9,450 words in 3.5 hours. What was his rate in words per minute? First convert 3.5 hours to 210 minutes. Then set up the rate with word on top and minutes on the bottom (because per means divided by): 9,450/210 minutes= 45 words per minute
How to solve a COMPOUND INTEREST problem
If interest is compounded, the interest is computed on the principle as well as on any interest earned. To compute compound interest: final balance= prinicpal x (1 + interest rate/ c)^(time)(c) where c is the number of times the interest is compounded annually Example: if $10,000 is invested at 8 percent annual interest, compounded semiannually, what is the balance after 1 year? Final balance= 10,000x(1 + 0.08/2)^(1)(2) 10,000x1.04^2 10,816 Semiannual interest is interest that is distributed twice a year. When an interest rate is given as an annual rate, divide by 2 to find the semiannual interest rate.
How to solve a COMBINATION problem
If the order or arrangement of the smaller group that's being drawn from the larger group does not matter, you are looking for the number of combinations, and a different formula is called for: nCk=n!/k!(n-k)! where n= (the number in the larger group) and k=(the number you're choosing) EXAMPLE: How many different ways are there to choose 3 delegates from 8 possible candidates? nCk= 8!/3!(8-3)! = 8!/3!x5! = 8x7 = 56 So there are 56 different possible combinations.
How to find the DISTANCE BETWEEN POINTS on the coordinate plane
If two points have the same x-coordinate or the same y-coordinates- that is, they make a line segment that is parallel to an axis- all you have to do is subtract the numbers that are different. Just remember that the distance is always positive. Example: what is the distance from (2,3) to (-7,3)? The ys are the same, so just subtract the xs: 2 - (-7)= 9 If the points have different x-coordinates and different y-coordinates, make a right triangle and use Pythagorean theorem or apply the special right triangle attributes if applicable.
How to find the MAXIMUM and MINIMUM lengths for a SIDE of a TRIANGLE
If you know the lengths of two sides of a triangle, you know that the third side is somewhere between the positive difference and the sum of the other two sides. EXAMPLE: the length of one side of a triangle is 7. The length of another side is 3. What is the range of possible lengths for the third side? The third side is greater than the positive difference (7-3=4) and less than the sum (7+3=10) of the other two sides
How to work with new SYMBOLS
If you see a symbol you've never seen before, don't be alarmed. It's just a made-up symbol whose operation is uniquely defined by the problem. Everything you need to know is in the question stem. Just follow the instructions.
How to solve a COMBINED WORK PROBLEM
In a combined work problem, you are given the rate at which people or machines perform work individually and you are asked to compute the rate at which they work together (or vise versa). The work formula states: The inverse of the time it would take everyone working together equals the sum of the inverses of the times it would take each working individual. In other words: 1/r + 1/s = 1/t where r and s are, for example, the number of hours it would take Rebecca and Sam, respectively, to complete a job working by themselves, and t is the number of hours it would take the two of them working together. Remember that all these variables must stand for units of time and must all refer to the amount of time it takes to do the same task. EXAMPLE: It it takes Joe 4 hours to paint a room and Pete twice as long to paint the same room, how long would it take the two of them, working together to paint the same room, if each of them works at his respective individual rate? Joe takes 4 hours, so Pete takes 8 hours; thus; 1/4 + 1/8 = 1/t 2/8 + 1/8 = 1/t 3/8 = 1/t t = 1/(3/8) = 8/3 So it would take them 8/3 hours, or 2 hours and 40 minutes, to paint the room together
How to solve DILUTION or MIXTURE problem
In dilution or mixture problems, you have to determine the characteristics of a resulting mixture when different substances are combined. Or, alternatively, you have to determine how to combine different substances to produce such problems- the straightforward setup and balancing method. EXAMPLE: If 5 pounds of raisins that cost $1 per pound are mixed with 2 pounds of almonds that cost $2.40 a pound, what is the cost per pound of the resulting mixture? The straightforward setup: ($1)(5) + ($2.40)(2) = $9.80 for the total cost for 7 pounds of the mixture The cost per pound is $9.80/7= $1.40 EXAMPLE: How many liters of a solution that is 10% alcohol by volume must be added to 2 liters of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume? The balancing method: Make the weaker and stronger (or cheaper and more expensive, etc.) substances balance. That is, (percent difference between the weaker solution and the desired solution) x (amount of weaker solution) +(percent difference between the stronger solution and the desired solution) x (amount of stronger solution). Make n the amount, in liters, of the weaker solution n(15-10) = 2(50-15) 5n=2(35) n= 70/5=14 So 14 liters of the 10% solution must be added to the original, stronger solution
How to work with SIMILAR TRIANGLES
In similar triangles, corresponding angels are equal and corresponding sides are proportional. If a GRE question tells you that triangles are similar, use the properties of similar triangles to find the length of a side or the measure of an angle.
