Arithmetic i Algebra
Multiplying exponents with same base [but ojo NOT true for addition e.g. 2squared plus 2 to the fourth is NOT 2 to the 6th]
you add the exponents
Raise a power to another power
you multiply the exponents
Distance/rate/time formulas
distance = rate x time
circle with its centre at the origins
x2 + y2 [squared] = r 2 [when r = radius]
circle with its centre at a specific point
"" except you subtract the points of the middle of the circle
Percent Word Problems general
% x a number = a part of number [ojo answer will always be one of the numbers divided by the other, unless it's a straight of find the part of a number x percent] so no need to memorize other formulas How to find what percent one number is one another = the part of the number / number e.g. 3 is what percent of 5? 3/5 = 60% How to find a number when the percent of it is known = the part of the number / the percent
Exponents and parentheses rules
*** when stuff inside the parentheses is multiplying or divided each other, then the exponent outside is multiplied to the exponent of each term inside.
if change is made within the parenthesis of x, then you are moving it sideways
**** ojo adding moves to the LEFT!!!! see example sort of counterintuitive; just memorize this
Ojo remember % = always the number in HUNDREDTHS
...Be careful not to confuse 0.01 with 0.01%. The percent symbol matters. For example, but 0.01 0.01 = 1% 0.01% = = 0.0001.
What is 0 a factor and multiple of?
0 is a multiple of every integer; 0 is not a factor of any integer except 0.
Steps for Dividing a Polynomial by a Binomial:
1. Remember that the terms in a binomial cannot be separated from one another when reducing. For example, in the binomial 2x + 3, the 2x can never be reduced unless the entire expression 2x + 3 is reduced. 2. Factor completely both the numerator and denominator before reducing. 3. Divide both the numerator and denominator by their greatest common factor.
Rounding to unit, tens, hundredth
15.239 rounded to units is 15 as the next digit (2) is less than 5 134.9 rounded to tens is 130 as the next digit (4) is less than 5 12,690 rounded to thousands is 13,000 as the next digit (6) is 5 or more
How to find percent increase?
18 is what percent of 3? Originally I wrote 6. I was WRONG. The answer is 600%. Use common sense 2x!
Multiple
We say that 60 is a multiple of each of its factors and that 60 is divisible by each of its divisors.
Simplifying basics:
8a = 7b ..... is a/b = 7/8
Quadratic Equation/Quadratic Formula
=
linear equation =
A linear equation is an equation involving one or more variables in which each term in the equation is either a constant term or a variable multiplied by a coefficient. [i.e. no exponents or roots!]
coefficient
A number that is multiplied by variables is called the coefficient of a term
constant term
A term that has no variable is called a constant term. A
area of a triangle
Area = .5 bh b= base h= height
Average (= same as the mean)
Average = total value / [number of values]
EG the average of 5 numbers is 30. After one of the numbres is removed, the average is now 32. What was the number?
Basically you know that 150 = 5 x 30 and you know (!) that 128 = 4 x 32 so you just subtract. the answer = 22.
Note on non integer fractions
Can be manipulated same way as fractions!
How to solve ratios ratio when you have total number
E.g. if a board is 18 ft and is cut into three sections that have a ration 1:3:5 then you solve this through the equation: 1x + 3x + 5x = 18. you ADD these: so 9x = 18; so x= 2. OJO can prob also just plug in
the slope of a perpendicular line is the negative reciprocal of the original line
In numbers, if the one line's slope is m = 4/5, then the perpendicular line's slope will be m = -5/4. If the one line's slope is m = -2, then the perpendicular line's slope will be m = 1/2. To answer this question, I'll find the slopes.
multiplying three algebraic expressions
OJO the trick is to do two parentheses FIRST; then you multiply the x in the leftover parentheses by EVERYTHING in first two; then you add this to the constant multiplied by everything
basics of standard deviation
One standard deviation away from the mean in either direction on the horizontal axis (the two shaded areas closest to the center axis on the above graph) accounts for somewhere around 68 percent of the people in this group. Two standard deviations away from the mean (the four areas closest to the center areas) account for roughly 95 percent of the people. And three standard deviations (all the shaded areas) account for about 99 percent of the people.
magoosh on what to study and not study for
Probably not. Meaning that parabolas can show up on the GRE, but probably won't show up on the GRE you are taking. That is my initial take based on forums and students experiences. Still, parabolas, along with absolute value graphs, are included in the Practicing to Take the Revised GRE book. ETS obviously didn't put them there to kill more trees. So, like compound interest and the quadratic equation, parabolas may show up.
