Math Review
Area of triangle
(1/2)(Base)(Height)
Sum of interior angles of triangle
180 degrees.
The sum of Angles
Sum of angles around a point: 360 Sum of angles along a straight line: 180 - two angles are supplementary to each other if their measures sum to 180.
Rates
Type of ratio. Relates one kind of quantity to a completely different kind. We use the word "per" meaning "for one" or "for each". So, we express the rates as ratios reduced to a denominator of 1. Ex: Speed or Cost per unit
Obtuse
between 90 and 180 degrees
Remainder
what is "left over" in a division problem. It is always smaller than the number you are dividing by.
Decimal System: Place Value 315.256
3- Hundreds 1- Tens Before decimal point: INTEGER PART 5- Units (ones) . - decimal point 2- Tenths 4- Hundredths After decimal point: DECIMAL PART 6- Thousandths
Numerator
A certain number of those equal parts (of denominator).
Decimal
A fraction written in decimal system format. To convert a fraction to a decimal, divide the numerator by the denominator. Ex: 5/8 is a fraction. To make it a decimal: 5 divided by 8= 0.625
Operation
A function or process performed on one or more numbers. Four basic arithmetic operations: addition, subtraction, multiplication, division.
Variable
A letter or symbol representing an unknown quantity.
Constant
A number not multiplied by any variable.
Cube
A number raised to the 3rd power. Ex: 4^3 = 4x4x4=64 64 is the cube of 4
Integer
A number without fractional or decimal part, includes positive and negative whole numbers and zero. All integers are multiples of 1. -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5........
Term
A numerical constant; also, the product of a numerical constant and one or more variables.
Prime Numbers
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. 2, 3, 5, 7, 11, 13.......
Equilateral triangle
A triangle whose three sides are all equal in length and whose three interior angles each measure 60 degrees
Isosceles triangle
A triangle with two equal sides, which are opposite two equal angles.
Scientific Notation
A way to express very large numbers. 123,000,000,000 would be written in scientific notation as 1.23 x 10^11 0.000000003 would be 3 x 10^-9 (multiplying by by a negative exponent like -9 is equivalent to dividing by positive 9)
Set
A well- defined collection of items, typically numbers, objects, or events. The bracket symbols { } are normally used to define sets of numbers. Ex: {2, 4, 6, 8}
Comparing Decimals
Add zeros after the last digit to the right of the decimal point in each decimal fraction until all the decimals you are comparing have the same digits. What you are actually doing is giving all fractions the same denomincator so you can just compare their numerators. .7, .77, .07, .707, .077 which one is greatest. (Add zeros until you reach thousandths place since .707 and .077 are already there. Which means all denominators will be 1,000. (700/1000)(770/1,000)(70/1000)(707/1,000)(77/1000) So which one is greatest? look at numerator. (770/1000). So decimal .77 is the largest.
Adding and Subtracting Numbers with the Same Sign
Adding: add the number and keep the sign. Subtracting: Change the minus sign to a plus and switch the sign of the second number. Add the number and keep the sign of the larger number.
Adding and Subtracting Numbers with Different Sign
Adding: find the difference and keep the sign of the number whose number part is larger. (-7) + (4) = -3 because 7-4 = 3 and 7> 3 so negative Subtracting: Change the minus sign to a plus and switch the sign of the second number. Add the number and keep the sign of the larger number. To subtract a positive from a negative or from a smaller positive, change the sign of the number that you are subtracting from positive to negative and follow the rules for addition of signed numbers. (-4) - 1 = (-4) + (-1) = -5
The Distributive Law
Allows you to "distribute" a factor over numbers that are added or subtracted. You do this by multiplying that favor by each number in the group. a(b+c)= ab+ac or a + b/c = a/c + b/c *** ***Works with numerator in division but not when the sum or difference is in the denominator. 9/ 4+5 is not equal to 9/4 + 9/5
Monomial
An algebraic expression with only one term. To multiply them, you multiply the coefficients and the variables separately. 2a x 3a = (2x3)(axa)= 6x^2
Algebraic Expression
An expression containing one or more variables, one more more constants, and possibly one or more operation symbols. It does not contain an equal sign. X, 3X+2x are both expressions.