How to work with ISOSCELES TRIANGLES
Isosceles triangles have at least two equal sides and two equal angles. If a GRE question tells you that a triangle is isosceles, you can bet that you'll need to use that information to find the length of a side or measure of an angle.
How to deal with STANDARD DEVIATION
Like the terms mean, mode, median, and range, standard deviation is a term used to describe sets of numbers. Standard deviation is a measure of how spread out a set of numbers is (how much the numbers deviate from the mean). The greater the spread, the higher the standard deviation. You'll rarely have to calculate standard deviation on the GRE. Here's how standard deviation is calculated: Find the average of the set Find the differences between the mean and each value in the set Square each of the differences Find the average of the squared difference Take the positive square root of the average In addition to the occasional question that asks you to calculate standard deviation, you may also be asked to compute standard deviation between sets of data or otherwise demonstrate that you understand what standard deviation mean. You can often handle these questions using estimation.
How to SIMPLIFY A RADICAL
Look for factors of the number under the radical sign that are perfect squares; then find the square root of those perfect squares. Keep simplifying until the term with the square root sign is as simplified as possible, that is, when there are no other perfect square factors inside the *square root*. Write the perfect squares as separate factors and "unsquare" them.
How to find the dimensions or area of an INSCRIBED or CIRCUMSCRIBED FIGURE
Look for the connection. Is the diameter the same as a side or a diagonal? EXAMPLE: If the area of the square is 36, what is the circumference of the circle? To get the circumference, you need the diameter or radius. The circle's diameter is also the square's diagonal. The diagonal of the square is 6radical2. This is because the diagonal of the square transforms it into two separate 45-45-90 triangles. So, the diameter of the circle is 6radical2. Circumference= pie(diameter)= 6(pie)radical2
How to TRANSLATE ENGLISH INTO ALGEBRA
Look for the key words and systematically turn phrases into algebraic expressions and sentences into equations. Key words: ADDITION- sum, plus, and, added to, more than, increased by, combined with, exceeds, total, greater than SUBTRACTION- difference between, minus, subtracted from, decreased by, diminished by, less than, reduced by MULTIPLICATION- of, product, times, multiplied by, twice, double, triply DIVISION- quotient, divided by, per, out of, half, ratio of _ to _ EQUALS- equals, is, was, will be, the result is, adds up to, costs, is the same as
How to solve certain QUADRATIC EQUATIONS
Manipulate the equation (if necessary) so that it is equal to 0, factor the left side (reverse FOIL by finding two numbers whose product is the constant whose sum is the coefficient of the term without the exponent), and break the quadratic into two simple expressions. Then find the value(s) for the variable that make either expression = 0.
How to determine a COMBINED RATIO
Multiply one or both ratios by whatever you need in order to get the terms they have in common to match EXAMPLE: The ratio of a to b is 7:3. The ratio of b to c is 2:5. What is the ratio of a to c? Multiply each member of a:b by 2 and multiply each member of b:c by 3, and you get a:b = 14:6 and b:c = 6:15. Now that the values of b match, you can write a:b:c = 14:6:15 and then say a:c = 14:15
How to handle a value with an EXPONENT RAISED TO AN EXPONENT
Multiply the exponents
How to solve an OVERLAPPING SETS problem involving EITHER/ OR CATEGORIES
Other GRE word problems involve groups with distinct "either/or" categories. The key to solving this type of problem is to organize the information into a grid.
How to solve a REMAINDERS problem
Pick a number that fits the given conditions and see what happens EXAMPLE: When n is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7? Find a number that leaves a remainder of 5 when divided by 7. You can find such a number by taking any multiple of 7 and adding 5 to it. A good choice would be 12. If n=12, then 2n=24, which when divided by 7 leaves a remainder of 3.
How to find the MEDIAN
Put the numbers in numerical order and take the middle number In a set with an even number of numbers, take the average of the two in the middle
How to deal with TABLES, GRAPHS, AND CHARTS
Read the question and all labels carefully. Ignore extraneous information and zero in on what the question asks for. Take advantage of the spread in the answer choices by approximating the answer whenever possible and choosing the answer choice closest to your approximation
How to use a ratio to determine an ACTUAL NUMBER
Set up a proportion using the given ratio EXAMPLE: The ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there? 3/4 = 135/g 3xg= 4x135 3g= 540 g=180
How to solve an OVERLAPPING SETS problem involving BOTH/ NEITHER
Some GRE word problems involve two groups with overlapping members and possibly elements that belong to neither group. It's easy to identify this type of question because the words both and/ or neither appear in the equation. These problems are quite workable if you memorize the following formula: GROUP 1 + GROUP 2 + NEITHER - BOTH = TOTAL EXAMPLE: Of the 120 students at a certain language school, 65 are studying French, 51 are studying Spanish, and 53 are studying neither language. How many are studying both French and Spanish? 65 + 51 + 53 - both = 120 169 - both = 120 both + 49
How to spot SPECIAL RIGHT TRIANGLES
Special right triangles are seen on the GRE with frequency. Recognizing them can streamline your problem solving. 3;4;5 5;12;13 These numbers represent the ratio of the side lengths of these triangles. In a 30-60-90 triangle, the side lengths are multiples of 1, the square root of 3, and 2, respectively. in a 45-45-90 triangle, the side lengths are multiples of 1, 1, and the square root of 2, respectively.
