GRE Math Reference Cheat Sheet

Math Reference Cheat Sheet

The math on the GRE covers a lot of ground— from number properties and arithmetic to basic algebra and symbol problems to geometry and statistics. Don’ t let yourself be intimidated.

We’ ve highlighted the 100 most important concepts that you need to know and divided them into three levels. The GRE Quantitative sections test your understanding of a relatively limited number of mathematical concepts, all of which you will be able to master.

Level 1 consists of foundation math topics. Though these topics may seem basic, review this list so that you are aware that these skills may play a part in the questions you will answer on the GRE. Look over the Level 1 list to make sure you’ re comfortable with the basics.

Level 2 is where most people start their review of math. Level 2 skills and formulas come into play quite frequently on the GRE. If the skills needed to handle Level 1 or 2 topics are keeping you from feeling up to the tasks expected on the GRE Quantitative section.

Level 3 represents the most challenging math concepts you’ ll find on the GRE. Don’ t spend a lot of time on Level 3 if you still have gaps in Level 2, but once you’ ve mastered Level 2, tackling Level 3 can put you over the top.

LEVEL 1

1. How to add, subtract, multiply, and divide WHOLE NUMBERS

You can check addition with subtraction.

You can check multiplication with division.

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 the denominator.

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 existing decimal points.

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 places. Divide as though there were no decimals, then place the decimal point in the quotient.

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.

5. 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 addends have different signs, subtract the absolute values; the sum has the sign of the greater absolute value.

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.

6. 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.

To plot the point − 2.5 on the number line, start at 0, go left to − 2.5, halfway between − 2 and − 3.

7. 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.

8. How to SOLVE a simple LINEAR EQUATION

Use algebra to isolate the variable. Do the same steps to both sides of the equation.

9. How to add and subtract LINE SEGMENTS

10. How to find the THIRD ANGLE of a TRIANGLE, given the other two angles

Use the fact that the sum of the measures of the interior angles of a triangle always sum to 180° .

LEVEL 2

11. How to use PEMDAS

When you’ re given a complex arithmetic expression, it’ s important to know the order of operations. Just remember PEMDAS (as in “ Please excuse my dear Aunt Sally” ). What PEMDAS means is this: Clean up Parentheses first (nested sets of parentheses are worked from the innermost set to the outermost set.); then deal with Exponents (or Radicals); then do the Multiplication and Division together, going from left to right; and finally do the Addition and Subtraction together, again going from left to right.

Example:

Begin with the parentheses:

Then do the exponent:

Now do multiplication and division from left to right:

Finally, do addition and subtraction from left to right:

12. How to use the PERCENT FORMULA

Identify the part, the percent, and the whole.

Find the part.

Example:

What is 12 percent of 25?

Setup:

Find the percent.

Example:

45 is what percent of 9?

Setup:

Find the whole.

Example:

15 is

percent of what number?

Setup:

13. How to use the PERCENT INCREASE/DECREASE FORMULAS

Identify the original whole and the amount of increase/decrease.

Example:

The price goes up from $80 to $100. What is the percent increase?

Setup:

14. How to predict whether a sum, difference, or product will be ODD or EVEN

Don’ t bother memorizing the rules. Just take simple numbers like 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 mn odd or even?

Setup:

Say m = 2 and n = 3.

2 × 3 is even, so mn is even.

15. 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 two 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 two digits are a multiple of 4

16. How to find a COMMON FACTOR of two numbers

Break both numbers down to their prime factors to see which they have in common. Then multiply the shared prime factors to find all common factors.

Example:

What factors greater than 1 do 135 and 225 have in common?

Setup:

First find the prime factors of 135 and 225; 135 = 3 × 3 × 3 × 5, and 225 = 3 × 3 × 5 × 5. The numbers share 3 × 3 × 5 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: 3 × 3 = 9, 3 × 5 = 15, and 3 × 3 × 5 = 45. To summarize, the common factors of 135 and 225 are 3, 5, 9, 15, and 45.