OJO probably WONT be quized on either compound interest OR parabolas on the GRE...
Probably not. Meaning that parabolas can show up on the GRE, but probably won't show up on the GRE you are taking. That is my initial take based on forums and students experiences. Still, parabolas, along with absolute value graphs, are included in the Practicing to Take the Revised GRE book. ETS obviously didn't put them there to kill more trees. So, like compound interest and the quadratic equation, parabolas may show up. What this means for your study plans is that you should only worry about parabolas if you are looking to score above 90%, and have already brushed up on the other concepts on the new GRE math. Meaning, if you are still struggling with number properties or circles, focus there.
How to write a decimal for repeating fraction
Put # over equivalent number of 9s. e.g. .25252525 = 25/99
Quadratic equations can have..
Quadratic equations have at most two real solutions, as in the example above. However, some quadratic equations have only one real solution. Other quadratic equations have no real solutions; for example, In this case, the expression under the 2 x + x + 5 = 0. square root symbol is negative, so the entire expression is not a real number.
Rounding to hundred vs hundredth
Rounding to tenths means to leave one number after the decimal point. Rounding to hundredths means to leave two numbers after the decimal point. etc. 3.1416 rounded to hundredths is 3.14
shifting a graph just moves it on the graph by a determined number of spots
Shifts A shift is a rigid translation in that it does not change the shape or size of the graph of the function. All that a shift will do is change the location of the graph. A vertical shift adds/subtracts a constant to/from every y-coordinate while leaving the x-coordinate unchanged. A horizontal shift adds/subtracts a constant to/from every x-coordinate while leaving the y-coordinate unchanged. Vertical and horizontal shifts can be combined into one expression. Shifts are added/subtracted to the x or f(x) components. If the constant is grouped with the x, then it is a horizontal shift, otherwise it is a vertical shift. e.g. f(x) = x^2 if you shift this by adding three, i.e. f(x) = x^2 +3 then you are moving the graph upwards three points, i.e. because you are changing the answer you will get for the y axis
what is a domain?
The domain of a function is the set of all permissible inputs, that is, all permissible values of the variable x --- .Without an explicit restriction, the domain is assumed to be the set of all values of x for which is a real number *** Asking for the domain of a function is the same as asking --- "What are all the possible x guys that I can stick into this thing?"
Example of two people going in the same direction at different miles per hour. How long does it take for them to catch up?
The takeaway: when two entities (trains, cyclists, cars) are headed in the same direction, the difference between the two rates is the amount the faster entity is outpacing the slower one per hour (in the case that the rate is expressed in terms of per hour). Two trains are headed along parallel tracks in the same direction at varying rates of 72 mph and 47 mph. In how many hours will the faster train be 100 miles in front of the slower train? (A) 2 hrs (B) 2.5 (C) 3.5 (D) 4 (E) Cannot be determined by information provided. The takeaway: when two entities (trains, cyclists, cars) are headed in the same direction, the difference between the two rates is the amount the faster entity is outpacing the slower one per hour (in the case that the rate is expressed in terms of per hour). Returning to the original problem, which asks for trains headed in the same direction, we want to take the difference of the two rates. This gives us 72 - 47 = 25. Therefore, for every hour the faster one is 25 miles ahead of the slower train. How long, then, will it take until the faster train is 100 miles ahead? 100/25 = 4 hours.
Solving systems of linear equations
There are two basic methods for solving systems of linear equations, by substitution or by elimination. In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation.
To find missing # in an average if you have total
To find missing # in an average if you have total.... subtract the total of the numbers you have from the product of the average x the # of #s:
Work rate problems = rare BUT...
To recap: to find the work rate, first find the hourly rates for each individual. Then, add these two rates together, and then flip, or take the reciprocal of, that fraction. It's that easy. As they say, it's nothing to get worked up about!
Systems of linear equations and simultaneous equations
Two equations with the same variables are called a system of equations, and the equations in the system are called simultaneous equations. To solve a system of two equations means to find an ordered pair of numbers that satisfies both equations in the system.
How to solve when two bodies heading towards each other distance problem?