How to determine COMBINED PERCENT INCREASE/ DECREASE when no original value is specified
Start with 100 as a starting value. EXAMPLE: A price rises by 10% one year and by 20% the next. What's the combined percent increase? Say the original price is $100. Year one: 100 + (10% of 100) = 100 + 10 = 110 Year two: 110 + (20% of 110) = 110 + 22 = 132 From 100 to 132 is a 32% increase
How to find the sum of all the ANGLES of a POLYGON and one angle measure of a REGULAR POLYGON
Sum of the interior angles in a polygon with n sides: (n-2)x180 The term regular means all angles in the polygon are of equal measure. Degree measure of one angle in a regular polygon with n sides: (n-2)x180/n
How to find the SUM OF CONSECUTIVE NUMBERS
Sum= (average) x (number of terms)
How to use the AVERAGE to find the SUM
Sum= (average) x (number of terms)
How to solve PROBABILITY problems where probabilities must be multiplied
Suppose that a random process if performed. Then there is a set of possible outcomes that can occur. An event is a set of possible outcomes. We are concerned with the probability of events. When all the outcomes are all equally likely, the basic probability formula is this: Probability= number of desired outcome/ number of total possible outcomes Many more difficult probability questions involve finding the probability that several events occur. Let's consider first the case of the probability that two events occur. Call these two events A and B. The probability that both events occur is the probability that event A occurs multiplied by the probability that event B occurs given that event A occurred. The probability that B occurs given that A occurs is called the conditional probability that B occurs given that A occurs. Except when events A and B do not depend on one another, the probability that B occurs given that A occurs is not the same as the probability that B occurs. The probability that three events, A, B, and C occur is the probability that A occurs multiplied by the conditional probability that B occurs given that A occurred multiplied by the conditional probability that C occurs given that both A and B have occurred. This can be generalized to any number of events.
How to find the MODE
Take the number that appears most often If there is a tie for most often, then there's more than one mode
How to find the RANGE
Take the positive difference between the greatest and least values
How to handle ABSOLUTE VALUES
The absolute value of a number n, is denoted by |n|, is defined as n if n is greater or equal to 0, and -n if n is less than 0. The absolute value of a number is the distance from zero to the number on the number line. The absolute value of a number or expression is always positive.
How to find the AVERAGE of CONSECUTIVE NUMBERS
The average of evenly spaced numbers is simply the average of the smallest number and the largest number. The average of all the integers from 13 to 77, for example, is the same as the average of 13 and 77.
How to handle CUBE ROOTS
The cube root of x is just the number that, when used as a factor 3 times, gives you x. Both positive and negative numbers have one and only one cube root, denoted by the symbol 3 square root, and the cube root of a number is always the same sign as the number itself.
How to solve a SEQUENCE problem
The notation used in sequence problems scares many test takers, but these problems aren't as bad as they look. In a sequence problem, the nth term in the sequence is generated by performing an operation, which will be defined for you, on either n or on the previous term in the sequence. The term itself is expressed as An. For instance, if you are referring to the fourth term in a sequence, it is called a, in sequence notation. Familiarize yourself with sequence notation and you should have no problem. EXAMPLE: what is the positive difference between the fifth and fourth terms in the sequence 0, 4, 18... whose nth term is n^2(n-1)? Use the definition given to come up with the values for your terms: a5= 5^2(5-1) = 25(4) = 100 a4= 4^2(4-1)= 16(3)= 48 So the positive difference between the fifth and fourth terms is 100-48= 52
How to fine a COMMON MULTIPLE of 2 numbers
The product of 2 numbers is the easiest common multiple to find, but it is not always the LCM. Example: What is the least common multiple of 28 and 42? Set up: 28= 2x2x7 42= 2x3x7 The LCM can be found by finding the prime factorization of each number, then seeing the greatest number of times each factor is used. Multiply each prime factor the greater number of times it appears. In 28, 2 is used twice. In 42, 2 is used once. In 28, 7 is used once. In 42, 7 is used once, and 3 is used once. So you multiply each factor the greatest number of times it appears in a prime factorization: LCM= 2x2x3x7= 84
How to find the x- and y-INTERCEPTS of a line
The x-intercepts of a line is the value of x where the line crosses the x-axis. In other words, it's the value of x when y=0. Likewise, the y-intercept is the value of y where the line crosses the y-axis. The y-intercept is also the value of b when the equation is in the form y=mx+b.
