GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT word problem note: Track mean, median, range, distribution, and how operations change spread.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. NUMBER THEORY Definition Number Theory is concerned with the properties of numbers in general, and in particular integers.


GMAT number properties note: GMAT Number Types GMAT deals with only Real Numbers:. Integers, Fractions and Irrational Numbers.


GMAT number properties note: INTEGERS Definition Integers are defined as: all negative natural numbers, zero, and positive natural numbers. Note that integers do not include decimals or fractions - just whole numbers.


GMAT number properties note: Even and Odd Numbers An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder. An even number is an integer of the form, where is an integer.


GMAT number properties note: An odd number is an integer that is not evenly divisible by 2. An odd number is an integer of the form, where is an integer.


GMAT number properties note: Addition / Subtraction: even +/- even = even. Multiplication: even * even = even.


GMAT number properties note: Division of two integers can result into an even/odd integer or a fraction.


GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.


GMAT fractions, decimals, and percents note: 0 is neither positive nor negative integer (the only one of this kind). 0 is divisible by EVERY integer.


GMAT fractions, decimals, and percents note: IRRATIONAL NUMBERS Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals terminating, non-repeating decimals are non-rational, so they are called the "irrationals". Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers.


GMAT fractions, decimals, and percents note: POSITIVE AND NEGATIVE NUMBERS A positive number is a real number that is greater than zero. A negative number is a real number that is smaller than zero.


GMAT fractions, decimals, and percents note: Zero is not positive, nor negative. Multiplication: positive * positive = positive positive * negative = negative negative * negative = positive Division: positive / positive = positive positive / negative = negative negative / negative = positive 2√ π =.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Prime Numbers A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself.


GMAT number properties note: Otherwise a number is called a composite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1.


GMAT number properties note: A number is prime if it cannot be written as a product of two factors and, both of which are greater than 1: n = ab. There are infinitely many prime numbers.


GMAT number properties note: The only even prime number is 2, since any larger even number is divisible by 2. Also 2 is the smallest prime.


GMAT number properties note: All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2 and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form or, because all other numbers are divisible by 2 or 3.


GMAT number properties note: Any nonzero natural number can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.


GMAT number properties note: Prime factorization: every positive integer greater than 1 can be written as a product of one or more prime integers in a way which is unique.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Factors A divisor of an integer, also called a factor of, is an integer which evenly divides without leaving a remainder.


GMAT number properties note: In general, it is said is a factor of, for non-zero integers and, if there exists an integer such that. 1 (and -1) are divisors of every integer.


GMAT number properties note: Every integer is a divisor of itself. Every integer is a divisor of 0, except, by convention, 0 itself.


GMAT number properties note: Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd. A positive divisor of n which is different from n is called a proper divisor.


GMAT number properties note: An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.


GMAT number properties note: Any positive divisor of n is a product of prime divisors of n raised to some power. If a number equals the sum of its proper divisors, it is said to be a perfect number.


GMAT number properties note: The proper divisors of 6 are 1, 2, and 3: 1+2+3 = 6, hence 6 is a perfect number. There are some elementary rules:.


GMAT number properties note: If is a factor of and is a factor of, then is a factor of. In fact, is a factor of for all integers and.


GMAT number properties note: If is a factor of and is a factor of, then or. If is a factor of, and, then a is a factor of.


GMAT number properties note: If is a prime number and is a factor of.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Power of the common factors).


GMAT number properties note: Every common divisor of a and b is a divisor of gcd(a, b). A*b = gcd(a, b)*lcm(a, b) Lowest Common Multiple - LCM The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b.


GMAT number properties note: Since it is a multiple, it can be divided by a and b without a remainder. If either a or b is 0, so that there is no such positive integer, then lcm(a, b) is defined to be zero.


GMAT number properties note: To find the LCM, you will need to do prime-factorization. Then multiply all the factors (pick the highest power of the common factors).


GMAT number properties note: Perfect Square A perfect square, is an integer that can be written as the square of some other integer. There are some tips about the perfect square:.


GMAT number properties note: The number of distinct factors of a perfect square is ALWAYS ODD. The sum of distinct factors of a perfect square is ALWAYS ODD.


GMAT number properties note: A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. Perfect square always has even number of powers of prime factors.


GMAT number properties note: Divisibility Rules 2 - If the last digit is even, the number is divisible by 2. 3 - If the sum of the digits is divisible by 3, the number is also.


GMAT number properties note: 4 - If the last two digits form.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Note: factorial of negative numbers is undefined.


GMAT number properties note: Trailing z eros are a sequence of 0's in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow. The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer, can be determined with this formula: , where k must be chosen such that.


GMAT number properties note: How many zeros are in the end (after which no other digits follow) of? (denominator must be less than 32, is less) Hence, there are 7 zeros in the end of 32!


GMAT number properties note: The formula actually counts the number of factors 5 in n!, but since there are at least as many factors 2, this is equivalent to the number of factors 10, each of which gives one more trailing zero. Finding the number of powers of a prime number, in the.


GMAT number properties note: Till What is the power of 2 in 25!? Finding the power of non-prime in n!:.


GMAT number properties note: How many powers of 900 are in 50! Make the prime factorization of the number:, then find the powers of these prime numbers in the n!.


GMAT number properties note: Find the power of 5: n Factorials Factorial of a non-negative integer, denoted by n!, is the product of all positive in.


GMAT number properties note: Track mean, median, range, distribution, and how operations change spread. We need all the prime {2,3,5} to be represented twice in 900, 5 can provide us with only 6 pairs, thus there is 900 in the power of 6 in 50!.


GMAT number properties note: Consecutive Integers Consecutive integers are integers that follow one another, without skipping any integers. 7, 8, 9, and -2, -1, 0, 1, are consecutive integers.


GMAT number properties note: Sum of consecutive integers equals the mean multiplied by the number of terms,. Given consecutive integers,, (mean equals to the average of the first and last terms), so the sum equals to.


GMAT number properties note: If n is odd, the sum of consecutive integers is always divisible by n. Given, we have consecutive integers.


GMAT number properties note: The sum of 9+10+11 = 30, therefore, is divisible by 3. If n is even, the sum of consecutive integers is never divisible by n.


GMAT number properties note: The sum of 9+10+11+12 = 42, therefore, is not divisible by 4. The product of consecutive integers is always divisible by.


GMAT number properties note: Given consecutive integers:. The product of 3*4*5*6 is 360, which is divisible by 4!


GMAT number properties note: Evenly Spaced Set Evenly spaced set or an arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant.


GMAT fractions, decimals, and percents note: Watch divisibility, factors, remainders, parity, and prime structure. Middle term multiplied by number of terms.


GMAT fractions, decimals, and percents note: There are five terms in the set {1, 7, 13, 19, 25}, middle term is 13, so the sum is 13*5 = 65. FRACTIONS Definition Fractional numbers are ratios (divisions) of integers.


GMAT fractions, decimals, and percents note: In other words, a fraction is formed by dividing one integer by another integer. Set of Fraction is a subset of the set of Rational Numbers.


GMAT fractions, decimals, and percents note: Fraction can be expressed in two forms fractional representation and decimal representation. Fractional representation Fractional representation is a way to express numbers that fall in between integers (note that integers can also be expressed in fractional form).


GMAT fractions, decimals, and percents note: A fraction expresses a part-to-whole relationship in terms of a numerator (the part) and a denominator (the whole). The number on top of the fraction is called numerator or nominator.


GMAT fractions, decimals, and percents note: The number on bottom of the fraction is called denominator. In the fraction,, 9 is the numerator and 7 is denominator.


GMAT fractions, decimals, and percents note: Fractions that have a value between 0 and 1 are called proper fraction. The numerator is always smaller than the denominator.


GMAT fractions, decimals, and percents note: Fractions that are greater than 1 are called improper fraction. Improper fraction can also be written as a mixed number.


GMAT fractions, decimals, and percents note: An integer combined with a proper fraction is called mixed number.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. Reciprocal Reciprocal for a number, denoted by or, is a number which when multiplied by yields.


GMAT fractions, decimals, and percents note: The reciprocal of a fraction is. To get the reciprocal of a number, divide 1 by the number.


GMAT fractions, decimals, and percents note: Operation on Fractions Adding/Subtracting fractions:. To add/subtract fractions with the same denominator, add the numerators and place that sum over the common denominator.


GMAT fractions, decimals, and percents note: To add/subtract fractions with the different denominator, find the Least Common Denominator (LCD) of the fractions, rename the fractions to have the LCD and add/subtract the numerators of the fractions Multiplying fractions:. To multiply fractions just place the product of the numerators over the product of the denominators.


GMAT fractions, decimals, and percents note: Change the divisor into its reciprocal and then multiply. Given, take the reciprocal of.


GMAT fractions, decimals, and percents note: Decimal Representation The decimals has ten as its base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating Reduced fraction (meaning that fraction is already reduced to its lowest term) can be expressed as terminating decimal if and only (denominator) is of the form, where and are non-negative integers.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. 2:, the number is of length so we have added two nines.


GMAT fractions, decimals, and percents note: Reducing it to lowest terms:. To convert a mixed-recurring decimal to fraction: 1.


GMAT fractions, decimals, and percents note: Write down the number consisting with non-repeating digits and r epeating digits. Subtract non-repeating number from above.