17. How to find a COMMON MULTIPLE of two numbers

The product of two numbers is the easiest common multiple to find, but it is not always the least common multiple.

Example:

What is the least common multiple of 28 and 42?

Setup:

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 greatest 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:

18. How to find the AVERAGE or ARITHMETIC MEAN

Example:

What is the average of 3, 4, and 8?

Setup:

19. How to use the AVERAGE to find the SUM

Example:

17.5 is the average (arithmetic mean) of 24 numbers.

What is the sum of the 24 numbers?

Setup:

20. 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:

21. 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?

Setup:

22. How to find the SUM OF CONSECUTIVE NUMBERS

Example:

What is the sum of the integers from 10 through 50, inclusive?

Setup:

Average 5 (10 + 50) ÷ 2 = 30

Number of terms = 50 − 10 + 1 = 41

Sum = 30 × 41 = 1,230

23. How to find the MEDIAN

Put the numbers in numerical order and take the middle number.

Example:

What is the median of 88, 86, 57, 94, and 73?

Setup:

First, put the numbers in numerical order, then take the middle number:

The median is 86.

In a set with an even number of numbers, take the average of the two in the middle.

Example:

What is the median of 88, 86, 57, 73, 94, and 100?

Setup:

First, put the numbers in numerical order.

Because 86 and 88 are the two numbers in the middle:

24. How to find the MODE

Take the number that appears most often. For example, if your test scores were 88, 57, 68, 85, 98, 93, 93, 84, and 81, the mode of the scores would be 93 because it appears more often than any other score. (If there’ s a tie for most often, then there’ s more than one mode. If there is only one of each number in a set, there is no mode.)

25. How to find the RANGE

Take the positive difference between the greatest and least values. Using the example under “ How to find the MODE” above, if your test scores were 88, 57, 68, 85, 98, 93, 93, 84, and 81, the range of the scores would be 41, the greatest value minus the least value (98 − 57 = 41).

26. How to use actual numbers to determine a RATIO

To find a ratio, put the number associated with of on the top and the word associated with to on the bottom.

The ratio of 20 oranges to 12 apples is

, or

Ratios should always be reduced to lowest terms.

27. 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?

Setup:

28. How to use actual numbers to determine a RATE

Identify the quantities and the units to be compared. Keep the units straight.

Example:

Anders typed 9,450 words in

hours. What was his rate in words per minute?

Setup:

First convert

hours to 210 minutes. Then set up the rate with words on top and minutes on bottom (because “ per” means “ divided by” ):

29. 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.

30. 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?

Setup:

Look at each digit individually. The first digit (or, the hundreds digit) has three possible numbers to plug in: 1, 3, or 5. The second digit (or, the tens digit) has two 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 × 2 × 1 = 6.

31. How to calculate a simple PROBABILITY

Example:

What is the probability of throwing a 5 on a fair six-sided die?

Setup:

There is one favorable outcome— throwing a 5. There are 6 possible outcomes— one for each side of the die.

32. 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 defined by the problem. Everything you need to know is in the question stem. Just follow the instructions.

33. 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. Multiply the First terms, then the Outer terms, followed by the Inner terms and the Last terms. Lastly, combine like terms.

Example:

34. 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 equations.

35. 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.

36. 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 reverse the inequality sign if you multiply or divide by a negative quantity.

Example:

Rewrite 7 − 3x > 2 in its simplest form.

Setup:

First, subtract 7 from both sides:

So − 3x > 25.

Now divide both sides by − 3, and remember to reverse the inequality sign:

37. How to handle ABSOLUTE VALUES

The absolute value of a number n, denoted by |n|, is defined as n if n≥ 0 and − n if n < 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.

If |x| = 3, then x could be 3 or − 3.

Example:

If |x− 3| < 2, what is the range of possible values for x?

Setup:

Represent the possible range for x− 3 on a number line.

38. How to TRANSLATE ENGLISH INTO ALGEBRA

Look for the key words and systematically turn phrases into algebraic expressions and sentences into equations.