Two trains starting from cities 300 miles apart head in opposite directions at rates of 70 mph and 50 mph, respectively. How long does it take the trains to cross paths? Let's go back, and attack the above problem the following way. When you have any two entities (trains, bicyclists, cars, etc.) headed towards each other you must add their rates to find the total rates. The logic behind this is the two trains (as is the case here) are coming from opposite directions straight into each other. This yields 120 mph, a very fast rate (which accounts for the severity of head-on collisions...don't worry the trains in the problem won't collide). To find the final answer, we want to employ our nifty old formula: D = RT, where D stands for distance, R stands for rate, and T stands for time. We've already found R, which is their combined rate of 120 mph. They are 300 miles apart so that is D. Plugging those values in, we get 300 = 120T. Dividing 120 by both sides, we get T = 2.5 hrs.
three kinds of symmetry
We've some fairly simply tests for each of the different types of symmetry. A graph will have symmetry about the x-axis if we get an equivalent equation when all the y's are replaced with y. A graph will have symmetry about the y-axis if we get an equivalent equation when all the x's are replaced with x. A graph will have symmetry about the origin if we get an equivalent equation when all the y's are replaced with y and all the x's are replaced with x.
Plugging in the Answers
When an algebra word problem gives you the answers, just plug in the answers! start with answer c always!
Important tip
When you are adding [ or multiplying] a series of number you can regroup then as you please
Plugging in
You can plug in in ANY question that has variables in the answer choices...
two angles are COMPLEMENTARY if they add up to 90
[ABCS] e.g. C = smaller than S, completentary = 90, supplementary = 180
stretching a graph
[maybe skip this!] you multiple a positive constant by x (which is same as saying f(xa)) this will stretch it AWAY from the x axis you multiple a fractional constant (i.e. divide) by x (which is same as saying f(x divided by a)) this will stretch it TOWARDS the x axis C > 1 stretches it 0 < C < 1 compresses it
distributive law
a (b + c) = ab + ac AND a (b - c) = ab - ac
rule ""
a product to a power can be written as the individual factors to that power
rule ""
a quotient to a power can be written as the numerator and the denominator to that same power
simplifying two opposition binomials
a tip: factor out -1 to be able to simplify
acute angle
angle between 0 i 90
obtuse
angle between 90 i 180
what is the range in this graph?
answer = f(x) greater/equal to -2
example of simplifying exponents (ojo have to do top first if possible)
answer = k5
e.g. permutation: A teacher is making a multiple choice quiz. She wants to give each student the same questions, but have each student's questions appear in a different order. If there are twenty-seven students in the class, what is the least number of questions the quiz must contain?
answer: If there were two questions on the quiz, we could prepare two quizzes with the questions in different order -- 2•1 = 2. If there were three questions, we could get 3•2•1 = 6 different orders. If there were four questions, we could get 4•3•2•1 = 24 different orders -- not quite enough for the class of 27 students. If there were five questions, we could get 5•4•3•2•1 = 120 different orders. The teacher will need at least 5 questions on the quiz.
average rate =
average miles per hour = total
interest problems
basic formula to find interest = I = Prt [i.e. I AM PRT - usec mexico] where I = interest, P = principal (amount you started with), r= rate (e.g. 4% = .o4) and t is the time (in years --eg. 18 months = 1.5)
how to multiply three binomials
basically do the first two, then multiply by the third
how to find the slope
change in y/change in x
how to determine if a set of numbers is a function? given an x, we get only and exactly one y.
definition
arc = a portion of a circumference of a circle
e.g. AB here
combined work formula = product of individual times to do the work over addition of total times between the two
e.g. If pipe 2 can fill a tank in 3 hours, and pipe B can fill a tank in 2 hours, how long does it take them to fill 2/3 of a tank
exterior angle = the supplementary angle outside the triangle along a
e.g. d
in ""
each angle = supplementary (i.e. adds to 180) with its adjacent angles
[prob skip... too hard] compound interest
formula = [main thing = percent rate over # of compounds to the power of times x the compounds
what is a function?
how to determine if a set of numbers is a function? given an x, we get only and exactly one y. [ojo: x can point to the same y though!]