How to find the LENGTH of an ARC
Thin of an arc as a fraction of the circle's circumference. Use the measure of an interior angle of a circle, which has 360 degrees around the central point, to determine the length of an arc Length of arc= n/360 x 2(pie)r
How to find the ORIGINAL WHOLE before percent increase/ decrease
Think of a 15% increase over x as 1.15x and set up an equation. EXAMPLE: After decreasing by 5%, the population is now 57,000. What was the original population? .95 x original population = 57,000 divide both sides by .95 original population= 60,000
How to find the AREA of a SECTOR
Think of a sector as a fraction of the circle's area. Again, set up the interior angle measure as a fraction of 360, which is the degree measure of a circle around the central point. Area of sector=n/360 x (pie)r^2
3. How to add, subtract, multiply, and divide DECIMALS
To add or subtract, align the decimal points and then add or subtract normally. Place the decimal point in the answer directly below the existing decimal point. To multiply with decimals, multiply the digits normally and count off decimal places (equal to the total number of places in the factors) from the right. To divide by a decimal, move the decimal point in the divisor to the right to form a whole number; move the decimal point in the dividend the same number of place. Divide as though there were no decimals, then place the decimal point in the quotient.
How to find an angle formed by a TRAVERSAL across PARALLEL LINES
When a traversal crosses parallel lines, all the acute angles formed are equal, and all the obtuse angles formed are equal. Any acute angle plus any obtuse angle equals 180 degrees.
How to solve MULTIPLE EQUATIONS
When you see two equations with two variables on the GRE, they're probably easy to combine in such a way that you get something closer to what you're looking for. EXAMPLE: If 5x-9=-9 and 3y-4x=6, what is the value of x+y? The question doesn't ask for x and y separately, so don't solve for them separately if you don't have to. Look what happens if you just rearrange a little and add the equations: 5x-2=-9 -4x+3y=6 = x+y = -3
How to solve a SIMPLE INTEREST problem
With simple interest, the interest is computed on the principal only and is given by interest= principle x rt In this formula, r is defined as the interest rate per payment period and t is defined as the number of payment periods. EXAMPLE: is $12,000 is invested at 6 percent simple annual interest, how much interest is earned after 9 months? Since the interest rate is annual and we are calculating how much interest accrues after 9 months, we will express the payment period as 9/12 12,000 x 6,000 x 9/12= 540
How to ADD, SUBTRACT, MULTIPLY, and DIVIDE ROOTS
You can add/ subtract roots only when the parts inside the *square root* are identical. To multiply/ divide roots, deal with what's inside the *square root* and outside the *square root* separately.
How to count the NUMBER OF POSSIBILITIES
You can use multiplication to find the number of possibilities when items can be arranged in various ways. EXAMPLE: How many three-digit numbers can be formed with the digits 1, 3, and 5 each used only once? Look at each digit individually. The first digit (or, the hundreds digit) has three possibly numbers to plug in: 1, 3, or 5. The second digit (or the tens digit) has 2 possible numbers, since one has already been plugged in. The last digit (or the ones digit) has only one remaining possible number. Multiply the possibilities together: 3 x 1 x 1= 6
How to work with FACTORIALS
You may see a problem involving factorial notation, which is indicated by the ! symbol. If n is an integer greater than 1, then n factorial, denoted by n!, is defined as the product of all the integers from 1 to n. For example: 2!= 2x1 = 2 4!= 4x3x2x1 = 24 By definition, 0! =1 Also note: 6! = 6 x 5! + 6 x 5 x 4!, etc. Most GRE factorial problems test your ability to factor and/ or cancel EXAMPLE: 8!/ (6!x2!) = 8x7x6/6!x2x1= 28
How to solve a FUNCTION problem
You may see function notation on the GRE. An algebraic expression of only one variable may be defined as a function, usually symbolized by f or g, of that variable. EXAMPLE: What is the minimum value of x in the function f(x)=x^2=1? In the function f(x)= x^2-1, if x is 1, then f(1)=1^2-1=0. In other words, by inputing 1 into the function, the output f(x)=0. Every number inputted has one and only one output (although the reverse is not necessarily true). You're asked to find the minimum value, so how would you minimize the expression f(x)=x^2-1? Since ^2 cannot be negative, in this case f(x) is minimized by making x=0: f(0)=0^2-1=-1, so the minimum value of the function is -1.
How to SIMPLIFY BINOMIALS
a binomial is a sum or difference of two terms. To simplify two binomials that are multiplied together, use the FOIL method. Multiple the First terms, then the Outer terms, followed by the Inner terms and the Last terms. Lastly, combine like terms.