GMAT fractions, decimals, and percents note: Divide 1-2 by the number with 9's and 0's: for every repeating digit write down a 9, and for every non- repeating digit write down a zero after 9's. The number consisting with non-repeating digits and repeating digits is 2512.


GMAT fractions, decimals, and percents note: Subtract 25 (non-repeating number) from above: 2512-25 = 2487. Divide 2487 by 9900 (two 9's as there are two digits in 12 and 2 zeros as there are two digits in 25): 2487/9900 = 829/3300.


GMAT fractions, decimals, and percents note: Rounding Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep.


GMAT fractions, decimals, and percents note: If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep. 3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.


GMAT fractions, decimals, and percents note: 3485 rounded to the nearest thousandth = 5.349, since the dropped.


GMAT word problem note: Watch divisibility, factors, remainders, parity, and prime structure. EXPONENTS Exponents are a "shortcut" method of showing a number that was multiplied by itself several times.


GMAT word problem note: For instance, number multiplied times can be written as, where represents the base, the number that is multiplied by itself times and represents the exponent. The exponent indicates how many times to multiple the base,, by itself.


GMAT word problem note: Any nonzero number to the power of 0 is 1. Any number to the power 1 is itself.


GMAT word problem note: If the exponent is positive, the power of zero is zero:, where. If the exponent is negative, the power of zero (, where) is undefined, because division by zero is implied.


GMAT word problem note: The integer powers of one are one. If n is an even integer, then.


GMAT word problem note: If n is an odd integer, then. Operations involving the same exponents:.


GMAT word problem note: Keep the exponent, multiply or divide the bases and not Operations involving the same bases:. Keep the base, add or subtract the exponent (add for multiplication, subtract for division) Fraction as power:.


GMAT word problem note: When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions. A n an a n n a = 1a0 = 150 (−3 = 1)0 = aa1 = 00n n > 0 0n n < 0.


GMAT sequences and patterns note: Watch divisibility, factors, remainders, parity, and prime structure. For instance, the two possible solutions are and.


GMAT sequences and patterns note: When solving equations with odd exponents, we'll have only one solution. For instance for, solution is and for, solution is.


GMAT sequences and patterns note: Exponents and divisibility: is ALWAYS divisible by. Is divisible by if is odd, and not divisible by a+b if n is even.


GMAT sequences and patterns note: LAST DIGIT OF A PRODUCT Last digits of a product of integers are last digits of the product of last digits of these integers. The last digit of 85945*89*58307 = 5*9*7 = 45*7 = 35 = 5?


GMAT sequences and patterns note: LAST DIGIT OF A POWER Determining the last digit of: 1. Last digit of is the same as that of.


GMAT sequences and patterns note: Determine the cyclicity number of. Find the remainder when divided by the cyclisity.


GMAT sequences and patterns note: When, then last digit of is the same as that of and when, then last digit of is the same as that of, where is the cyclisity number. Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base.


GMAT sequences and patterns note: Integers ending with 2, 3, 7 and 8 have a cyclicity of 4. Integers ending with 4 (eg.) have a cyclisity of 2.


GMAT sequences and patterns note: When n is odd will end with 4 and when n is even will end with 6. Integers ending with 9 (eg.) have a cyclisity of 2.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. , when, then and when, then When the GMAT provides the square root sign for an even root, such as or, then the only accepted answer is the positive root.


GMAT fractions, decimals, and percents note: In contrast, the equation has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.


GMAT fractions, decimals, and percents note: Odd roots will have the same sign as the base of the root. For GMAT it's good to memorize following values:.


GMAT fractions, decimals, and percents note: PERCENTS A percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, "%", or the abbreviation "pct".


GMAT fractions, decimals, and percents note: Since a percent is an amount per 100, percents can be represented as fractions with a denominator of 100. A percent can be represented as a decimal.


GMAT fractions, decimals, and percents note: The following relationship characterizes how percents and decimals interact. What is 2% represented as a decimal?


GMAT fractions, decimals, and percents note: Percent Form / 100 = Decimal Form: 2%/100 = 0.02 Percent change General formula for percent increase or decrease, (percent change):. A company received $2 million in royalties on the first $10 million in sales and then $8 million in royalties on the next $100 million in sales.


GMAT word problem note: Percent problems usually relate part, whole, rate, and percent change. So the royalties decreased by 60%.


GMAT word problem note: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months? If C = 1, meaning that interest is compounded once a year, then the formula will be: , where time is number of years.


GMAT word problem note: If $20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 year? ORDER OF OPERATIONS - PEMDAS Perform the operations inside a Parenthesis first (absolute value signs also fall into this category), then Exponents, then Multiplication and Division, from left to right, then Addition and Subtraction, from left to right - PEMDAS.


GMAT word problem note: An exclamation mark indicates that one should compute the factorial of the term immediately to its left, before computing any of the lower-precedence operations, unless grouping symbols dictate otherwise. A factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power.


GMAT word problem note: If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus: and not.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. N the exponent applies to the entire power.


GMAT word problem note: If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus: and not Percent = ∗ 100 = Change Original = ∗ 100 = 60 Balance(final) = principal∗ (1 + interest C)time∗C Balance(final) = principal∗ (1 + interest)time Balance = 20, 000 ∗ (1 + = 20, 000 ∗ (1.03 = $25, 335.40.12 ()!


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Https://gmatclub.com/forum/root-80-root-139868.html Medium: 2.


GMAT number properties note: Https://gmatclub.com/forum/topic-143744.html 6. Https://gmatclub.com/forum/what-is-63908.html Hard.


GMAT quant note: Identify the concept first, then apply the matching rule or setup.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. GMAT Club Forum Remainders https://gmatclub.com/forum/remainders-144665.html REMAINDERS created by:.


GMAT number properties note: This formula can also be written as. Properties When is divided by the remainder is 0 if is a multiple of.


GMAT number properties note: When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. X y q r y = divisor∗ quotient+ remainder = xq+ r 0 ≤ r < x 15 = 6 ∗ 2 + 3 0 ≤ r < x = q+y x r x y x y x 12 = 3 ∗ 4 + 0 7 = 11 ∗ 0 + 7 y x x− 1 y.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If a number is divided by 10, its remainder is the last digit of that number.


GMAT number properties note: If it is divided by 100 then the remainder is the last two digits and so on. 4 When positive integer n is dived by 18 the remainder is 7:.


GMAT number properties note: Now, since the first term (18q) is divisible by 6, then the remainder will only be from the second term, which is 7. 7 divided by 6 yields the remainder of 1.


GMAT number properties note: 5 There are several algebraic ways to solve this question, but the easiest way is as follows: since we cannot have two correct answers just pick a prime greater than 3, square it and see what would be the remainder upon division of it by 12. The remainder upon division 25 by 12 is 1.


GMAT statistics note: Watch divisibility, factors, remainders, parity, and prime structure. (1) x divided by 100 has a remainder of 30.


GMAT statistics note: (2) x divided by 110 has a remainder of 30. We have that: 30, 130, 230, (2) x divided by 110 has a remainder of 30.


GMAT statistics note: We have that: 30, 140, 250, Answer:. (1) n is multiple of 5 (2) n is a multiple of 12 (1) n is multiple of 5.


GMAT statistics note: If n = 5, then n yields the remainder of 5 when divided by 6 but if n = 10, then n yields the remainder of 4 when divided by 6. We already have two different answers, which means that this statement is not sufficient.


GMAT statistics note: Every multiple of 12 is also a multiple of 6, thus n divided by 6 yields the remainder of 0. 45 divided by yields the remainder of can always be expressed as: (which is the same as), where is the quotient and is the remainder.


GMAT statistics note: Given that, so according to the above, which means that must be a multiple of 3. Only option E offers answer which.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. If n is greater than 30, what is the remainder that n leaves after division by 30?


GMAT number properties note: 28 Positive integer n leaves a remainder of 4 after division by 6:. Thus n could Positive integer n leaves a remainder of 3 after division by 5:.


GMAT number properties note: Thus n could There is a way to derive general formula for (of a type, where is a divisor and is a remainder) based on above two statements:. Divisor would be the least common multiple of above two divisors 5 and 6, hence Remainder would be the first common integer in above two patterns, hence.


GMAT number properties note: Therefore general formula based on both statements is. Hence the remainder when positive integer n is divided by 30 is 28.


GMAT number properties note: (1) When 3x is divided by 2, there is a remainder. (2) x = 4y + 1, where y is an integer.


GMAT number properties note: , notice that we have the product of three consecutive integers. Now, notice that if, then and are consecutive even integers, thus one of them will also be divisible by 4, which will make divisible by 2*4 = 8 (basically if then will be divisible by 8*3 = 24).


GMAT algebra note: Watch divisibility, factors, remainders, parity, and prime structure. 0 , now if we expand this expression all terms but the last one will have in them, thus will leave no remainder upon division by 13, the last term will be.


GMAT algebra note: Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12:.


GMAT algebra note: And when x is divided by 7, the remainder is 4. When positive integer y is divided by 5, the remainder is 3.


GMAT algebra note: And when y is divided by 7, the remainder is 4. If x > y, which of the following must be a factor of x - y?


GMAT algebra note: 35 When the positive integer x is divided by 5 and 7, the remainder is 3 and 4, respectively:. Divisor will be the least common multiple of above two divisors 5 and 7, hence 35.


GMAT algebra note: Remainder will be the first common integer in above two patterns, hence 18. So, to satisfy The same for y (as the same info is given about y):.


