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MAGOOSH GRE MATH FLASHCARDS
What's the acronym for remembering the order of operations? - "GEMDAS" G= Grouping Symbols E= Exponents M= Multiplication D= Division A= Addition S= Subtraction Division and Multiplication are at the same level. what's the technique for multiplying and dividing decimals? - Multiplication: count decimal places before; this will indicate how many decimal places in product from left decimal place. So you first find total decimal places. Remove them to multiple. then count from right depending how many total decimal places from start. Division: slide numerator and denominator decimal place to the right one, until you get a manageable number. What's the exponent of 10,000? What's the rule here? - 10^ for numbers that are factors of 10, the number of zeros = the exponent. How do we convert decimals to exponents in terms of values less than 1? for example, .00001= what? - the amount of decimal places = the exponent. 10^- 1/10x10x10x10x what are the tricks we can use when dividing or multiplying by a multiply of 10? - count up decimal places and move to right or left. How do you convert a mixed fraction to an improper fraction? - A mixed fraction is separated by a plus sign. First separate, then multiple whole number by a whole number that will allow it to add to proper fraction. then just add and you will have an improper fraction. How do you convert a particular fraction decimal pair to another one? multiply each side by desired number - 1/5=.2 ---> multiple both sides by two to get 2/5=. How do you estimate whether a fraction is equal to, less than, or greater than another fraction? - cross multiply and compare. What should you always try to do if you are multiplying fractions? - simplify via cross- eliminating.
How do you divide two fractions? - multiply by one of the reciprocals. In a proportion, what's considered a legal cancellation? How about an illegal cancellation? How about in terms of multiplication? - Proportions: Vertical and horizontal cancellation is acceptable. Cancelling diagonally is illegal. Fractions: cancelling diagonally is legal. Cancelling horizontally is illegal. In word problems, what does "of" and "is" mean? - "of"= multiply (3/5 of 400= 3/5x400/1) "is"= equal ab/c and a+b/c. Which can be separated? - only if the top is adding can you separate to a/c+b/c. In multiplication you CANNOT separate to (a/c)(b/c) What's the difference between percent of vs. percent greater? - percent greater= new- old/old x 100% Percent of= compare the two numbers to 100%. what's the formula for % increase/decrease? - new-old/old x 100% what's the compound interest formula? - do not memorize it! use logic to reconstruct What are the important skills for recreating the compound interest? - -multipliers. -exponents How do you recreate the compound interest formula? - 1.) take the principle. 2.) find the multiplier of the interest. 3.) take the exponent of the multiplier, depending on how many years you are compounding. 4.) multiply this by the principle. How to you recreate the compound interest if it is quarterly? - 1.) divide annual interest by n (n= 4 if quarterly, n=365 if daily, n=12 if monthly). 2.) find the multiplier of this percent 3.) multiply the years by the period of compounded; this is your exponent 4.) take the interest from before and set to the exponent found before. 5.) multiply this by the principle. what's the first thing one should do when comparing ratios? - make the denominator common. What's the equilateral triangle area formula? - square root of 3)/4 x side^ square root of y^16? - y^
Is one a prime number? - no what's a prime number? - divisible by 1 and itself How do you test if a large number is prime? - Divide this large number by 2,3,5,7. If it is not divisible by these numbers, then it is a prime number. What's the fundamental theorem of arithmetic? - for every non-prime number over 1, one can express this number in terms of the product of prime numbers. How do you find the prime factorization of a number and how is it useful? - you create a prime factorization tree. If we are told which numbers are factors of the original number, then any combination of these prime factors will work fine. But, it can only be a combo of these factors. How does one count the factors of a large number? - 1.) find the prime factorization of the number 2.) make a list of the prime factor's exponents 3.) add one to each one of these numbers 4.) multiply these numbers together this calculates the amount of factors within a large number. How do you find the odd factors of a large number? - 1.) find the prime factorization of the number 2.) make a list of the prime factor's exponents 3.) ignore any factors of 2. 4.) add one to each of the remaining non-even numbers 5.) multiply these numbers together How do you find the even factors of a large number? - first you find all the factors in the large number. then, you find the odd factors of the number. then subtract the second from the first. Let's say we break a number down into its prime factors, how do we know if it is a square? how about a perfect square? - square: if the exponents are all even. perfect square: if it has an odd number of factors (why? the perfect square will have two identical factors= the original number; for example, 6^2= 36) How do you determine the amount of integers between two numbers? - Number 1- Number 2 +1= answer
how do you calculate the greatest common factor of two numbers? - first, find the factors of both numbers. then, see which are common. the greatest of the common is the greatest common factor. How do you calculate the greatest common factor of two large numbers? - 1.) calculate the prime factors of both 2.)see which prime factors are in common 3.) building all the common prime factors together. this is your greatest common factor. How do you find the least common multiple? - you make a list of the multiples, find which multiples are in common, and then the least multiple is your answer. How do you create the least common multiple of large numbers? - 1.) calculate prime factors of both 2.) calculate the greatest common factor. 3.) use the greatest common factor to calculate each number with the integers. 4.) Multiple the greatest common factor by each of these numbers. 5.) this is your least common multiple When do we use Least Common Multiples? - when we are trying to matches sets. buns come in packs of 7, hotdogs come in packs of 6. how many packs of hotdogs and buns do we need to buy? Use LCM to calculate this. Least common multiples are useful for finding a common denominator (which is an LCM) game to study GCF or LCM? - pick two numbers randomly and go! give the possibilities of even and odd based on two numbers and addition/subtraction - O+O= E E+E=E O+E=O give the possibilities of even and odd based on two numbers and multiplication - OxO=O ExE=E OxE=E is 0 even or odd? - even. Can you assume that numbers are integers? - no, it must be stated explicitly How do you convert a decimal into a fraction? - You first find the fraction of the decimal, reduce to the lowest terms. The numerator is the remainder.