Converting Decimals
Any decimal fraction is equivalent to some common fraction with a power of 10 in the denominator. To convert a decimal to a fraction, determine the place value of the last nonzero digit and set that value as denominator. Then use all digits of the decimal place as the numerator, ignoring the decimal point. Then, reduce. Ex: Convert 0.875 into fraction. Last nonzero digit is 5 which is in thousandths place so the denominator is 1,000. the numerator is 875. (875/1,000). Which is reduced to (7/8). To convert faction to decimal, divide numerator by denominator.
Mixed Number
Consist of an integer and a fraction. To change into fraction, multiply the integer portion by the denominator and add the numerator. this new number is your numerator. the denominator willl not change. 2(3/7)-- 7x(2)x3 =17 so (17/7)
Distinct
Different from each other. 12 has three prime factors (2, 2, 3) but only 2 DISTINCT factors.
Fraction into Decimal
Divide the numerator by the denominator.
Speed
Expressed in miles or kilometers per hour. Formula: Speed= Distance/Time OR Distance=(Speed)(time) OR Time=Distance/Speed All rates can be solved using the speed formula by changing Speed to Rate and Distance as Quantity.
FOIL method
First, Outer, Inter, Last term. For multiplying binomials.
Reciprocals
Flip the fraction upside down so that the numerator becomes the denominator and visa versa. If the numerator is 1, then the reciprocal becomes equivalent to an integer. 1/25? flip= 25/1 which is 25.
Right Triangle
Has one right angle. Right angle is always the largest angle; therefore, the hypotenuse, which lies opposite the right angle, is always the longest side.
Even
Integers (and only integers, positive or negative) that are evenly divisible by 2. Integers that end in 2, 4, 6, 8, 0.
odd
Integers (and only integers, positive or negative) that are not evenly divisible by 2. integers that end in 1, 3, 5, 7, or 9.
Sequence
Lists that have an infinite number of terms, in order. The terms of a sequence are often indicated by a letter with a subscript indicating the position of the number in the sequence. a3 denotes the third number in a sequence while aN indicates the Nth term in a sequence.
Perimeter of Triangle
No formula, just sum of the lengths of the sides.
Consecutive Numbers
Numbers of a certain type, following one another without interruption. Can be in ascending or descending order. -4, -2, 0, 2, 4.... consecutive even numbers 2, 3, 5, 11.... consecutive prime numbers
Element
One of the members of a set.
Pythagorean Theorem
Only for right triangles. States that the square of the hypotenuse is equal to the sum of the square of the legs. (leg1)^2 + (legs2)^2 = (Hypotenuse)^2 Useful when you've been given the lengths of any two sides of a right triangle- as long as you know whether the remaining sides is a leg or the hypotenuse, you can find its length using this theorem.
Radicals and Rules of operations
Only like radicals can be added to or subtracted from one another. Until simplification. To multiply or divide one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs. Ex: (6√3) 2√5 = (6)(2) = 12√(3)(5)= 12√15 Ex: 12√15 / 2√5 = (12/2)=6√(15/5)= 6√3 Simplifying Radicals: if the number inside the radical is a multiple of a perfect square, the expression can be simplified by factoring out the perfect square. Ex: √72 = (√36) √2= 6√2
Rules of Operation
PEMDAS "Please Excuse My Dear Aunt Sally" Parentheses Exponents Multiplication Division Addition Subtraction Multiplication and Division - simultaneously from left to right. Addition and Subtraction- simultaneously from left to right.
Properties of Power
Raising a fraction between 0 and 1 to a power produces a smaller result. (1/2)^2= (1/2)(1/2)= (1/4) Raising a negative number to an even power produces a positive result. (-2)^2= 4 Raising a negative number to an odd power gives a negative result. (-2)^3 =-8 Rising an even number to any exponent gives an even number. Raising an odd number to any exponent gives an odd number. 8^5 = 32,768 5^8= 390,625
Properties of Fractions
Reciprocal of positive fraction is greater than both the original fraction and 1. Reciprocal of a negative fraction is less than both the original fraction and -1. A square of a positive fraction is less than the original fraction A square of any negative fraction is greater than the original fraction (because multiplying two negative numbers gives you a positive product which is always positive which is always more than a negative). Multiplying any positive number by a positive fraction gives a product smaller than the original (not the fraction). Multiplying any negative number by a negative fraction gives a product greater than the original number (not the fraction).