Here’ s a table of key words that you may have to translate into mathematical terms:

Operation	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, triple, half
Division	quotient, divided by, per, out of, ratio of ___ to ___
Equals	equals, is, was, will be, the result is, adds up to, costs, is the same as

39. How to find an ANGLE formed by INTERSECTING LINES

Vertical angles are equal. Angles along a line add up to 180° .

40. How to find an angle formed by a TRANSVERSAL across PARALLEL LINES

All the acute angles are equal. All the obtuse angles are equal. An acute plus an obtuse equals 180° .

Example:

ℓ₁ is parallel to ℓ₂

41. How to find the AREA of a TRIANGLE

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 the base.

Example:

Setup:

42. How to work with ISOSCELES TRIANGLES

Isosceles triangles have 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 a measure of an angle.

43. How to work with EQUILATERAL TRIANGLES

Equilateral triangles have three equal sides and three 60° 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 a measure of an angle.

44. How to work with SIMILAR TRIANGLES

In similar triangles, corresponding angles 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.

45. How to find the HYPOTENUSE or a LEG of a RIGHT TRIANGLE

For all right triangles, Pythagorean theorem: a² + b² = c², here a and b are the legs and c is the hypotenuse.

46. How to spot SPECIAL RIGHT TRIANGLES

These numbers (3, 4, 5 and 5, 12, 13) represent the ratio of the side lengths of these triangles.

Likewise, the numbers

and

represent the ratios of the side lengths of these special triangles.

47. How to find the PERIMETER of a RECTANGLE

Example:

Setup:

48. How to find the AREA of a RECTANGLE

Example:

Setup:

49. How to find the AREA of a SQUARE

Example:

Setup:

50. How to find the AREA of a PARALLELOGRAM

Example:

Setup:

51. 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:

Example:

Setup:

Area of rectangle = 8 × 5 = 40

Area of triangle

Area of trapezoid = 40 + 10 = 50

or Area of trapezoid

52. How to find the CIRCUMFERENCE of a CIRCLE

Example:

Setup:

53. How to find the AREA of a CIRCLE

Example:

Setup:

54. How to find the DISTANCE BETWEEN POINTS on the coordinate plane

If two points have the same x coordinates 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 distance is always positive.

Example:

What is the distance from (2, 3) to (− 7, 3)?

Setup:

The y’ s are the same, so just subtract the x’ s: 2 − (− 7) = 9.

If the points have different x coordinates and different y coordinates, make a right triangle and use the Pythagorean theorem or apply the special right triangle attributes if applicable.

Example:

What is the distance from (2, 3) to (− 1, − 1)?

Setup:

55. How to find the SLOPE of a LINE

Example:

What is the slope of the line that contains the points (1, 2) and (4, − 5)?

Setup:

LEVEL 3

56. How to determine COMBINED PERCENT INCREASE/DECREASE

Start with 100 as a starting value.

Example:

A price rises by 10 percent one year and by 20 percent the next. What’ s the combined percent increase?

Setup:

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— that’ s a 32 percent increase.

57. How to find the ORIGINAL WHOLE before percent increase/decrease

Think of a 15 percent increase over x as 1.15x and set up an equation.

Example:

After decreasing by 5 percent, the population is now 57,000. What was the original population?

Setup:

0.95 × (Original population) = 57,000

Divide both sides by 0.95.

Original population = 57,000 ÷ 0.95 = 60,000

58. How to solve a SIMPLE INTEREST problem

With simple interest, the interest is computed on the principal only and is given by:

Example:

If $12,000 is invested at 6 percent simple annual interest, how much interest is earned after 9 months?

Setup:

59. How to solve a COMPOUND INTEREST problem

If interest is compounded, the interest is computed on the principal as well as on any interest earned. To compute compound interest:

where C = the number of times compounded annually

Example:

If $10,000 is invested at 8 percent annual interest, compounded semiannually, what is the balance after 1 year?

Setup:

Final balance

60. 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?

Setup:

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.