GMAT algebra note: Thus must be a multiple of 35.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Since given that, then --> So,.


GMAT number properties note: Now, if then and if then , so in any case --> --> in order to be multiple of 4 must be multiple of 16 but as we see it's not, so is not multiple of 4. (2) x – y = 3 --> --> but not sufficient to say whether it's multiple of 4.


GMAT number properties note: Is the remainder of bigger than the remainder of? (2) The remainder of is 2 First of all any positive integer can yield only three remainders upon division by 3: 0, 1, or Since, the sum of the digits of and is always 1 then the remainders of and are only dependent on the value of the number added to and.


GMAT number properties note: There are 3 cases: the digits of and will be 1 more than a multiple of 3). Of the digits of and will be 2 more than a multiple of 3).


GMAT number properties note: Of the digits of and will be a multiple of 3). P = 8 q+ 5 = +x2 y2 y = odd = 2 k+ 1 8q+ 5 = + (2 k+ 1x2)2 = 8 q+ 4 − 4 − 4 k = 4(2 q+ 1 − − k)x2 k2 k2 = 4(2 q+ 1 − − k)x2 k2 k = odd 2q+ 1 − − k = even+ odd− odd− o.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. Means that the remainder of is 0.


GMAT number properties note: Now, the question asks whether the remainder of, which is 0, greater than the reminder of, which is 0, 1, or 2. Obviously it cannot be greater, it can be less than or equal to.


GMAT number properties note: So, the answer to the question is NO. Check more PS questions on remainders here n 3 2 n +n10m +n10m +m10n.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation.


GMAT algebra note: Algebra https://gmatclub.com/forum/algebra-101576.html Algebra 101 Scope Manipulation of various algebraic expressions Equations in 1 & more variables Dealing with non-linear equations Algebraic identities Notation & Assumptions In this document, lower case roman alphabets will be used to denote variables such as a,b,c,x,y,z,w In general it is assumed that the GMAT will only deal with real numbers () or subsets of such as Integers (), rational numbers () etc Concept of variables A variable is a place holder, which can be used in mathematical expressions.


GMAT algebra note: To represent unknown quantities in known relationships. These are generalized relationships such as, which says for any number, if you square it and take the root, you get the absolute value back.


GMAT algebra note: So the variable acts like a true placeholder, which may be replaced by any number. Basic rules of manipulation A.


GMAT algebra note: When switching terms from one side to the other in an algebraic expression + becomes - and vice versa. When switching terms from one side to the other in an algebraic.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Degree of an expression The degree of an algebraic expression is defined as the highest power of the variables present in the expression.


GMAT algebra note: Bi-quadratic Egs: the degree is 1 the degree is 3 the degree of x is 3, degree of z is 5, degree of the expression is 5 Solving equations of degree 1:. LINEAR Degree 1 equations or linear equations are equations in one or more variable such that degree of each variable is one.


GMAT algebra note: One variable Such equations will always have a solution. General form is and solution is One equation in Two variables This is not enough to determine x and y uniquely.


GMAT algebra note: There can be infinitely many solutions. Two equations in Two variables If you have a linear equation in 2 variables, you need at least 2 equations to solve for both variables.


GMAT algebra note: If then there are infinite solutions. Any point satisfying one equation will always satisfy the second If then there is no such x and y which will satisfy both equations.


GMAT algebra note: No solution In all other cases, solving the equations is straight forward, multiply eq (2) by a/d and subtract from (1). More than two equations in Two variables Pick any 2 equations and try to solve them:.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Solving equations of degree 2:.


GMAT algebra note: QUADRATIC The general form of a quadratic equation is The equation has no solution if The equation has exactly one solution if This equation has 2 solutions given by if The sum of roots is The product of roots is If the roots are and, the equation can be written as A quick way to solve a quadratic, without the above formula is to factorize it:.


GMAT algebra note: Step 1> Divide throughout by coeff of x^2 to put it in the form Step 2> Sum of roots = -d and Product = e. Search for 2 numbers which satisfy this criteria, let them be f,g Step 3> The equation may be re-written as (x-f)(x-g) = 0.


GMAT algebra note: And the solutions are f,g Eg. The sum is -11 and the product is 30.


GMAT algebra note: So numbers are -5,-6 Solvingequations with DEGREE>2 You will never be asked to solved higher degree equations, except in some cases where using simple tricks these equations can either be factorized or be reduced to a lower degree or both. What you need to note is that an equation of degree n has at most n unique solutions.


GMAT algebra note: Factorization This is the easiest approach to solving higher degree equations. Though there is no general rule to do this, generally a knowledge of algebraic identities helps.


GMAT algebra note: The basic idea is that if you can write an equation in the form where each of A,B,C are algebraic expressions.


GMAT number properties note: Watch divisibility, factors, remainders, parity, and prime structure. So the solution is y = 1 or 2 or x^3 = 1 or 2 or x = 1 or Other tricks Sometimes we are given conditions such as the variables being integers which make the solutions much easier to find.


GMAT number properties note: When we know that the solutions are integral, often times solutions are easy to find using just brute force.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. ABSOLUTE VALUE (Modulus) Definition The absolute value (or modulus) of a real number x is x's numerical value without regard to its sign.


GMAT algebra note: How to approach equations with moduli It's not easy to manipulate with moduli in equations. There are two basic approaches that will help you out.


GMAT algebra note: Both of them are based on two ways of representing modulus as an algebraic expression. This approach might be helpful if an equation has × and /.


GMAT algebra note: 2) |x| equals x if x> = 0 or -x if x<0. It looks a bit complicated but it's very powerful in dealing with moduli and the most popular approach too (see below).


GMAT algebra note: General approach to solving equalities and inequalities with absolute value: 1. Open modulus and set conditions.


GMAT algebra note: To solve/open a modulus, you need to consider 2 situations to find all roots:. Positive (or rather non-negative) Negative |x| |3| = 3 | − 12| = 12 | − 1.3| = 1.3 |x| ≥ 0.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. We can also think about conditions like graphics.


GMAT algebra note: Is a key point in which the expression under modulus equals zero. All points right are the first condition and all points left are second condition.


GMAT algebra note: Check conditions for each solution: a) has to satisfy initial condition.. Otherwise, we would have to reject x = 5.


GMAT algebra note: B) has to satisfy initial condition.. Otherwise, we would have to reject x = -3.


GMAT algebra note: How many solutions does the equation have? There are 3 key points here: -8, -3, 4.


GMAT algebra note: So we have 4 conditions: a). We reject the solution because our condition is not satisfied (-1 is not less than -8) b).


GMAT algebra note: We reject the solution because our condition is not satisfied (-15 is not within (-8,-3) interval.) c). We reject the solution because our condition is not satisfied (9 is not within (-3,4) interval.) d).


GMAT algebra note: We reject the solution because our condition is not satisfied (-1 is not more than 4) (Optional) The following illustration may help you understand how to open modulus at different conditions.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. X e {, } and both solutions satisfy the condition.


GMAT word problem note: (Optional) The following illustration may help you understand how to open modulus at different conditions. At the same time, often there are shortcuts and tricks that allow you to solve absolute value problems in 10-20 sec.


GMAT word problem note: Thinking of inequality with modulus as a segment at the number line. What inequality represents this condition?


GMAT word problem note: Traditional 3-steps method is too time-consume technique. First of all we find length (9-1) = 8 and center (1+8/2 = 5) of the segment represented by 1<x<9.


GMAT word problem note: Now, let’s look at our options. Only B and D has 8/2 = 4 on the right side and D had left site 0 at x = 5.


GMAT word problem note: Converting inequalities with modulus into a range expression. In many cases, especially in DS problems, it helps avoid silly mistakes.


GMAT word problem note: |x+3|>3 is equal to x e (-inf,-6)&(0,+inf) III. Thinking about absolute values as the distance between points at the number line.


GMAT word problem note: We can think of absolute values here as the distance between points.


GMAT coordinate geometry note: Https://gmatclub.com/forum/4-272003.html Medium:.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. It is an ordered list of objects.


GMAT sequences and patterns note: It can be finite or infinite. The elements may repeat themselves more than once in the sequence, and their ordering is important unlike a set Arithmetic Progressions Definition It is a special type of sequence in which the difference between successive terms is constant.


GMAT sequences and patterns note: General Term is the ith term is the common difference is the first term Defining Properties Each of the following is necessary & sufficient for a sequence to be an AP:. Constant If you pick any 3 consecutive terms, the middle one is the mean of the other two For all i,j > k > = 1:.


GMAT sequences and patterns note: Summation The sum of an infinite AP can never be finite except if & The general sum of a n term AP with common difference d is given by The sum formula may be re-written as = + d = + (n− 1)dan an−1 a1 ai d a1 − = ai ai−1 −ai ak i−k −aj ak j−k = 0a1 d = 0 (2a+ (n− 1)d)n n∗ Avg(,) = ∗ (FirstTerm + LastTerm)a1 an n.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Constant If you pick any 3 consecutive terms, the middle one is the geometric mean of the other two For all i,j > k > = 1:.


GMAT sequences and patterns note: Track mean, median, range, distribution, and how operations change spread. Important Properties Of any three consecutive terms of a HP, the middle one is always the harmonic mean of the other two, where the harmonic mean (HM) is defined as:.


GMAT sequences and patterns note: Linkage Each progression provides us a definition of "mean":. OR For all non-negative real numbers:.


