GRE Most Important Math Concepts

0! = what?

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Example of probability problem where probabilities must be multiplied: If a fair coin is tossed 4 times, what's the probability that at least 3 of the 4 tosses will be heads? 2 ways to solve this problem.

1. List the different possible sequences where at least 3 of the 4 tosses are heads then do # desired outcomes/# total possible outcomes. 2. Use combinations formula. Probability of a head is 1/2 and probability of a tail is 1/2. Probability of any particular sequence of heads and tails resulting from 4 tosses is 1/2x1/2x1/2x1/2 = 1/16. Supposed that the result of each of the 4 tosses is recorded in 4 spaces: _____ _____ _____ _____ 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: 4C3 = 4!/[3!(4-3)!] = 4!/[3!(1!)] = (4x3x2x1)/(3x2x1x1) = 4. 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: 4C4 = 4!/[4!(4-4)!] = 4!/[4!(0)!] = (4x3x2x1)/(4x3x2x1x1) = 1 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 5/16.

2 kinds of special right triangles by side. What do these numbers refer to? 2 kinds of special triangles by angle. What are the side lengths for these triangles respectively?

3:4:5; 5:12:13; ratio of side lengths of these triangles; 30-60-90; 45-45-90; 1, sqrt(3), 2; 1, 1, sqrt(2)

Solving multiple equations example: If 5x - 2y = -9 and 3y - 4x = 6, what is the value of x + y?

5x - 2y = -9 +[-4x + 3y = 6] = x + y = -3

How to find area of a trapezoid

A trapezoid is a quadrilateral having only two parallel sides. You can either divide the trapezoid into triangles and a rectangle or use the formula: Area = average of parallel sides x height

Formula to find area of a circle

Area = (pi)r^2

Formula for area of a triangle

Area = 1/2(base)(height)

Formula to find area of a parallelogram

Area = average of parallel sides x height

Formula to find an average rate. Example: If the first 500 pages have an average of 150 words per page and the remaining 100 pages have an average of 150 words per page, what is the average number of words per page for the entire 600 pages?

Average A per B = Total A/Total B Total pages = 500+100 = 600 Total words = (500x150) + (100x450) = 12,000 Average words per page = 120,000/600 = 200

How to solve probability problems where probabilities must be multiplied

Consider the case of the probability that several events 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 don't depend on each other, 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.

How to work with similar triangles

Corresponding angles are equal and corresponding sides are proportional. If GRE question tells you that triangles are similar, use properties of similar triangles to find the length of a side or the measure of an angle.

How to handle fractional powers

Fractional exponents relate to roots. For instance, x^(1/2) is sqrt(x) and x^(2/3) is third root(x^2)

How to solve a compound interest problem. Example: If $10,000 is invested at 8 percent annual interest, compounded semiannually, what is the balance after 1 year?

If interest is compounded, the interest is computed on the principal as well as on any interest earned. Formula: A = P(1+r/n)^(nt) Final balance = 10,000 x (1 + 0.08/2)^(2x1) = 10,000 x 1.04^2 = $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 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 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. Aka, 1/r+1/s=1/t, where r and s are individual rates and t is the rate together. All variables stand for units of time and must all refer to amount of time it takes to do the same task.

How to solve simple interest problem. Example: If $12,000 is invested at 6 percent simple annual interest, how much interest is earned after 9 months?

Interest computed on principal only and is given by interest = principal x rt, where r is defined as the interest rate per payment period, and t is defined as the number of payment periods. Since interest rate is annual and we are calculating how much interest accrues after 9 months, we will express payment period as 9/12: 12,000 x 0.06 x 9/12 = $540

How to work with isosceles triangles

Isosceles triangles have at least two equal sides and two equal angles. If GRE question says that triangle is isosceles, can bet that you'll need to use that info to find the length of a side or a measure of an angle.

How to simplify a radical. Example: Simplify sqrt(48)

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 sqrt. Write the perfect squares as separate factors and unsquare them. sqrt(48) = sqrt(16) x sqrt(3) = 4sqrt(3)

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 diagonal?

How do you divide decimals?

Move the decimal point in the divisor to the right to form a whole number and 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

How to solve an overlapping sets problem involving either/or categories

Organize the info in a grid then use simple arithmetic to fill in the remaining boxes.

Rule for solving complex arithmetic expressions

PEMDAS

Dilution problem: 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?

Percent difference between weaker solution and desired solution x amount of weaker solution = percent difference between 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.

Example of probability problem where probabilities must be multiplied: 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?

Probability that 1st student chosen will be a girl: 5/10=1/2. 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 is a girl) is 4/9. Thus, the probability that both students chosen will be girls is 1/2 x 4/9 = 2/9.

Formula for the surface area of a rectangular solid

SA = 2(lw) + 2(wh) + 2(lh)

Formula for the surface area of a cylinder

SA = 2(pi)r^2 + 2(pi)rh

How to determine combined percent increase/decrease when no original value is specified. Example: A price rises by 10 percent one year and by 20 percent the next. What's the combined percent increase?

Start with 100 as starting value; Year 1: $100 + (10% of 100) = 100 + 10 = 110. Year 2: 110 + (20% of 110) = 110 + 22 = 132. From 100 to 132 is a 32 percent 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) x 180 Term regular means all angles in polygon are of equal measure. Degree measure of one angle in a regular polygon with n sides: [(n-2) x 180]/n

How to find the original whole before percent increase/decrease. Example: After decreasing by 5 percent, the population is now 57,000. What was the original population?