What happens if a system of equations results in two different answers? - There are no solutions. This means that the two lines are parallel to each other. What do you do if there is a three system equation problem? - alter the equations so that they will work together via the elimination method. How do you calculate the percent difference and percent increase and percent error? - percent difference= difference between two of the same kind of number/ their average x 100% percent error/increase/change= difference between the two/the original number x 100% How do you solve for the absolute value of a variable? - Make this variable set to the positive and negative equivalent. What does f(x) mean? - f(x) is a placeholder for an x within an expression. X can be anything. You take it and put it inside the expression. What do you do if there is a strange operator? - Along with the strange operator, there will be a rule that defines this operator. What are the operations we can do to an inequality? - we can always add and subtract the same number from both sides. We can multiply or divide the same positive number from both sides. We cannot divide or multiply by the same negative number. Why? it reverses the inequality because it changes the sign. What do you do to an inequality if you multiply/divide both sides by positive integer? what about if you multiply/divide it by a negative integer? - positive integer= nothing. the inequality remains. negative integer= the inequality is reversed. What do you do if you have two inequality notations within you equation? -4<5-3x<17 - when you alter any "subsection" of an inequality make sure you do the same to all the other subsections. if you subtract 5 from the middle, you do this to the 17 and -4. If you divide -3, then you do that to each (remember, for division, you invert the inequality sign). What are the rules for adding and subtracting inequalities? - adding inequalities with same sign is allowed. Adding inequalities with different signs is not allowed. Subtracting inequalities with same signs is not allowed. subtracting inequalities with different signs is allowed.
What should we remember when looking at numbers? - remember you can use decimals, not only integers. How do you simplify equations that have lots of operations around a subsection with a variable? - Make a subsection into a variable x. Then remove this subsection and put x inside instead. Solve the whole equation and then put the subsection back into x and solve again. This helps you divide up any complex part of an equation. How do you find the angle of a circular arc? or the length of a circular arc? - Name three of the most popular Pythagorean sets? - These sets are called Pythagorean Triplets, numbers that satisfy a2+b2=c2. {3, 4, 5} and {5, 12, 13} and {8, 15, 17}, as well as multiples of these, are good triplets to know. What is the length of the diagonal between two interior vertices of a square? - 3Squarroot of s What are the properties of a 30,60,90 triangle? - short leg opposite 30= long leg opposite 60=square root of 3 hypotenuse= 2 What's the area of a parallelogram? - Area=bh How do you find the sum of all angles of an n-polygon? - (n-2) How do you find the angle of a a circular sector (a slice in the pie)? - What is the length of the interior diagonal between two opposite vertices? - What is the formula to solve for the area of a trapezoid? - What are the properties of a 45, 45, 90 triangle? - each leg= 1, hypotenuse= square root of 2 How do you solve for the age word problems? - First, choose variables to represent the age now. From here, apply the specifications of the question to the variables you choose and solve. So whenever you have a variable about a person's age , you plug in what the age was at the present time. So, if they are telling us that in 8 years bob is x/Tom's age. for bob's and tom's age, you plug in what it was at present. What's the mneumonic to remember for rate problems? - DiRT Distance= (Rate)(Time) What's the trap for average speed problems? - The test makers want the takers to just quickly average the two rates together as to get the average rate. But, remember, due to the fact that the two rates are different, they are not the same. You need to find a common feature and work to unify them this way.