GMAT sequences and patterns note: AM > = GM > = HM In particular for 2 numbers:.


GMAT sequences and patterns note: The HM = (2*50*2)/(52) = 3.85 AM > GM > HM AM*HM = 100 = GM^2 Misc Notes A subsequence (any set of consequutive terms) of an AP is an AP A subsequence (any set of consequutive terms) of a GP is a GP A subsequence (any set of consequutive terms) of a HP is a HP If given an AP, and I pick out a subsequence from that AP, consisting of the terms such that are in AP then the new subsequence will also be an AP 1) = 2n-1 Pick out the subsequence of terms If given a GP, and I pick out a subsequence from that GP, consisting of the terms such that are in AP then the new subsequence will also be a GP a b HM(a,b) HM(a, b) = 2ab a+b a+b a1+..+an n ab− −√ (a1∗..


GMAT sequences and patterns note: ∗an) 1n 2ab a+b n a1 an ,,,...ai1 ai2 ai3 i1, i2, i3 = 1, d = 2a1 ,,,...a5 a10 a15 = 9a1 d = 10 ,,,...bi1 bi2 bi3 i1, i2, i3.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. Pick out the subsequence of terms The special sequence in which each term is the sum of previous two terms is known as the fibonacci sequence.


GMAT sequences and patterns note: It is neither an AP nor a GP. In a finite AP, the mean of all the terms is equal to the mean of the middle two terms if n is even and the middle term if n is odd.


GMAT sequences and patterns note: Solution We know that P(H) = P(T) = 0.5 This is just a finite GP, with first term = 0.5, n = 5 and ratio = 0.5. (1) a1 = 8 (2) a12 = 24 Solution (1) a1 = 8, does not tell us anything about the common difference, so impossible to say how many terms are greater than 24 (2) a12 = 24, and we know common difference is non-zero.


GMAT sequences and patterns note: So either all the terms below a12 are greater than 24 and the terms above it less than 24 or the other way around. In either case, there are exactly 11 terms either side of a12.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation. If T is the sum of the first 10 terms in the sequence then T is a)greater than 2 b)between 1 and 2 c)between 1/2 and 1 d)between 1/4 and 1/2 e)less than 1/4.


GMAT sequences and patterns note: Solution The sequence given has first term 1/2 and each subsequent term can be obtained by multiplying with -1/2. What is the sum of the first 15 terms of the arithmetic progression?


GMAT sequences and patterns note: 270 Solution Now we need the sum of first 15 terms, which is given by:.


GMAT sequences and patterns note: Focus on equations, expressions, variables, and step-by-step manipulation.


GMAT coordinate geometry note: Algebra Functions & Coordinate Geometry https://gmatclub.com/forum/algebra-functions-87652.html FUNCTIONS & COORDINATE GEOMETRY Definition Coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.


GMAT coordinate geometry note: The Coordinate Plane In coordinate geometry, points are placed on the "coordinate plane" as shown below. The coordinate plane is a two-dimensional surface on which we can plot points, lines and curves.


GMAT coordinate geometry note: It has two scales, called the x-axis and y-axis, at right angles to each other. The plural of axis is 'axes' (pronounced "AXE-ease").


GMAT coordinate geometry note: A point's location on the plane is given by two numbers, one that tells where it is on the x-axis and another which tells where it is on the y-axis. Together, they define a single, unique position on the plane.


GMAT coordinate geometry note: These are the coordinates of the point A, sometimes referred to as its.


GMAT coordinate geometry note: X axis The horizontal scale is called the x-axis and is usually drawn with the zero point in the middle. Values to the right are positive and those to the left are negative.


GMAT coordinate geometry note: Y axis The vertical scale is called the y-axis and is also usually drawn with the zero point in the middle. Values above the origin are positive and those below are negative.


GMAT coordinate geometry note: Origin The point where the two axes cross (at zero on both scales) is called the origin. Quadrants When the origin is in the center of the plane, they divide it into four areas called quadrants.


GMAT coordinate geometry note: The first quadrant, by convention, is the top right, and then they go around counter- clockwise. It is conventional to label them with numerals but we talk about them as "first, second, third, and fourth quadrant".


GMAT coordinate geometry note: Point (x,y) The coordinates are written as an "ordered pair". The letter P is simply the name of the point and is used to distinguish it from others.


GMAT coordinate geometry note: The two numbers in parentheses are the x and y coordinate of the point. The first number (x) specifies how far along the x (horizontal) axis the point is.


GMAT coordinate geometry note: The second is the y coordinate and specifies how far up or down the y axis to go. It is called an ordered pair because the order of the two numbe.


GMAT coordinate geometry note: Negative numbers mean to go left (x) or down (y). Distance between two points Given coordinates of two points, distance D between two points is given by: (where is the difference between the x-coordinates and is the difference between the y-coordinates of the points) As you can see, the distance formula on the plane is derived from the Pythagorean theorem.


GMAT coordinate geometry note: Above formula can be written in the following way for given two points and Vertical and horizontal lines If the line segment is exactly vertical or horizontal, the formula above will still work fine, but there is an easier way. For a horizontal line, its length is the difference between the x- coordinates.


GMAT coordinate geometry note: For a vertical line its length is the difference between the y-coordinates. Distance between the point A (x,y) and the origin As the one point is origin with coordinate O (0,0) the formula can be simplified to:.


GMAT coordinate geometry note: Find the distance between the point A (3,-1) and B (-1,2) Solution:. Substituting values in the equation we'll get D = d + dx2 y2− − − − − − − −√ dx dy (,)x1 y1 (,)x2 y2 D = (− + (−x2 x1)2 y2 y1)2− −− − − − − − − − − − − − − − − − − D = +x2 y2− − − − − −√ D = (− + (−x2 x1)2 y2 y1)2− −− − − − − − − − − − − − − − − − −√.


GMAT coordinate geometry note: Midpoint of a Line Segment A line segment on the coordinate plane is defined by two endpoints whose coordinates are known. The midpoint of this line is exactly halfway between these endpoints and it's location can be found using the Midpoint Theorem, which states:.


GMAT coordinate geometry note: The x-coordinate of the midpoint is the average of the x-coordinates of the two endpoints. Likewise, the y-coordinate is the average of the y-coordinates of the endpoints.


GMAT coordinate geometry note: Coordinates of the midpoint of the line segment AB, (and ) are and Lines in Coordinate Geometry In Euclidean geometry, a line is a straight curve. In coordinate geometry, lines in a Cartesian plane can be described algebraically by linear equations and linear functions.


GMAT coordinate geometry note: Every straight line in the plane can represented by a first degree equation with two variables.


GMAT coordinate geometry note: There are several approaches commonly used in coordinate geometry. It does not matter whether we are talking about a line, ray or line segment.


GMAT coordinate geometry note: In all cases any of the below methods will provide enough information to define the line exactly. The general form of the equation of a straight line is Where, and are arbitrary constants.


GMAT coordinate geometry note: This form includes all other forms as special cases. For an equation in this form the slope is and the y intercept is.


GMAT coordinate geometry note: Where: is the slope of the line. Is the y-intercept of the line.


GMAT coordinate geometry note: Is the independent variable of the function. Using two points In figure below, a line is defined by the two points A and B.


GMAT coordinate geometry note: By providing the coordinates of the two points, we can draw a line. No other line could pass through both these points and so the line they define is unique.


GMAT coordinate geometry note: Ax+ by+ c = 0 a b c − a b − c b y = mx+ b m b x y.


GMAT algebra note: The equation of a straight line passing through points and is:. Find the equation of a line passing through the points A (17,4) and B (9,9).


GMAT algebra note: Substituting the values in equation we'll get: --> --> OR if we want to write the equation in the slope-intercept form: 4. Using one point and the slope Sometimes on the GMAT you will be given a point on the line and its slope and from this information you will need to find the equation or check if this line goes through another point.


GMAT algebra note: You can think of the slope as the direction of the line. So once you know that a line goes through a certain point, and which direction it is pointing, you have defined one unique line.


GMAT algebra note: In figure below, we see a line passing through the point A at (14,23). We also see that it's slope is +2 (which means it goes 2 up for every one across).


GMAT algebra note: With these two facts we can establish a unique line. (,)P1 x1 y1 (,)P2 x2 y2 y−y1 x−x1 −y1 y2 −x1 x2 y−y1 x−x1 −y1 y2 −x1 x2 = y−4 x−17 = y−4 x−17 8 8y− 32 = −5 x+ 85 8y+ 5x− 117 = 0 y = − x+5.


GMAT algebra note: The equation of a straight line that passes through a point with a slope m is:. Find the equation of a line passing through the point A (14,23) and the slope 2.


GMAT algebra note: Substituting the values in equation we'll get 4. The equation of a straight line whose x and y intercepts are a and b, respectively, is:.


GMAT algebra note: Find the equation of a line whose x intercept is 5 and y intercept is 2. Substituting the values in equation we'll get --> OR if we want to write the equation in the slope-intercept form:.


GMAT algebra note: Slope of a Line The slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline.


GMAT algebra note: The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two points on the line. (,)P1 x1 y1 y− = m(x−)y1 x1 y− = m(x−)y1 x1 y− 23 = 2(x− 14) y = 2 x− 5 + = 1x a y b + = 1x a y b + = 1x y 5y+ 2x− 10 = 0 y = − x+ 22.


GMAT algebra note: Given two points and on a line, the slope of the line is:. If the equation of the line is given in the Point-intercept form:, then is the slope.