61. How to solve a DIGITS problem

Use a little logic— and some trial and error.

Example:

If A, B, C, and D represent distinct digits in the addition problem below, what is the value of D?

Setup:

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 be 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 give you the same value for D, which is 2:

62. 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?

Setup:

HINT: Don’ t just average the averages.

63. 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 four tests is 80. If he scores 100 on the fifth test, what’ s his new average?

Setup:

Find the original sum from the original average:

Add the fifth score to make the new sum:

Find the new average from the new sum:

64. How to use the ORIGINAL AVERAGE and NEW AVERAGE to figure out WHAT WAS ADDED OR DELETED

Use the sums.

Example:

The average of five numbers is 2. After one number is deleted, the new average is − 3. What number was deleted?

Setup:

Find the original sum from the original average:

Find the new sum from the new average:

The difference between the original sum and the new sum is the answer.

65. How to find an AVERAGE RATE

Convert to totals.

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?

Setup:

To find an average speed, you also convert to totals.

Example:

Rosa drove 120 miles one way at an average speed of 40 miles per hour and returned by the same 120-mile route at an average speed of 60 miles per hour. What was Rosa’ s average speed for the entire 240-mile round trip?

Setup:

To drive 120 miles at 40 mph takes 3 hours. To return at 60 mph takes 2 hours. The total time, then, is 5 hours.

66. 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 asked to compute the rate at which they work together (or vice 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 individually. In other words:

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:

If 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?

Setup:

Joe takes 4 hours, so Pete takes 8 hours; thus:

So it would take them

hours, or 2 hours 40 minutes, to paint the room together.

67. How to determine a COMBINED RATIO

Multiply one or both ratios by whatever you need to 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?

Setup:

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 bs match, you can write a:b:c = 14:6:15 and then say a:c = 14:15.

68. How to solve a 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 a desired mixture. There are two approaches to such problems— the straightforward setup and the balancing method.

Example:

If 5 pounds of raisins that cost $1 per pound are mixed with 2 pounds of almonds that cost $2.40 per pound, what is the cost per pound of the resulting mixture?

Setup:

The straightforward setup:

($1)(5) + ($2.40)(2) = $9.80 = total cost for 7 pounds of the mixture

The cost per pound is

Example:

How many liters of a solution that is 10 percent alcohol by volume must be added to 2 liters of a solution that is 50 percent alcohol by volume to create a solution that is 15 percent alcohol by volume?

Setup:

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) × (amount of weaker solution) = (percent difference between the stronger solution and the desired solution) × (amount of stronger solution). Make n the amount, in liters, of the weaker solution.

So 14 liters of the 10 percent solution must be added to the original, stronger solution.

69. How to solve a GROUP 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 neitherappear in the question. These problems are quite workable if you just memorize the following formula:

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?

Setup:

70. How to solve a GROUP problem involving EITHER/OR CATEGORIES

Other GRE word problems involve groups with distinct “ either/or” categories (male/female, blue-collar/white-collar, etc.). The key to solving this type of problem is to organize the information in a grid.

Example:

At a certain professional conference with 130 attendees, 94 of the attendees are doctors, and the rest are dentists. If 48 of the attendees are women and

of the dentists in attendance are women, how many of the attendees are male doctors?

Setup:

To complete the grid, use the information in the problem, making each row and column add up to the corresponding total:

	Doctors	Dentists	Total
Male	55	27	82
Female		9	48
Total	94	36	130

After you’ ve filled in the information from the question, use simple arithmetic to fill in the remaining boxes until you get the number you are looking for— in this case, that 55 of the attendees are male doctors.

71. How to work with FACTORIALS

You may see a problem involving factorial notation. 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. In other words:

By definition, 0! = 1.

Also note: 6! = 6 × 5! = 6 × 5 × 4!, etc. Most GRE factorial problems test your ability to factor and/or cancel.

Example:

72. How to solve a PERMUTATION problem

Factorials are useful for solving questions about permutations (i.e., the number of ways to arrange 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 so on— in other words: 7 × 6 × 5 × 4 × 3 × 2 × 1, or 7!.