Associative Laws of Addition and Multiplication
Regrouping the numbers does not affect the results. (a+b)+c=a+(b+c) (ab)c=a(bc)
Comparing positive fractions
Same denominator? Fraction with larger numerator is larger. Same numerator, Different denominator? Fraction with smaller denominator is larger. If there are neither same numerator or denominator: turn into decimals and determine larger, find the common denominator and see which has larger numerator, or cross multiply the numerator of each fraction by the denominator of the other and the greater result will end up next to the greater fraction EX: which is greater (5/6) or (7/9)? 45- (5/6)(7/9)- 42 (5x9) and (6x7) SO the answer is (5/6)
Multiplication and Division of Positive and Negative Numbers
Same signs- positive result Different signs- negative result
Interest
Simple interest Principle Compound interest
Symbolism on GRE
Sometimes you will see strange symbols like a star and a diamond. Just means substitue the numbers and follow the following two steps: 1. find the reciprocal of the second quantity 2. subtract the reciprocal from the first. x @ y = x-(1/y) What is the value of 2 @ 7? Sub x with 2. Sub y with 7. 2- (7) Find reciprocal of 7 which is (7/1) Solve equation 2- (1/7) = 13/7 OR if it does not include an equation, it usually is testing your understanding of number properties. @x is defined as the largest even number that is less than the negative square root of x. What is the value of @81. Answer is -10.
Commutative Laws of Addition and Multiplication
Switching the order of any two numbers being added or multiplied together does not affect the results. a + b=b + a abc=cba
Fraction
The division of a part by a whole. Part/Whole
Sales Price
The final price after discount or decrease.
Polynomial
The general name for an algebraic expression with more than one term. If two, its called binomial.
Greatest Common Factor
The greatest common factor (GCF) of a pair of integers is the largest factor that they both share. To do this, you need to break down both integers into their prime factorizations and multiply all the prime factors they have in common. GCF of 36 and 48? 36: 2, 2, 3, 3, 48: 2, 2, 2, 2, 3 What they have in cmmon is the two 2s and one 3, so (2)(2)(3)=12
Hypotenuse
The longest side of the triangle. With a triangle with a 90 degree angle, the side opposite of the 90 degree angle is always the longest side of that triangle
Denominator
The number of equal parts into which a whole is divided.
Exponents
The number that denotes the power to which another number or variable is raised. Written as superscript. Can be referred to as a "power". Can be positive or negative integers or fractions, and may include variables. 5^3 "5 to the 3rd power" =125 125 is "the 3rd power of 5"
Coefficient
The numerical constant by which one or more variables are multiplied. The coefficient of 3x is 3. A variable without a numerical coefficient, such as z or xy, is understood to have a coefficient of 1.
Squared
The product of a number multiplied by itself. A squared number has been raised to the 2nd power. 4 squared = (4)(4)= 16 16 is the square of 4.
Denominator
The quantity in the bottom of a fraction, representing the whole.
Numerator
The quantity in the top of a fraction, representing the part.
Product
The result of multiplication.
Difference
The result of subtraction.
Legs (triangle)
The two equal sides of an isosceles triangle or the two shorter sides of a right triangle (the ones forming the right angle). NOTE: the third, unequal side of an isosceles triangle is called the base.
Absolute Value
The value of a number without its sign. Written as two vertical lines. The distance from zero on the number line. -3 and 3 are both 3 units from zero so each has an absolute value of 3. [x]= 5, x could equal 5 or -5
Converting Percents
To change a fraction to its percent equivalent, multiply by 100%. Ex: % equivalent of (5/8)? (5/8)(100%)= (500/8%) which is equal to 62(1/2)% To change decimal fraction to percent, use rule for multiplying by powers of 10. Move decimal point two places to the right and add percent sign. Ex: % equivalent of 0.17? 0.17 x100%= 17% To change a percent to its fraction equivalent, divide by 100%. Ex: what is common fraction equivalent of 32%? 32%= 32%/100%= 8/25 To convert a % to its decimal equivalent, use the rules for dividing by powers of ten, move two places to the left. Ex: 32% = .32 When you divide a % by another %, the percent sign "drps out" Ex: 100%/5% = 100/5 = 20 -> There are 20 groups of 5% in 100%. When you divide a % by a regular number and not a percent, the percent sign remains. Ex: 100%/5 = 20% --> One-fifth of 100% is 20%.