GMAT algebra note: This form of a line's equation is called the slope-intercept form, because can be interpreted as the y-intercept of the line, the y-coordinate where the line intersects the y-axis. If the equation of the line is given in the General form:, then the slope is and the y intercept is.


GMAT algebra note: SLOPE DIRECTION The slope of a line can be positive, negative, zero or undefined. (,)x1 y1 (,)x2 y2 m m = −y2 y1 −x2 x1 y = mx+ b m b ax+ by+ c = 0 − a b − c b.


GMAT algebra note: Positive slope Here, y increases as x increases, so the line slopes upwards to the right. The slope will be a positive number.


GMAT algebra note: The line below has a slope of about +0.3, it goes up about 0.3 for every step of 1 along the x-axis. Negative slope Here, y decreases as x increases, so the line slopes downwards to the right.


GMAT algebra note: The slope will be a negative number. The line below has a slope of about -0.3, it goes down about 0.3 for every step of 1 along the x-axis.


GMAT algebra note: Zero slope Here, y does not change as x increases, so the line in exactly horizontal. The slope of any horizontal line is always zero.


GMAT algebra note: The line below goes neither up nor down as x increases, so its slope is zero. Undefined slope When the line is exactly vertical, it does not have a defined slope.


GMAT algebra note: The two x coordinates are the same, so the difference is zero. The slope calculation is then something like When you divide anything by zero the result has no meaning.


GMAT algebra note: The line above is exactly vertical, so it has no defined slope. If the slope of a line is negative, the line WILL intersect quadrants II and IV.


GMAT algebra note: X and Y intercepts of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I.


GMAT algebra note: If the slope is 1 the angle formed by the line is degrees. Given a point and slope, equation of a line can be found.


GMAT algebra note: The equation of a straight line that passes through a point with a slope is:. Vertical and horizontal lines A vertical line is parallel to the y-axis of the coordinate plane.


GMAT algebra note: All points on the line will have the same x-coordinate. A vertical line has no slope.


GMAT algebra note: Or put another way, for a vertical line the slope is undefined. The equation of a vertical line is:.


GMAT algebra note: Where: x is the coordinate of any point on the line. A is where the line crosses the x-axis (x intercept).


GMAT algebra note: Notice that the equation is independent of y. Any point on the vertical line satisfies the equation.


GMAT algebra note: A horizontal line is parallel to the x-axis of the coordinate plane. All points on the line will have the same y-coordinate.


GMAT algebra note: (,)x1 y1 m y− = m(x−)y1 x1 x = a.


GMAT algebra note: A horizontal line has a slope of zero. The equation of a horizontal line is:.


GMAT algebra note: Where: y is the coordinate of any point on the line. B is where the line crosses the y-axis (y intercept).


GMAT algebra note: Notice that the equation is independent of x. Any point on the horizontal line satisfies the equation.


GMAT algebra note: Parallel lines Parallel lines have the same slope. The slope can be found using any method that is convenient to you: y = b.


GMAT algebra note: From two given points on the line. From the equation of the line in slope-intercept form From the equation of the line in point-slope form The equation of a line through the point and parallel to line is:.


GMAT algebra note: Distance between two parallel lines and can be found by the formula:. One line is defined by two points at (5,5) and (25,15).


GMAT algebra note: The other is defined by an equation in slope-intercept form form y = 0.52x - 2.5. For the top line, the slope is found using the coordinates of the two points that define the line.


GMAT algebra note: For the lower line, the slope is taken directly from the formula. Recall that the slope intercept formula is y = mx + b, where m is the slope.


GMAT algebra note: So looking at the formula we see that the slope is 0.52. So, the top one has a slope of 0.5, the lower slope is 0.52, which are not equal.


GMAT algebra note: Therefore, the lines are not parallel. Define a line through a point C parallel to a line passes through the points A and B.


GMAT algebra note: (,)P1 x1 y1 ax+ by+ c = 0 a(x−) + b(y−) = 0x1 y1 y = mx+ b y = mx+ c D = |b−c| Slope = = 0.515−5.


GMAT algebra note: For the line to be parallel to AB it will have the same slope, and will pass through a given point, C(12,10). We therefore have enough information to define the line by it's equation in point-slope form form:.


GMAT algebra note: Perpendicular lines For one line to be perpendicular to another, the relationship between their slopes has to be negative reciprocal. In other words, the two lines are perpendicular if and only if the product of their slopes is.


GMAT algebra note: SlopeAB = = −0.5220−7 y = −0.52(x− 12) + 10 y = −0.52x + 16.24 m.


GMAT algebra note: The two lines and are perpendicular if The equation of a line passing through the point) and perpendicular to line is:. Are the two lines below perpendicular?


GMAT algebra note: To answer, we must find the slope of each line and then check to see if one slope is the negative reciprocal of the other or if their product equals to -1. X+ y+ = 0a1 b1 c1 x+ y+ = 0a2 b2 c2 + = 0a1a2 b1b2 (,P1 x1 y1 ax+ by+ c = 0 b(x−) − a(y−) = 0x1 y1 SlopeAB = = = 0.3585−19.


GMAT algebra note: If the lines are perpendicular, each will be the negative reciprocal of the other. It doesn't matter which line we start with, so we will pick AB:.


GMAT algebra note: Negative reciprocal of 0.358 is So, the slope of CD is -2.22, and the negative reciprocal of the slope of AB is -2.79. These are not the same, so the lines are not perpendicular, even though they may look as though they are.


GMAT algebra note: Define a line passing through the point E and perpendicular to a line passing through the points C and D on the graph above. The point E is on the y-axis and so is the y-intercept of the desired line.


GMAT algebra note: Once we know the slope of the line, we can express it using its equation in slope-intercept form y = mx+b, where m is the slope and b is the y-intercept. First find the slope of line CD:.


GMAT algebra note: The line we seek will have a slope which is the negative reciprocal of:. Since E is on the Y-axis, we know that the intercept is 10.


GMAT algebra note: Plugging these values into the line equation, the line we need is described by the equation This is one of the ways a line can be defined and so we have solved the problem. If we wanted to plot the line, we would find anoth.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. To find the intersection of two straight lines: 1.


GMAT algebra note: First we need their equations 2. Then, since at the point of intersection, the two equations will share a point and thus have the same values of x and y, we set the two equations equal to each other.


GMAT algebra note: This gives an equation that we can solve for x 3. We substitute the x value in one of the line equations (it doesn't matter which) and solve it for y.


GMAT algebra note: This gives us the x and y coordinates of the intersection. Find the point of intersection of two lines that have the following equations (in slope- intercept form):.


GMAT algebra note: At the point of intersection they will both have the same y-coordinate value, so we set the equations equal to each other:. This gives us an equation in one unknown (x) which we can solve:.


GMAT algebra note: To find y, simply set x equal to 10 in the equation of either line and solve for y:. Equation for a line (Either line will do) Set x equal to 10:.


GMAT coordinate geometry note: Intercept form): and (A vertical line) Solution:. When one of the lines is vertical, it has no defined slope.


GMAT coordinate geometry note: We find the intersection slightly differently. On the vertical line, all points on it have an x-coordinate of 12 (the definition of a vertical line), so we simply set x equal to 12 in the first equation and solve it for y.


GMAT coordinate geometry note: Equation for a line Set x equal to 12 So the intersection point is at (12,33). If both lines are vertical or horizontal, they are parallel and have no intersection Distance from a point to a line The distance from a point to a line is the shortest distance between them - the length of a perpendicular line segment from the line to the point.


GMAT coordinate geometry note: The distance from a point to a line is given by the formula:. When the line is horizontal the formula transforms to:.


GMAT coordinate geometry note: Where: is the y-coordinate of the given point P. Is the y-coordinate of any point on the given vertical line L.


GMAT coordinate geometry note: | | the vertical bars mean "absolute value" - make it positive even if it calculates to a negative. When the line is vertical the formula transforms to:.


GMAT coordinate geometry note: Where: is the x-coordinate of the given point P. Is the x-coordinate of any point on the given vertical line L.


GMAT geometry note: If the circle is centered at the origin (0, 0), then the equation simplifies to:. Number line A number line is a picture of a straight line on which every point corresponds to a real number and every real number to a point.


GMAT geometry note: On the GMAT we can often see such statement: is halfway between and on the number line. Remember this statement can ALWAYS be expressed as:.


GMAT geometry note: Also on the GMAT we can often see another statement:. The distance between and on the number line is the same as the distance between and.


GMAT geometry note: Parabola A parabola is the graph associated with a quadratic function, i.e. A function of the form + = x2 y2 r2 k m n = km+n p m p n |p− m| = |p − n|.


GMAT coordinate geometry note: Focus on equations, expressions, variables, and step-by-step manipulation. The general or standard form of a quadratic function is, or in function form,, where is the independent variable, is the dependent variable, and,, and are constants.


GMAT coordinate geometry note: The larger the absolute value of, the steeper (or thinner) the parabola is, since the value of y is increased more quickly. If is positive, the parabola opens upward, if negative, the parabola opens downward.


GMAT coordinate geometry note: The x-intercepts, if any, are also called the roots of the function. The x- intercepts are the solutions to the equation and can be calculated by the formula: and Expression is called discriminant:.


GMAT coordinate geometry note: If discriminant is positive parabola has two intercepts with x-axis. If discriminant is negative parabola has no intercepts with x-axis.