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:

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?

Setup:

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 total of 5 × 4 × 3 = 60 possible outcomes for gold, silver, and bronze medal winners. Or, using the formula:

73. 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 numbers of combinations, and a different formula is called for:

Example:

How many different ways are there to choose 3 delegates from 8 possible candidates?

Setup:

So there are 56 different possible combinations.

74. How to solve PROBABILITY problems where probabilities must be multiplied

Suppose that a random process is 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:

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 Boccurs.

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 Coccurs given that both A and B have occurred.

This can be generalized to any number of events, where the number of events is an integer greater than 3.

Example:

If 2 students are chosen at random to run an errand from a class with 5 girls and 5 boys, what is the probability that both students chosen will be girls?

Setup:

The probability that the first student chosen will be a girl is

, and since there would be 4 girls and 5 boys left out of 9 students, the probability that the second student chosen will be a girl (given that the first student chosen is a girl) is

. Thus the probability that both students chosen will be girls is

. There was conditional probability here because the probability of choosing the second girl was affected by another girl being chosen first.

Now let’ s consider another example where a random process is repeated.

Example:

If a fair coin is tossed 4 times, what’ s the probability that at least 3 of the 4 tosses will be heads?

Setup:

There are 2 possible outcomes for each toss, so after 4 tosses, there are 2 × 2 × 2 × 2 = 16 possible outcomes.

We can list the different possible sequences where at least 3 of the 4 tosses are heads. These sequences are

Thus, the probability that at least 3 of the 4 tosses will come up heads is:

We could have also solved this question using the combinations formula. The probability of a head is

, and the probability of a tail is

. The probability of any particular sequence of heads and tails resulting from 4 tosses is

, which is

Suppose that the result each of the four tosses is recorded in each of the four spaces.

Thus, we would record an H for head or a T for tails in each of the 4 spaces.

The number of ways of having exactly 3 heads among the 4 tosses is the number of ways of choosing 3 of the 4 spaces above to record an H for heads.

The number of ways of choosing 3 of the 4 spaces is

The number of ways of having exactly 4 heads among the 4 tosses is 1.

If we use the combinations formula, using the definition that 0! = 1, then

Thus, ₄C₃ = 4 and ₄C₄ = 1. So the number of different sequences containing at least 3 heads is 4 + 1 = 5.

The probability of having at least 3 heads is

75. How to deal with STANDARD DEVIATION

Like 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 the standard deviation on Test Day (although this skill may be necessary for some high-difficulty questions). Here’ s how standard deviation is calculated:

Find the average (arithmetic mean) 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 differences.
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 compare standard deviations between sets of data or otherwise demonstrate that you understand what standard deviation means. You can often handle these questions using estimation.

Example:

High temperatures, in degrees Fahrenheit, in two cities over five days:

September	1	2	3	4	5
City A	54	61	70	49	56
City B	62	56	60	67	65

For the five day period listed, which city had the greater standard deviation in high temperatures?

Setup:

Even without trying to calculate them out, one can see that City A has the greater spread in temperatures and, therefore, the greater standard deviation in high temperatures. If you were to go ahead and calculate the standard deviations following the steps described above, you would find that the standard deviation in high temperatures for City

, while the standard deviation for City

76. How to MULTIPLY/DIVIDE VALUES WITH EXPONENTS POWERS

Add/subtract the exponents.

Example:

77. How to handle a value with an EXPONENT RAISED TO AN EXPONENT

Multiply the exponents.

Example:

78. How to handle POWERS with a base of ZERO and POWERS with an EXPONENT of ZERO

Zero raised to any nonzero exponent equals zero.

Example:

Any nonzero number raised to the exponent 0 equals 1.

Example:

The lone exception is 0 raised to the 0 power, which is undefined.

79. How to handle NEGATIVE POWERS

A number raised to the exponent − x is the reciprocal of that number raised to the exponent x.