Note: The exponent only refers to the x NOT the whole thing unless there's a parenthesis
i.e. 3x squared = 3 (x squared) but (3x) squared = 3x times 3x because in PEMDAS you ALWAYS do power before multiplication
vertical angles
in this a and c, i b and d = vertical angles will always be same angle
transversal = a line that intersects with two parrallell lines
in this case (of a transversal), all acute angles are the same all obtuse angles are the same and all acute angles are supplementary with any obtuse angel
Example of solving hard distance problem
see two sections below for full answer: basically subtract distance you have from total distance then solve average for what remains...
like terms
like terms =have the same variables, e.g. 5z squared i z squared
the ratio of the areas of two similar triangles is the square of the ration of the corresponding lengths
maybe too complicated?
how to find weighted average/mean...
number x percent + number x percent / total amount of numbers
simplifying polynomials with exponent examples
ojo remember: you add and subtract the exponents (top i bottom), but divided or multiply the # e.g.
if you're given the hypotenuse and one of the legs in a 30,60, 90 triangle, you can find the other leg using the ratio 1, [squareroot of 3], 2
ojo the weird square root leg is just the other leg's length times [square root of 3]
tip on above
ojo: this means that negative exponent in numerator gets turn positive if put in the denominator; and it means that a negative base in the denominator becomes positive if moved to the denominator
A fractional exponent = a root
only worry about square roots on GRE!
INTRANET: things that I need to study/practice more =
prime factorization, definition of positive divisor, prime number up through 100, ratios, three part ratios [answer = make middle number have a common denominator w both], simplying square roots, solving for ratios when there's an addition or subtraction, *** simplifying algebraic expression, rules of exponents, exponents and parentheses, minimum and maximum value in a function (easy plug in 0), definition of complementary and supplementary angles, finding slope, def of ver tical angles, formula for surface area of a cylinder, interquartile range (ojo median of all number BELOW median of set!), probability question, permutation and combination, slope of a triangle, 4 quadrants!, 30-60-90 triangle areas, have to practice picking numbers/when to realize I can pick numbers!, i need to know 45, 45, 90 angles in a triangle, slopes, combined work formula, quadratic equations (difficult ones; must practice more!), solving circle areas, diameters, etc. practice tests: http://www.majortests.com/gre/problem_solving.php
how to find probability of random people winning prizes
question=
slope-intercept form
the "slope-intercept" form: y = mx + b [will always be a straight line] b= the y-intercept (i.e. point where line hits the y axis) *** ojo when solving for this make sure there is no constant (other than 1) next to y!
Pythagorean theorem
the Pythagorean theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. a squared plus b squared = c 2 (if c = hypotenuse)
rule ""
to raise a power by another power, multiply the exponents
A negative exponent means a reciprocal
to simplify, you take the reciprocal of the base and change the sign of the exponent negative exponent = one over the number squared, etc. [without the negative sign!]
multiplying binomials
use the nmeumonic FOIL: first; outer; inner; last
rule ""
when the bases are the same in a product (i.e. two number multiplied by eachother) add the exponents
rule ""
when the bases are the same in a quotient (i.e. two numbers divided by eachother), subtract the exponent
when given a system of linear equations, if you do the solution it will show the point where the two lines meet
you substitute, which in practice just means setting the functions as equal to each other then solving for x
Dividing exponents with same base
you subtract the exponent
How to find greatest common divisor?
"" except you only multiple primes that are common in both. OJO use prime to LEAST high power: e.g.
ETS summary of key things
Arithmetic topics include properties and types of integers, such as divisibility, factorization, prime numbers, remainders and odd and even integers; arithmetic operations, exponents and roots; and concepts such as estimation, percent, ratio, rate, absolute value, the number line, decimal representation and sequences of numbers. Algebra topics include operations with exponents; factoring and simplifying algebraic expressions; relations, functions, equations and inequalities; solving linear and quadratic equations and inequalities; solving simultaneous equations and inequalities; setting up equations to solve word problems; and coordinate geometry, including graphs of functions, equations and inequalities, intercepts and slopes of lines. Geometry topics include parallel and perpendicular lines, circles, triangles — including isosceles, equilateral and 30°-60°-90° triangles — quadrilaterals, other polygons, congruent and similar figures, three-dimensional figures, area, perimeter, volume, the Pythagorean theorem and angle measurement in degrees. The ability to construct proofs is not tested. Data analysis topics include basic descriptive statistics, such as mean, median, mode, range, standard deviation, interquartile range, quartiles and percentiles; interpretation of data in tables and graphs, such as line graphs, bar graphs, circle graphs, boxplots, scatterplots and frequency distributions; elementary probability, such as probabilities of compound events and independent events; conditional probability; random variables and probability distributions, including normal distributions; and counting methods, such as combinations, permutations and Venn diagrams. These topics are typically taught in high school algebra courses or introductory statistics courses. Inferential statistics is not tested
Simplifying factions with exponents example
Basically subtract/add the units w exponents on in the numerator [any other numbers simplify on their own] ... e.g.