GMAT coordinate geometry note: If discriminant is zero parabola has one intercept with x-axis (tangent point). Given, the y-intercept is, as y intercept means the value of y when x = 0.


GMAT coordinate geometry note: The vertex represents the maximum (or minimum) value of the function, and is very important in calculus. Y = a + bx+ cx2 y = a + bx+ cx2 f(x) = a + bx+ cx2 x y a b c a a 0 = a + bx+ cx2 = x1 2a = x2 2a − 4acb2 y = a + bx+ cx2 c.


GMAT coordinate geometry note: The vertex of the parabola is located at point. Note: typically just is calculated and plugged in for x to find y.


GMAT coordinate geometry note: Hard: (−,b 2a c−)b2 4a −,b 2a.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. It deals primarily with the translation of word problems into equations.


GMAT word problem note: Discussions relating to specific types of word problems will be dealt with separately (see end of post). The Following Points Outline a General Approach to Word Problems: 1) Read the entire question carefully and get a feel for what is happening.


GMAT word problem note: Identify what kind of word problem you're up against. 2) Make a note of exactly what is being asked.


GMAT word problem note: 3) Simplify the problem - this is what is usually meant by 'translating the English to Math'. Sometimes a simple illustration makes the problem much easier to approach.


GMAT word problem note: 4) It is not always necessary to start from the first line. Invariably, you will find it easier to define what you have been asked for and then work backwards to get the information that is needed to obtain the answer.


GMAT word problem note: 5) Use variables (a, b, x, y, etc.) or numbers (100 in case of percentages, any common multiple in case of fractions, etc.) depending on the situation. Think for a moment and choose the best possible value that would help.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. 7) Once you have the equations written down it's time to do the math!


GMAT word problem note: This is usually quite simple. Be very careful so as not to make any silly mistakes in calculations.


GMAT word problem note: 8) Lastly, after solving, cross check to see that the answer you have obtained corresponds to what was asked. The makers of these GMAT questions love to trick students who don’t pay careful attention to what is being asked.


GMAT word problem note: My recommendation is to refer to them only in the initial phases of study. With practice, decoding a word problem should come naturally.


GMAT word problem note: If, on test day, you still have to try and remember what the math translations to some English term is, you haven’t practiced enough! Plus SUBTRACTION: decreased by.


GMAT word problem note: Subtracted from MULTIPLICATION: of. Increased/decreased by a factor of (this type can.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. The ‘how much is left’ construction Sometimes, the question will give you a total amount that is made up of a number of smaller amounts of unspecified sizes.


GMAT word problem note: In this case, just assign a variable to the unknown amounts and the remaining amount will be what is left after deducting this named amount from the total. A hundred-pound order of animal feed was filled by mixing products from Bins A, B and C, and that twice as much was added from Bin C as from Bin A.


GMAT word problem note: Then the amount from Bin C was "2a", and the amount taken from Bin B was the remaining portion of the hundred pounds: 100 – a – 2a. The length of a rectangular garden is 2 meters more than its width.


GMAT word problem note: Express its length in terms of its width. Key words: more than (implies addition).


GMAT word problem note: Is (implies equal to) Thus, t.


GMAT fractions, decimals, and percents note: Focus on equations, expressions, variables, and step-by-step manipulation. The ratio of the length of a rectangular garden to its width is 2.


GMAT fractions, decimals, and percents note: Express its length in terms of its width. Key words: ratio of (implies division).


GMAT fractions, decimals, and percents note: Is (implies equal to) Thus, the phrase ‘ratio of length to width is 2’ becomes:. The length of a rectangular garden surrounded by a walkway is twice its width.


GMAT fractions, decimals, and percents note: If difference between the length and width of just the rectangular garden is 10 meters, what will be the width of the walkway if just the garden has width 6 meters? Key words: and (implies addition).


GMAT fractions, decimals, and percents note: Twice (implies multiplication). Difference between (implies subtraction where order is important).


GMAT fractions, decimals, and percents note: Is, will be (imply equal to) Since this is a slightly more complicated problem, let us first define what we want. 'What will be the width of the walkway' implies that we should assign a variable for width of the walkway and find its value.


GMAT fractions, decimals, and percents note: Thus, let width of the walkway be ‘x’. Now, in order to find the width of walkway, we need to have some relation between the total length/width of the rectangular garden + walkway and the length/width of just the gard.


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Now let’s see what the question gives us.


GMAT algebra note: ‘Garden with width 6 meters’ translates to:. ‘Length of a rectangular garden surrounded by walkway is twice its width’ translates to:.


GMAT algebra note: Length of garden + length of walkway = 2*(width of garden + width of walkway) L = 2*W ‘Difference between the length and width of just the rectangular garden is 10 meters’ translates to:. Length of garden – width of garden = 10 (L – 2x) – (W – 2x) = 10 L – W = 10 Now, since we have two equations and two variables (L and W), we can find their values.


GMAT algebra note: With practice, writing out word problems in the form of equations will become second nature. How much you need to practice depends on your own individual ability.


GMAT algebra note: It could be 10 questions or it could be 100. But once you’re able to effortlessly translate word problems into equations, more than half your battle will already be won.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In case you are not familiar with translating word problems into equations please go through THIS POST FIRST.


GMAT word problem note: What is a ‘Work’ Word Problem? It involves a number of people or machines working together to complete a task.


GMAT word problem note: We are usually given individual rates of completion. We are asked to find out how long it would take if they work together.


GMAT word problem note: Sounds simple enough doesn’t it? There is just one simple concept you need to understand in order to solve any ‘work’ related word problem.


GMAT word problem note: The ‘Work’ Problem Concept STEP 1:. Calculate how much work each person/machine does in one unit of time (could be days, hours, minutes, etc).


GMAT word problem note: If we are given that A completes a certain amount of work in X hours, simply reciprocate the number of hours to get the per hour work. Thus in one hour, A would complete of the work.


GMAT word problem note: But what is the logic behind this? Assume we are given that Jack paints a wall in 5 hours.


GMAT word problem note: This means that in every hour, he completes a fraction of the work so that at the end of 5 hours, the fraction of work he has completed will becom.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Completed will be (1*1)/5 STEP 2:.


GMAT word problem note: Add up the amount of work done by each person/machine in that one unit of time. This would give us the total amount of work completed by both of them in one hour.


GMAT word problem note: Calculate total amount of time taken for work to be completed when all persons/machines are working together. The logic is similar to one we used in STEP 1, the only difference being that we use it in reverse order.


GMAT word problem note: This means that in one hour, A and B working together will complete of the work. Therefore, working together, they will complete the work in Z hours.


GMAT word problem note: DON'T go about these problems trying to remember some formula. Once you understand the logic underlying the above steps, you will have all the information you need to solve any ‘work’ related word problem.


GMAT word problem note: (You will see that the formula you might have come across can be very easily and logically deduced from this concept). Now, lets go through a few problems so that the above-mentioned concept becomes crystal clear.


GMAT word problem note: Lets start off with a simple one:. Jack can paint a wall in 3 hours.


GMAT word problem note: John can do the same job in 5 hours.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Working, independently X takes 12 hours to finish a certain work.


GMAT word problem note: He finishes 2/3 of the work. The rest of the work is finished by Y whose rate is 1/10 of X.


GMAT word problem note: In how much time does Y finish his work? Now the only reason this is trickier than the first problem is because the sequence of events are slightly more complicated.


GMAT word problem note: The concept however is the same. So if our understanding of the concept is clear, we should have no trouble at all dealing with this.


GMAT word problem note: ‘Working, independently X takes 12 hours to finish a certain work’ This statement tells us that in one hour, X will finish of the work. ‘He finishes 2/3 of the work’ This tells us that of the work still remains.


GMAT word problem note: ‘The rest of the work is finished by Y whose rate is (1/10) of X’ Y has to complete of the work. ‘Y's rate is (1/10) that of X‘.


GMAT word problem note: We have already calculated rate at which X works to be. Therefore, rate at which Y works is.


GMAT word problem note: ‘In how much time does Y finish his work?’ If Y completes of the work in 1 hour, then he will complete of the work in 40 hours. So as you can see, even though the question might have been a little difficult to follow at first reading, the solution was in fact quite simple.


GMAT word problem note: We didn’t use any new concepts. All we did was apply our knowledge of the concept we learnt earlier to.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Calculate the rate at which B works:.


GMAT word problem note: ‘B prints 5 pages a minute more than printer A’ This means that the difference between the amount of work B and A complete in one minute corresponds to 5 pages. So, let us calculate that difference.


GMAT word problem note: It will be ‘How many pages does the task contain?’ If of the job consists of 5 pages, then the 1 job will consist of pages. Machine A and Machine B are used to manufacture 660 sprockets.


GMAT word problem note: It takes machine A ten hours longer to produce 660 sprockets than machine B. Machine B produces 10% more sprockets per hour than machine A.


GMAT word problem note: How many sprockets per hour does machine A produce? The rate of A is sprockets per hour.


GMAT word problem note: The rate of B is sprockets per hour. We are told that B produces 10% more sprockets per hour than A, thus --> --> the rate of A is sprockets per hour.


GMAT word problem note: As you can see, the main reason the 'tough' problems are 'tough' is because they test a number of other concepts apart from just the ‘work’ concept. However, once you manage to form the equations, they are really not all that tough.