Rules of Operations with Exponents
To multiply two powers with same base, keep base and add exponents. 2^2x2^3 = 2^5 To divide two powers with same base, keep base and subtract exponent of the denominator from exponent of numerator. 4^5 / 4^2 = 4^3 To raise a power to another power, multiply the exponent. (3^2)^4 =3^8 To multiply two powers with different bases by the same power, multiply the bases together and raise to the power. 3^2 x 5^2 = 15 ^2 A base with a negative exponent indicates the reciprocal of that base to the positive value of the exponent. 5^-3 = (1/5^3) = 1/125 Raising any non-zero number to an exponent of zero equals 1. 5^0=1 161^0=1 (-6)^0 =1
Algebraic expression
Two algebraic expressions separated by an equal sign or one algebraic expression separated from a number by an equal sign.
Powers of 10
When 10 is raised to an exponent that is a positive integer, that exponent tells how many zeros the number would contain if it were written out. Ex:Write 10^6 in ordinary notation - 1,000,000 10x10x10x10x10x10= 1,000,000 To multiply a number by a power of 10, move the decimal point the same number of places to the right as the value of the exponent. Ex: Multiply 0.029 by 10^3 -The exponent is 3, so move the decimal point three places to the right. = 0029. = 29 Ex. Multiply 0.029 by 10^6 = 0029000.= 29,000 To divide by a power of 10, move the decimal point to the corresponding number of places to the left.
Percent Formula in Word Problems
When you have a word problem that involves percents, it usually gives you two of the three variables and ask you for the third. Look for the "part" and the "whole" and plug it into the "Percent Formula" which is (Part / Whole) times (100%). Just turn the fraction into a decimal. If the problem ask for increase or decrease, use the formula: Percent increase = Increase (100%)/ original Percent decrease= decrease (100%)/ original to find the increase or decrease, just take the difference between the original and the new.
Permutation
Within any group of items or people, there are multiple arrangements. This differs from combinations because permutations are ordered. With the combination ABC there are six permutations (ABC, ACB, BAC, BCA, CAB, CBA) To find, the first place can be filled with any of the items in the group. The second place can be filled with any except the one used to fill first place. The third place can be filled with any item in the group except for the two used int he first two places. The multiply. Spelling bee, winner recieves a gold, second receives silver, third bronzes, 4th blue ribbon. if there are seven people in the contest, how many different arrangements of award winners are there? Gold metal could be won by any 7 people. Silver by any of the remaining 6. Bronze by the remaining 5 Ribbon 4. (7)(6)(5)(4) =840
Improper fraction
a fractions whose numerator is greater than its denominator. To convert into a mixed number, divide the numerator by the denominator. The number of whole times that the denominator does into the numerator will be the integer and the remainder will the numerator. (23/4)-- 23 /4=5 with a remainder of 3. 5(3/4)
Transversal
a line that intersects two parallel lines. Each of the parallel lines intersects the third line at the same angle.
Square root
a square root of any non-negative number X is a number that, when multiplied by itself, yields X. Every positive number has two square roots, one is positive and one is negative. Positive square root of 25 is 5 (5x5=25) and -5 (-5x-5=25) The square root sign √ only stands for the positive square root. (The five and not the negative five.
Quadratic Equation
an equation in the form of ax^2+ bx+c=0 Many of these equations have two solutions. So, the equation will be true for two different values of x. You solve it by factoring the algebraic expression, setting each of the factors equal to zero, and solving the resulting equation. x^2 - 3x +2 =0. Factor into (x-2)(x-1) =0. Each part needs to equal to zero so (x-2)=0 which means x can be 2 and (x-1)= 0 which means x can be 1 also.
Prime number
an integer greater than 1 that has only two factors: it's self and one. The number one is not considered a prime because its only divisible by itself. The number 2 is the smallest prime number and the only even prime. first ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Division by zero
answer is undefined. For GRE purposes, that translates to "It can't be done" Any fraction with zero as denominator is also undefined since fractions are essentially division. -- If given an fraction that has algebraic expression in denominator, be sure the expression cannot equal zero.