How to solve for ratio
Bella's grade consists of 5 tests and 1 exam; if the exam counts for double that of a each test, then what fraction of the final grade is determined by the exam?
How to simplify a fraction to get x/y, in a ratio problem e.g.
Multiply by reciprocal
How to solve for ratio when answer = in fractions
Multiply each side by lowest common denominator e.g.
Is 1 a prime number?
NO! The integer 1 is not a prime number
Greatest common divisor/factor
The greatest common divisor (or greatest common factor) of two nonzero integers a and b is the greatest positive integer that is a divisor of both a and b. For example, the greatest common divisor of 30 and 75 is 15. This is because the positive divisors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30, and the positive divisors of 75 are 1, 3, 5, 15, 25, and 75. Thus, the common positive divisors of 30 and 75 are 1, 3, 5, and 15, and the greatest of these is 15.
What is the only even primer number?
The integer 2 is the only prime number that is even.
Integer
The integers are the numbers 1, 2, 3, and so on, together with their negatives, −1, −2, −3,..., and 0. Thus, the set of integers is {...., −3, −2, −1, 0, 1, 2, 3,...}.
Least common multiple
The least common multiple of two nonzero integers a and b is the least positive integer that is a multiple of both a and b. For example, the least common multiple of 30 and 75 is 150. This is because the positive multiples of 30 are 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, etc., and the positive multiples of 75 are 75, 150, 225, 300, 375, 450, etc. Thus, the common positive multiples of 30 and 75 are 150, 300, 450, etc., and the least of these is 150.
Ten squared, or to the 3 or 4, etc. power =
= 1 with that number of zeros... e.g. 10 to the 4th power = 10,000
Irrational numbers
=Decimals that don't terminate or repeat
Rational Number
A Rational Number is a real number that can be written as a simple fraction (i.e. as a ratio). Most numbers we use in everyday life are Rational Numbers. Example: 1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)
PEMDAS
A common technique for remembering the order of operations is the abbreviation "PEMDAS", which is turned into the phrase "Please Excuse My Dear Aunt Sally". It stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". This tells you the ranks of the operations: Parentheses outrank exponents, which outrank multiplication and division (but multiplication and division are at the same rank), and these two outrank addition and subtraction (which are together on the bottom rank). When you have a bunch of operations of the same rank, you just operate from left to right. For instance, 15 ÷ 3 × 4 is not 15 ÷ 12, but is rather 5 × 4, because, going from left to right, you get to the division first.
Note: negative in denominator = same as negative in numerator
A fraction with a negative sign in either the numerator or denominator can be written with the negative sign in front of the fraction;
Note: decimals with patterns = rational ONLY if pattern terminates
A number is rational if and only if it has an eventually repeating decimal representation. So a number like 4.212121... 4.212121... or 4.999212121... 4.999212121... is rational, while the number 4.212112111211112... 4.212112111211112... is not.
Prime Number
A prime number is an integer greater than 1 that has only two positive divisors: 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101
Another common/easy example
After spending 1/4 of paycheck, Ron has $150 dollars left: equation to solve this =
What are (non zero) numbers to negative power ?
Always 1 over that number...
What are (non zero) numbers to zero power equal to?
Always 1.
Composite Number
An integer greater than 1 that is not a prime number is called a composite number. The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18.
How to solve for ratio usually?
Cross multiply!
Absolute Value
Distance from zero
Proportion
Equation relating two ratios: e.g. 3/4 = 9/12
How to find least common multiple?
Find prime factors. Then multiply each factor the greatest number of times it occurs in either number. If the same factor occurs more than once in both numbers, you multiply the factor the greatest number of times it occurs. e.g. LCM for 30 i 45: E.g.: 30 = 2 × 3 × 5 45 = 3 × 3 × 5 2: one occurrence 3: two occurrences 5: one occurrence ** 2 × 3 × 3 × 5 = 90. 90 = Lowest common multipler?
Example of when simplify when you have various radical on the top and bottoms
In this case, you should simplify top part, then multiple/divide with bottom
How to solve for ratio example
Lin finishes first half of exam in 1/3 the time it takes her to finish the second half. If the whole exam takes 60 minutes, then how long does she spend on first half of exam?
How to solve for ratio example: when given two ratios plus a change in quantity and have to solve.