Think of a 15 percent increase over x as 1.15x and set up an equation, a 5 percent decrease over x as .95x, etc. 0.95 x original population = 57,000 original population = 57,000/0.95 = 60,000

How to solve an inequality

Treat it like an equation - adding, subtract, multiplying, and dividing both sides by the same thing. Remember to reverse inequality sign if you multiply or divide by a negative quantity.

How to count number of possibilities. Example: How many three-digit numbers can be formed with the digits 1, 3, and 5 each used only once?

Use multiplication to find number of possibilities when items can be arranged in various ways; Look at each digit individually. Hundreds digit has 3 possible numbers to plug in: 1, 3, or 5. Tens digit has two possible numbers. Ones digit has only one remaining possible number. Multiply the possibilities together: 3x2x1=6.

How to find angle formed by transversal across parallel lines

When transversal crosses parallel lines, all acute angles formed are equal and all obtuse angles formed are equal. Any acute angle plus any obtuse angle equals 180 degrees.

How to add, subtract, multiply, and divide roots. Examples: sqrt(2) + 3sqrt(2) and sqrt(2) - 3sqrt(2)

You can add/subtract roots only when the parts inside the root are identical. sqrt(2) + 3sqrt(2) = 4sqrt(2) sqrt(2) - 3sqrt(2) = -2sqrt(2)

How to handle negative powers

a number raised to the exponent -x is the reciprocal of that number raised to the exponent x

Example of how to solve a sequence problem: 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)?

a5 = 5^2(5-1) = 100 a4 = 4^2(4-1) = 48 100 - 48 = 52

How to multiply/divide values with exponents

add/subtract the exponents

Formula to find the area of a sector (fraction of the circle's area)

area of sector = n/360 x (pi)r^2, where n is the interior angle of the arc

How do you find the average of consecutive terms?

average of evenly spaced numbers is the average of the smallest number and the largest number

2 formulas for circumference of a circle

circumference = 2(pi)r; circumference = (pi)d

If absolute value of x = 3, what does x equal?

could be 3 or -3

Most GRE factorial problems test your ability to ____ and/or ____

factor; cancel

How to calculate standard deviation (5 steps)

find average of the set; find differences between mean and each value in the set; square each of the differences; find average of squared differences; take positive square root of the average

Formula for how to solve an overlapping sets problem involving both/neither

group 1 + group 2 + neither - both = total

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 is somewhere between the positive difference and the sum of the other two sides

Formula to find the length of an arc

length of an arc = n/360 x 2(pi)r, where n is the interior angle of the arc

When finding the distance between points on a coordinate plane, if the points have different x and coordinates and different y coordinates, what do you do?

make a right triangle and use the Pythagorean theorem or apply the special right triangle attributes if applicable

How do you divide fractions?

multiply by its reciprocal

How do you multiply decimals?

multiply digits normally and count off decimal places (equal to total number of places in the factors) from the right

How do you multiply fractions?

multiply numerators first then denominators and simplify if necessary

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

How to handle a value with an exponent raised to an exponent

multiply the exponents

Combination formula

nCk = n!/[k!(n-k)!]

Permutation formula

nPk = n!/[(n-k)!] n is the number in the larger group k is the number you're arranging

How to use the original average and new average to figure out what was added or deleted (2 formulas). Example: The average of 5 numbers is 2. After one number is deleted, the new average is -3. What number was deleted?

number added = new sum - original sum number deleted = original sum - new sum original sum = 5x2 = 10 new sum = 4x(-3) = -12 number deleted = 10-(-12) = 22

How do you count consecutive numbers?

number of integers from A to B inclusive is B-A+1

What is a permutation problem?

number of ways to arrange elements sequentially

Percent formula

part = percent x whole

formula for percent decrease

percent decrease = amount of decrease/original whole x 100

formula for percent increase

percent increase = amount of increase/original whole x 100

Formula for perimeter of a rectangle

perimeter = 2(length + width)

How to solve a remainders problem

pick a number that fits the given conditions and see what happens

How to answer questions where asks you to choose which equation the parabola is describing

pick out obvious points on the graph, plug these values into the answer choices, and eliminate answer choices that don't work until only one is left

When finding the distance between points on a coordinate plane, the answer is always ____

positive

Formula to calculate a simple probability

probability = number of desired outcomes/number of total possible outcomes

How to write ratio as a fraction. Example: 20 oranges to 12 apples = ____ = ____.

ratio=of/to; 20:12; 20/12

How to use a ratio to determine an actual number. Example: Setup for the ratio of boys to girls is 3 to 4. If there are 135 boys, how many girls are there?

setup a proportion; 3/4 = 135/g

Formula for slope

slope = rise/run

How to find the sum of consecutive numbers, based on average

sum = average x number of terms

How to use average to find sum

sum = average x number of terms

The higher the standard deviation, ____

the greater the spread

How to find the diagonal of a rectangular solid

use Pythagorean theorem twice, unless you spot "special" triangles

How to find angle formed by intersecting lines

vertical angles are equal and angles along a line add up to 180 degrees

Formula for volume of a cylinder

volume = area of the base x height = (pi)r^2h

Formula for the volume of a rectangular solid

volume = length x width x height

Example of weighted average problem: 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 avg. = [(1x30) + (2x24)]/3 = 78/3 = 26

What is a combination problem?

when the order or arrangement of the smaller group that's being drawn from the larger group does not matter and you're looking for the numbers of combinations

What is the linear equation?

y = mx + b, where m is the slope and b is the y-intercept.

The x-intercept of a line is the value of x when ____. The y-intercept of a line is the value of y when ____.

y=0; x=0

How to handle exponents with a base of zero and bases with an exponent zero

zero raised to any nonzero equals zero; any nonzero number raised to the exponent zero equals 1; 0 raised to the 0 power is undefined