Reducing a Fraction
converting a fraction into its lowest terms. To do this, just divide any common factors out of both the numerator and denominator. fastest way is to find the greatest common factor and divide it out of both the top and the bottom.
Sum of exterior angle of triangle
equal to the sum of the remote interior angle. the three exterior angles are equal to 360.
Acute
less than 90 degrees
Factoring Algebraic Expressions- polynomial
means to express it as a product of two or more simpler expressions. Common factors can be factored out by using the distributive laws. Ex: Factor the expression 2a + 6ac. Greatest common factor is 2a. Using distributive law, you can factor out 2a so the expression becomes 2a(1+3c)
Dividing Fractions
multiply by the reciprocal of the fraction that follows the division sign. (1/2)/(3/5) === (1/2)x(5/3)= 5/6 If top number is more than bottom, divide.
Multiplying fractions
multiply the numerator and the denominator. Do not need to have common denominator. Just multiply.
Profit
seller's price minus the cost to the seller. Profit/original selling price --- for percent word problems
Radicals
square roots written with the radical symbol √
Averages
the average of a group of numbers is defined as the sum of the terms divided by the number of terms. Equation can be written as: Number of terms= sum of terms/average Sum of terms= (Number of terms)(Average) --so if you have two out of the three, you can figure it out.
Discount
the original price minus the reduce price. Discount/ original price --- for percent word problems.
Vertex
the point at which two lines or line segments intersect to form an angle which is measured in degrees.
Factors (divisors) of an integer
the positive integers by which it is evenly divisible (leaving no remainder). What are the factors of 36? Answer: There are 9 factors- 1, 2, 3, 4, 6, 9, 12, 18, 36
Prime Factorization
the prime factorization of a number is the expression of the number as the product of its prime factors (the factors that are prime numbers). Two ways to do this: Prime of 210? Start with 2 and work your way through the prime numbers. 210= (2)(105). Next work with 105. its odd so it cant be 2 so work with the next prime number -3. 3 times 35 makes 105. 210=(2)(3)(35). Next work with 35. The next prime number is 5 (because 3 doesnt work). 5 times 7 is 35. 210= (2)(3)(5)(7) . You cannot break this down anymore since all numbers are now prime. You can also just pick one pair of factors and then determine their factors until you are left with only prime numbers. Make a tree. When writing these, write in increasing order. If there is a number twice, you can write it with exponent.
Multiple
the product of a specified number and an integer. Multiples do not have to be of integers but all multiples must be the product of a specific number and an integer. 3, 12, 90 are all multiples of 3. 3x1, 3x4, 3x30 2.4, 12, 132 are multiples of 1.2 (1.2 is not an integer).
Ratio
the proportional relationship between two quantities. It is expressed with a colon between two numbers or as a fraction. Fraction: to turn a ratio from words to numbers, use whatever number follows the word OF as the numerator and whatever follows the word TO as the denominator. Should be reduced to lowest terms.
The least common multiple of two integers
the smallest number that is a multiple of each of the integers. Steps: 1. determine prime factorization of each integer. 2. Write out each prime number the maximum numbers of times that it appears in any one of the prime factorizations. 3. multiply those prime numbers together to get the least common multiple of the original integers. Least common multiple of 6 and 8? 6=(2)(3) 8=(2)(2)(2) (2)(2)(2)(3)= 24.
Complementary
two angles are complementary to each other if their measures sum to 90 degrees.
Supplementary
two angles are supplementary if their measures sum to 180 degree.
The Combination Formula
used in a combination question which ask you how many unordered subgroups can be formed from a larger group. n!/k!(n-k)! where ! means factorial (for example, 5! = (5)(4)(3)(2)(1)=120, n means the number of items in the group as a whole, and k is the the number of an items in each subgroup formed. The 4 finalists in a spelling bee contest win commemorative plaques. If there are 7 entrants in the spelling contest, how many possible groups of winners are there? 7!/4!(7-4)! = 7!/4!3! So, since 7 factorial contains all the factors of 4 factorial, we can write 7! as (7)(6)(5)(4!) which means we can cancel out the 4! in the numerator and denominator. So then we are left with (7)(6)(5)/(3)(2)(1). We can reduce further by canceling out the (6) and the (3)(2) which leaves us with (7)(5)/(1) which is = to 35. There are 35 potential groups of spelling contest finalists.