Method = new ratio = #x (plus or minus change)/#x [the numbers here are the original ratio] then you solve for x [but OJO you have to divide In pet store, the ratio of the number of puppies to kittens is 4:7. When 7 more puppies are received, the ration of the number of puppies to kittens changes to 5:7. How many puppies does the pet store now have?
Do you do exponent with or without negative sign?
Multiple first; then add negative.
Divide both sides of the equation to find x
The "undo" of multiplication is division. If something is multiplied on the x, you undo it by dividing both sides (that is, dividing each term on both sides) of the equation by whatever is multiplied on the x: Solve 2x = 5 Since the x is multiplied by 2, I need to divide both sides by 2: 2x/2 = 5/2, so x = 5/2 Then the solution is x = 5/2 or x = 2.5
Ratio
The ratio of one quantity to another is a way to express their relative sizes, often in the form of a fraction, where the first quantity is the numerator and the second quantity is the denominator. E.g 2/3
Real numbers
The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. The set of real numbers can be represented by a number line called the real number line.
How add fractions with different denominators?
To add two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator. For example, to add the fractions and 1/3 and - 2/5 use the common denominator 15:
Online summary of key things
To adequately prepare for the GRE quantitative reasoning portion of the test, it's important that you memorize a few basic equations, particularly for the geometry section. Know how to calculate the area of basic shapes, including triangle, square, rectangle, and parallelogram. Also memorize equations for the area and circumference of a circle, and volume equations for basic shapes. Don't forget to look at the side ratios of all triangles. https://ironline.american.edu/cracking-the-gre-quantitative-reasoning/
How to divide fractions?
To divide one fraction by another, first invert the second fraction—that is, find its reciprocal—then multiply the first fraction by the inverted fraction.
Quotient/Remainder
When 19 is divided by 7, the result is greater than 2, since (2)(7) < 19, but less than 3, since 19 < (3)(7). Because 19 is 5 more than (2)(7), we say that the result of 19 divided by 7 is the quotient 2 with remainder 5, or simply "2 remainder 5." When an integer a is divided by an integer b, where b is a divisor of a, the result is always a divisor of a. For example, when 60 is divided by 6 (one of its divisors), the result is 10, which is another divisor of 60. If b is not a divisor of a, then the result can be viewed in three different ways. The result can be viewed as a fraction or as a decimal, both of which are discussed later, or the result can be viewed as a quotient with a remainder, where both are integers. Each view is useful, depending on the context. Fractions and decimals are useful when the result must be viewed as a sin- gle number, while quotients with remainders are useful for describing the result in terms of integers only.
Factor/Divisor
When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product. For example, (2)(3)(10) = 60, so 2, 3, and 10 are factors of 60.
What is 1 a factor and multiple of?
YES! 1 is a factor of every integer; 1 is not a multiple of any integer except 1 and −1.
Can a remainder be zero?
YES! 24 divided by 4 is 6 remainder 0, since the greatest multiple of 4 that's less than or equal to 24 is 24 itself, which is 0 less than 24. In general, the remainder is 0 if and only if a is divisible by b.
Can negative integers be factors of positive numbers?
Yes ! The positive factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60. The negatives of these integers are also factors of 60, since, for example, (−2)(−30) = 60.
How to solve in three-part ratios?
e.g. if they give you two different ratios, and want to see what the ratio between the 1st and 3rd number is: make the one unit that is common to both ratios the same number (lowest common denominator), then solve...
ojo greatest common divisor can be the number itself
e.g GCD of 8 and 40 = 8 e.g. GCD of 4 and 12 = 12
How to solve in three-part ratios?
e.g. If they give you middle # then... solve each in relation to the middle # : A clothing store A sells T-shirts in only three colors: red, blue and green. The colors are in the ratio of 3 to 4 to 5. If the store has 20 blue T-shirts, how many T-shirts does it have altogether? Solution: Step 1: Assign variables : Let x = red shirts y = green shirts Write the items in the ratios as fractions. red/blue green/blue Step 2: Solve the equation Cross Multiply both equations 3 × 20 = x × 4 60 = 4x x = 15 5 × 20 = y × 4 100 = 4y y = 25 The total number of shirts would be 15 + 25 + 20 = 60 Answer: There are 60 shirts.
intr anet: read this later, seems like a GOOD summary of basics for math
http://www.txstate.edu/slac/stad-test-prep/gre/quantitative.html