GMAT word problem note: And as far as the concept of ‘work’ word problems is concerned – it is always the same!


GMAT algebra note: Focus on equations, expressions, variables, and step-by-step manipulation. Age to form the equations, they are really not all that tough.


GMAT algebra note: And as far as the concept of ‘work’ word problems is concerned – it is always the same! T+10 t t+10 t t = 100 = 6660 t+10.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In case you are not familiar with translating word problems into equations please go through this post first: https://gmatclub.com/forum/word-problem...


GMAT word problem note: Html What is a ‘D/S/T’ Word Problem? Usually involve something/someone moving at a constant or average speed.


GMAT word problem note: Out of the three quantities (speed/distance/time), we are required to find one. Information regarding the other two will be provided in the question stem.


GMAT word problem note: Distance = Speed x Time I’m sure most of you are already familiar with the above formula (or some variant of it). But how many of you truly understand what it signifies?


GMAT word problem note: When you see a ‘D/S/T’ question, do you blindly start plugging values into the formula without really understanding the logic behind it? If then answer to that question is yes, then you would probably have noticed that your accuracy isn’t quite where you’d want it to be.


GMAT word problem note: My advice here, as usual, is to make sure you understand the concept behind the formula rather than just using it blindly. The Distance = Speed x Time formula is just a way of s.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Travel for 2 hours at that speed, you would have traveled 100 miles.


GMAT word problem note: 3 hours would be 150 miles, etc. If you were to double the speed, then you would have traveled 100 miles in the first hour and 200 miles at the end of the second hour.


GMAT word problem note: We can figure out any one of the components by knowing the other two. Similarly, if a friend visits you from 100 miles away and tells you that it took him 4 hours to reach, you will know that he AVERAGED 25 mph.


GMAT word problem note: All calculations depend on AVERAGE SPEED. Supposing your friend told you that he was stuck in traffic along the way and that he traveled at 50 mph whenever he could move.


GMAT word problem note: Therefore, although practically he never really traveled at 25 mph, you can see how the standstills due to traffic caused his average to reduce. Now, if you think about it, from the information given, you can actually tell how long he was driving and how long he was stuck due to traffic (assuming.


GMAT word problem note: What is false but what they never worry about in these problems. That he was either traveling at 50 mph or 0 mph).


GMAT word problem note: If he was traveling constantly at 50 mph, he should have reached in 2 hours.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. About how to use the formula.


GMAT word problem note: So the trick is to be able to understand EXACTLY what they are giving you and EXACTLY what it is that is missing, but you do that from thinking, not from the formula, because the formula only works for the COMPONENTS of any trip where you are going an average speed for a certain amount of time.


GMAT word problem note: ONCE the conditions deal with different speeds or different times, you have to look at each of those components and how they go together. And that can be very difficult if you are not methodical in how you think about the components and how they go together.


GMAT word problem note: The formula doesn't tell you which components you need to look at and how they go together. For that, you need to think, and the thinking is not always as easy or straightforward as it seems like it ought to be.


GMAT word problem note: In the case of your friend above, if we call the time he spent driving 50 mph, T1. Then the time he spent standing still is (4 - T1) hours, since the whole trip took 4 hours.


GMAT word problem note: So we have 100 miles = (50 mph x T1) + (0 mph x [4 - T1]) which is equivalent then to: 100 miles = 50 mph x T1 So, T1 will equal 2 hours. And, since the time he spent going zero is (4 - 2), it also turns out to be 2 hours.


GMAT word problem note: Sometimes the right answers will seem counter-intuitive, so it is really important to think ab.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In the next section we will learn how to do just that.


GMAT word problem note: Resolving the Components When you first start out with these problems, the best way to approach them is by organizing the data in a tabular form. Use a separate column each for distance, speed and time and a separate row for the different components involved (2 parts of a journey, different moving objects, etc.).


GMAT word problem note: The last row should represent total distance, total time and average speed for these values (although there might be no need to calculate these values if the question does not require them). Assign a variable for any unknown quantity.


GMAT word problem note: If there is more than one unknown quantity, do not blindly assign another variable to it. Look for ways in which you can express that quantity in terms of the quantities already present.


GMAT word problem note: Assign another variable to it only if this is not possible. In each row, the quantities of distance, speed and time will always satisfy d = s*t.


GMAT word problem note: The distance and time column can be added to give you the values of total distance and total time but you CANNOT add the speeds. If you drive 20 mph on one street, and 40 mph on another street, does that mean you averaged 60 mph?


GMAT word problem note: Once the table is ready, form the equations and solve for what has been asked!


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Make sure that the units for time and distance agree with the units for the rate.


GMAT word problem note: For instance, if they give you a rate of feet per second, then your time must be in seconds and your distance must be in feet. Sometimes they try to trick you by using the wrong units, and you have to catch this and convert to the correct units.


GMAT word problem note: A Few More Points to Note Motion in Same Direction (Overtaking):. The first thing that should strike you here is that at the time of overtaking, the distances traveled by both will be the same.


GMAT word problem note: Motion in Opposite Direction (Meeting):. The first thing that should strike you here is that if they start at the same time (which they usually do), then at the point at which they meet, the time will be the same.


GMAT word problem note: In addition, the total distance traveled by the two objects under consideration will be equal to the sum of their individual distances traveled. The key thing here is that the distance going and coming back is the same.


GMAT word problem note: All values in black have been given in the question stem. All values in blue have been calculated.


GMAT word problem note: To qualify for a race, you need to average 60 mph driving two laps around a 1.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In the first row, we are given the distance and the speed.


GMAT word problem note: Thus it is possible to calculate the time. Time(1) = Distance(1)/Speed(1) = 1/30 In the second row, we are given just the distance.


GMAT word problem note: Since we have to calculate speed, let us give it a variable 'x'. Now, by using the 'D/S/T' relationship, time can also be expressed in terms of 'x'.


GMAT word problem note: Time(2) = Distance(2)/Speed(2) = 1/x In the third row, we know that the total distance is 2 miles (by taking the sum of the distances in row 1 and 2) and that the average speed should be 60 mph. Thus we can calculate the total time that the two laps should take.


GMAT word problem note: Time(3) = Distance(3)/Speed(3) = 2/60 = 1/30 Now, we know that the total time should be the sum of the times in row 1 and 2. Thus we can form the following equation:.


GMAT word problem note: Time(3) = Time(1) + Time(2) ---> 1/30 = 1/30 + 1/x From this, it becomes clear that '1/x' must be 0. Since 'x' is the reciprocal of 0, which does not exist, there can be no speed for which the average can be made up in the second lap.


GMAT word problem note: An executive drove from home at an average speed of 30 mph to an airport where a helicopter was waiting. The executive boarded the helicopter and flew to the corporate offices at an average speed of 60 mph.


GMAT word problem note: The entire distance was 150 miles. The entire trip took three hours.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. Thus, the distance he spent driving will be '150 - x' Now, in the first row, we have the distance in terms of 'x' and we have been given the speed.


GMAT word problem note: Thus we can calculate the time he spent driving in terms of 'x'. Time(1) = Distance(1)/Speed(1) = (150 - x)/30 Similarly, in the second row, we again have the distance in terms of 'x' and we have been given the speed.


GMAT word problem note: Thus we can calculate the time he spent flying in terms of 'x'. Time(2) = Distance(2)/Speed(2) = x/60 Now, notice that we have both the times in terms of 'x'.


GMAT word problem note: Also, we know the total time for the trip. Thus, summing the individual times spent driving and flying and equating it to the total time, we can solve for 'x'.


GMAT word problem note: Time(1) + Time(2) = Time(3) --> (150 - x)/30 + x/60 = 3 --> x = 120 miles Answer: 120 miles Note:. In this problem, we did not calculate average speed for row 3 since we did not need it.


GMAT word problem note: Remember not to waste time in useless calculations! A passenger train leaves the train depot 2 hours after a freight train left the same depot.


GMAT word problem note: The freight train is traveling 20 mph slower than the passenger train. Find the speed of the passenger train, if it overtakes the freight train in three hours.


GMAT word problem note: Since this is an 'overtaking' problem, the first thing that should strike us is that the dist.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. From the second row, we can form the following equation:.


GMAT word problem note: Distance(2) = Speed(2) * Time(2) = (x - 20)*5 Now, equating the distances because they are equal we get the following equation: 3*x = 5*(x - 20) --> x = 50 mph. Two cyclists start at the same time from opposite ends of a course that is 45 miles long.


GMAT word problem note: One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?


GMAT word problem note: Since this is a 'meeting' problem, there are two things that should strike you. First, since they are starting at the same time, when they meet, the time for which both will have been cycling will be the same.


GMAT word problem note: Second, the total distance traveled by the will be equal to the sum of their individual distances. Since we are asked to find the time, let us assign it as a variable 't'.


GMAT word problem note: (which is same for both cyclists) In the first row, we know the speed and we have the time in terms of 't'. Thus we can get the following equation:.


GMAT word problem note: Distance(1) = Speed(1) * Time(1) = 14*t In the second row, we know the speed and again we have the time in terms of 't'. Distance(2) = Speed(2) * Time(2) = 16*t Now we know that the total distance traveled is 45 miles and it is equal to the sum of the two.


GMAT word problem note: Translate the wording into variables, relationships, and equations before calculating. In the first row, we have the speed of the boat in terms of 'b' and we are given the time.