What's one way we know that there is an arithmetic sequence? - "any evenly spaced list" they can be evenly spaced due to the same remainder as well. What does common difference mean? - fixed amount between numbers in a list. It is designated as "d" What's the formula that we need to know for determining the next number in the set? - don't memorize. understand What does a recursive definition of patterns mean? - It relies on the previous number to compose the current number. The General format of a recursive definition of patterns is. you can also have only one. Then the next number in the pattern is reliant on only what's dictated. But, it can be reliant on as many as possible. Usually there will be a limit associated with this formula so that the number doesn't become negative. How do you add up the sum of numbers between x-->y. Why does this work? - first add x and y. then multiply it by half of the total numbers. This works because if you take the ends of each list of numbers, they will add up to the same thing. What's the best way to think about exponents? - Think about its definition What are the 4 different exponential patterns? what is it reliant on? - It is reliant on the base. The 4 patterns are: --positive base= normal exponential growth. --negative base= exponential growth with alternating extremes of negative and positive (odd= negative, even=positive) --negative 0-->1= getting smaller and smaller with alternating extremes of negative and positive --positive 0-->1= getting smaller and smaller with only positive. What are the laws of exponents? Understand how these are derived based on the definition of an exponent - multiplying two exponents with same base= adding them. dividing two exponents with same base= subtracting them. Raising a power to a power= multiplying the powers together. What is the requirement of applying the laws of exponents? - the bases must be the same. Is there a pattern for 5^6+5^3? - no. the exponential laws only pertain to multiplying, dividing, powering the same bases
What can you do with exponents in terms of being in the numerator and denominator? - you can move a base with a negative exponent from the denominator to the numerator and changing the exponent to positive. Or you can move a base with a negative exponent from the numerator to the denominator and changing the exponent to positive. How does the distributive law work with exponents? - exponents distribute just like numbers do in multiplication and division. What process is illegal when distributing exponents? what is legal? - b^n=b^m what is the relationship of n and m? - n=m How do you solve unit digit problems? 57^123 - 1.) all you need to focus on is the units. 2.) look for repeating pattern of 7^1, 7^2, 7^3, etc... 3.) extend the patterns using multiples of the period. How do we know if a square root problem is asking us to give only the positive answer or both positive and negative answer? - only the positive= if the question gives you the square root. positive and negative= if you have to initiate the square root function Can you take the square root of zero? - yes. 0^ How do cubes and squares differ? - you cannot square a negative number, you can cube a negative or positive number. There will always be only one answer with cubes. There will always be only one answer with squares. What are the general rules for odd roots and even roots? - odd roots of positive number= positive answer. odd roots of negative number= negative answer. (follow same format as cubing). Even roots result in same format as squares. caveat with roots (decimals ) - if there are roots of integers, then the numbers will be lesser if the roots are higher. If there are roots of decimals, then the numbers will be higher if the roots are higher. How do you simplify non-perfect squares? - see if there are perfect squares within this number that can be rooted down to its square. How do you simplify roots that are subtracting each other? - Simplify each root individually and see if you can make the roots match. You should see each different root as a different variable. For example, you cannot subtract 3x-2y. But you can subtract 3x-2x. This latter position is what you want your roots to look like.
What's the most important exponential equation rule? - When you have 49^x, how can you simplify it? - you can simplify it to 7^2x How do you solve exponential equations? - you need to form unified bases. Try to find common bases on each side. Remember, when you do that, the exponents can merge with preexisting exponents: How do you rationalize a root? - How do you rationalize a root if there is subtraction/addition in the numerator? - How do you rationalize a root if there is subtraction/addition in the denominator? - you multiply the denominator by its reciprocal. How do you solve for 1/p+1/q=1/f? - you can multiply the right side components by whatever is one.
How do we determine possible triangle sides if we have the length of two sides? - large- small= x; the unknown side must be larger than x. large+small=y; the unknown side must be smaller than y. how do you find the triangle's base? height? how do you do this with an obtuse triangle?
- either of the sides. from the base to the vertex opposite to the base. With an obtuse triangle, if you cannot get a perpendicular line internally to the triangle from the vertex, you can draw it from the vertex to the outside line: how can you use areas to solve for length of lines in triangles? - if two lines are touching the same triangle, then the triangle's area can be used to equate them: What are the right triangle Pythagorean triplets? How do we derive other triplets? - 3,4, ; 5,12,13 ; 8,15, How do you deal with large numbers with the Pythagorean theorem? - Try to take out a GCF. when you make the numbers smaller, perhaps you will see it is a pythagorean triplte what's a common way to show similar triangles? what does similar triangle mean? (angles are the same) -
How do we prove that two triangles are similar? - if two angles are the same as two angles of another triangle, then they are similar. If their sides are proportional. What is the significance of the proportionality? If you take the smaller side and multiply it by the scale factor, then you will get the larger side, or if you divide the larger side by the scale factor, then you will get the smaller side. how do we relate the area of two similar triangles to the scale factor? - multiply/divide the area of the smaller/larger triangle by the scale factor^2. What is the isosceles right triangle? - What is the 1-2-square root 3 triangle? - What's the area of an equilateral triangle? - How can we easily calculate the sides of a 30,60,90 right triangle? -