Example:

80. How to handle FRACTIONAL POWERS

Fractional exponents relate to roots. For instance,

Likewise,

, and so on.

Example:

81. How to handle CUBE ROOTS

The cube root of x is just the number that when used as a factor 3 times (i.e., cubed) gives you x. Both positive and negative numbers have one and only one cube root, denoted by the symbol

, and the cube root of a number is always the same sign as the number itself.

Example:

82. How to ADD, SUBTRACT, MULTIPLY, and DIVIDE ROOTS

You can add/subtract roots only when the parts inside the

are identical.

Example:

To multiply/divide roots, deal with what’ s inside the

and outside the

separately.

Example:

83. 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: there are no other perfect square factors (4, 9, 16, 25, 36… ) inside the

. Write the perfect squares as separate factors and “ unsquare” them.

Example:

84. How to solve certain QUADRATIC EQUATIONS

Manipulate the equation (if necessary) into the “ _____ = 0” form, factor the left side (reverse FOIL by finding two numbers whose product is the constant and 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.

Example:

85. 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− 2y = 29 and 3y− 4x = 6, what is the value of x + y?

Setup:

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:

86. 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. For instance, if you are referring to the fourth term in a sequence, it is called n₄ 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 n²(n− 1)?

Setup:

Use the definition given to come up with the values for your terms:

So the positive difference between the fifth and fourth terms is 100 − 48 = 52.

87. 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) = x² − 1?

Setup:

In the function f (x) = x²− 1, if x is 1, then f (1) = 1² − 1 = 0. In other words, by inputting 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) = x²− 1? Since x² cannot be negative, in this case f (x) is minimized by making x = 0: f(0) = 0² − 1 = − 1, so the minimum value of the function is − 1.

88. 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) = x²− 1, you’ ve already determined 2 points, (1, 0) and (0, − 1). If you were to keep plugging in numbers to determine more points and then 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 (it could be upside down or more narrow or wider than the one shown), 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 jibe with those values until only one answer choice is left.

89. How to handle LINEAR EQUATIONS

You may also encounter linear equations on the GRE. A linear equation is often expressed in the form

For instance, a slope of 3 means that the line rises 3 steps for every 1 step it makes to the right. A line with positive slope slopes up from left to right. A line with negative slope slopes down from left to right. A slope of zero (e.g., y = 5) is a flat (horizontal) line.

Example:

The graph of the linear equation

Note:

The equation could also be written in the form 3x + 4y = 12, but this form does not readily describe the slope and y-intercept of the line.

To get a better handle on an equation written in this form, you can solve for y to write it in its more familiar form. Or, if you’ re asked to choose which equation the line is describing, you can pick obvious points, such as (0, 3) and (4, 0) in this example and use these values to eliminate answer choices until only one answer is left.

90. How to find the x - and y -INTERCEPTS of a line

The x-intercept of a line is the value of x where the line crosses the x-axis. In other words, it’ s the value ofx when y = 0. Likewise, the y-intercept is the value of y where the line crosses the y-axis (i.e., the value ofy when x = 0). The y-intercept is also the value b when the equation is in the form: y = mx + b. For instance, in the line shown in the previous example, the x-intercept is 4 and the y-intercept is 3.

91. How to find the MAXIMUM and MINIMUM lengths for a SIDE of a TRIANGLE

If you know n 5 the lengths of two sides of a triangle, you know that the third side is somewhere between the positive difference and the sum.

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?

Setup:

The third side is greater than the difference (7 − 3 = 4) and less than the sum (7 + 3 = 10).

92. How to find one angle or the sum of all the ANGLES of a REGULAR POLYGON

The term “ regular” means all angles in the polygon are of equal measure.

Sum of the interior angles in a polygon with n sides =

Degree measure of one angle in a regular polygon with n sides =

Example:

What is the measure of one angle of a regular pentagon?

Setup:

Since a pentagon is a five-sided figure, plug n = 5 into the formula:

Degree measure of one angle

93. How to find the LENGTH of an ARC

Think 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.

94. 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.