GMAT word problem note: Thus we can get the following equation:. Distance(1) = Speed(1) * Time(1) = (b + 3)*3 In the second row, we again have the speed in terms of 'b' and we are given the time.


GMAT word problem note: Distance(2) = Speed(2) * Time(2) = (b - 3)*4 Since the two distances are equal, we can equate them and solve for 'b'. Distance(1) = Distance(2) --> (b + 3)*3 = (b - 3)*4 --> b = 21 mph.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Overlapping Sets https://gmatclub.com/forum/overlapping-sets-144260.html ADVANCED OVERLAPPING SETS Some hard GMAT quantitative questions will require you to know and understand the formulas for set theory, presenting three sets and asking various questions about them.


GMAT sets, probability, and counting note: There are two main formulas to solve questions involving three overlapping sets. FIRST FORMULA Let's see how this formula is derived.


GMAT sets, probability, and counting note: When we add three groups A, B, and C some sections are counted more than once. For instance: sections d, e, and f are counted twice and section g thrice.


GMAT sets, probability, and counting note: Hence we need to subtract sections d, e, and f ONCE (to count section g only once) and subtract section g TWICE (again to count section g only once). In the formula above,, where AnB means intersection of A and B (sections d, and g), AnC means intersection of A and C (sections e, and g), and BnC means intersection of B and C (sections f, and g).


GMAT sets, probability, and counting note: Now, when we subtract (d, and g), (e, and g), and (f, and g) from, we are subtract sections d, e, and f ONCE BUT section g THREE TIMES (and we need to subtract section g only twice), therefor we should add only section g, which is intersection of A, B and C (AnBnC) again to get. SECOND FORMULA Notice that EXACTLY (only) 2-group overlaps is not the same as 2-group ov.


GMAT geometry note: Translate the wording into variables, relationships, and equations before calculating. Again: when we add three groups A, B, and C some sections are counted more than once.


GMAT geometry note: For instance: sections d, e, and f are counted twice and section g thrice. Hence we need to subtract sections d, e, and f ONCE (to count section g only once) and subtract section g TWICE (again to count section g only once).


GMAT geometry note: When we subtract from A+B+C we subtract sections d, e, and f once (fine) and next we need to subtract ONLY section g () twice. Now, how this concept can be represented in GMAT problem?


GMAT geometry note: Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team.


GMAT geometry note: 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?


GMAT geometry note: Translating: "are placed on at least one team": members of none = 0. "20 are on the marketing team":.


GMAT geometry note: "40 are on the Vision team":. "5 workers are on both the Marketing and Sales teams":.


GMAT statistics note: Track mean, median, range, distribution, and how operations change spread. Of 20 Adults, 5 belong to A, 7 belong to B, and 9 belong to C.


GMAT statistics note: If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations? "5 belong to A, 7 belong to B, and 9 belong to C":.


GMAT statistics note: "2 belong to all three organizations":. "3 belong to exactly 2 organizations": (sum of EXACTLY 2-group overlaps) = 3, so the sum of sections d, e, and f is given to be 3, (among these 3 adults there are no one who is a member of ALL 3 clubs) Question::.


GMAT statistics note: Apply second formula: --> Answer: 6. This semester, each of the 90 students in a certain class took at least one course from A, B, and C.


GMAT statistics note: If 60 students took A, 40 students took B, 20 students took C, and 5 students took all the three, how many students took exactly two courses? "of the 90 students in a certain class took at least one course from A, B, and C":.


GMAT statistics note: "60 students took A, 40 students took B, 20 students took C":. "5 students took all the three courses":.


GMAT statistics note: Apply second formula: --> Answer: 20.


GMAT fractions, decimals, and percents note: Percent problems usually relate part, whole, rate, and percent change. "4 names were on all 3 rosters":.


GMAT fractions, decimals, and percents note: Apply first formula: --> Answer: 50. There are 50 employees in the office of ABC Company.


GMAT fractions, decimals, and percents note: Of these, 22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course. Nine of the employees have taken exactly two of the courses and 1 employee has taken all three of the courses.


GMAT fractions, decimals, and percents note: How many of the 50 employees have taken none of the courses? Translating: "There are 50 employees in the office of ABC Company":.


GMAT fractions, decimals, and percents note: "22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course". "Nine of the employees have taken exactly two of the courses": (sum of EXACTLY 2-group overlaps) = 9.


GMAT fractions, decimals, and percents note: "1 employee has taken all three of the courses":. Apply second formula: --> Answer: 10.


GMAT fractions, decimals, and percents note: In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3.


GMAT fractions, decimals, and percents note: If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?


GMAT fractions, decimals, and percents note: Translate the wording into variables, relationships, and equations before calculating. Apply first formula: {Total} = {Hockey}+{Cricket}+{Football}-{HC+CH+HF}+{All three}+{Neither} 50 = 20+15+11-(7+4+5)+{All three}+18 --> {All three} = 2.


GMAT fractions, decimals, and percents note: Those who play ONLY Hockey and Cricket are 7-2 = 5. Those who play ONLY Cricket and Football are 4-2 = 2.


GMAT fractions, decimals, and percents note: Those who play ONLY Hockey and Football are 5-2 = 3. Hence, 5+2+3 = 10 students play exactly two of these sports.


GMAT fractions, decimals, and percents note: A student has decided to take GMAT and TOEFL examinations, for which he has allocated a certain number of days for preparation. On any given day, he does not prepare for both GMAT and TOEFL.


GMAT fractions, decimals, and percents note: How many days did he allocate for the preparation? (1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days.


GMAT fractions, decimals, and percents note: (2) He prepared for either GMAT or TOEFL on 14 days We have: {Total} = {GMAT } + {TOEFL} - {Both} + {Neither}. Since we are told that "on any given day, he does not prepare for both GMAT and TOEFL", then {Both} = 0, so {Total} = {GMAT } + {TOEFL} + {Neither}.


GMAT fractions, decimals, and percents note: We need to find {Total} (1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days --> {Total} - {GMAT } = 10 and {Total} - {TOEFL} = 12.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Notice that max range for the original question is not limited when the max # of people in all 3 groups for revised question is 17+28+35 (in case there is 0 overlap between the 3 groups).


GMAT sets, probability, and counting note: For more questions on overlapping sets check our Question Banks Problem Solving Questions on Overlapping Sets.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Probability https://gmatclub.com/forum/probability-87244.html PROBABILITY Definition A number expressing the probability (p) that a specific event will occur, expressed as the ratio of the number of actual occurrences (n) to the number of possible occurrences (N).


GMAT sets, probability, and counting note: A number expressing the probability (q) that a specific event will not occur:. Therefore, the probability of getting head is 50% or and the probability of getting tail is 50% or.


GMAT sets, probability, and counting note: All possibilities: {H,T} Dice p = n N q = = 1 − p(N−n) N.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. There are 6 equally possible outcomes when we roll a die.


GMAT sets, probability, and counting note: The probability of getting any number out of 1-6 is. All possibilities: {1,2,3,4,5,6} Marbles, Balls, Cards...


GMAT sets, probability, and counting note: Let's assume we have a jar with 10 green and 90 white marbles. If we randomly choose a marble, what is the probability of getting a green marble?


GMAT sets, probability, and counting note: N = 10 + 90 = 100 The number of green marbles: n = 10 Probability of getting a green marble:. There is one important concept in problems with marbles/cards/balls.


GMAT sets, probability, and counting note: When the first marble is removed from a jar and not replaced, the probability for the second marble differs (vs.). Whereas in case of a coin or dice the probabilities are always the same (and).


GMAT sets, probability, and counting note: Usually, a problem explicitly states: it is a problem with replacement or without replacement. Independent events Two events are independent if occurrence of one event does not influence occurrence of other events.


GMAT sets, probability, and counting note: Watch divisibility, factors, remainders, parity, and prime structure. After one flip and one toss, what is the probability of getting heads and a "4"?


GMAT sets, probability, and counting note: Tossing a coin and rolling a die are independent events. The probability of getting heads is and probability of getting a "4" is.


GMAT sets, probability, and counting note: Therefore, the probability of getting heads and a "4" is:. If there is a 20% chance of rain, what is the probability that it will rain on the first day but not on the second?


GMAT sets, probability, and counting note: The probability of rain is 0.2. Therefore probability of sunshine is q = 1 - 0.2 = 0.8.


GMAT sets, probability, and counting note: This yields that the probability of rain on the first day and sunshine on the second day is:. If Robert chooses randomly one integer from the first set and one integer from the second set, what is the probability of getting two odd integers?


GMAT sets, probability, and counting note: There is a total of 5 integers in the first set and 3 of them are odd: {1, 3, 7}. Therefore, the probability of getting odd integer out of first set is.


GMAT sets, probability, and counting note: There are 3 integers in the second set and 2 of them are odd: {3, 5}. Therefore, the probability of getting an odd integer out of second set is.


GMAT sets, probability, and counting note: Finally, the probability of of getting two odd integers is:. Mutually exclusive events Shakespeare's phrase "To be,.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. Combination of independent and mutually exclusive events Many probability problems contain combination of both independent and mutually exclusive events.


GMAT sets, probability, and counting note: To solve those problems it is important to identify all events and their types. One of the typical problems can be presented in a following general form:.


GMAT sets, probability, and counting note: If the probability of a certain event is p, what is the probability of it occurring k times in n-time sequence? (Or in English, what is the probability of getting 3 heads while tossing a coin 8 times?) Solution:.


GMAT sets, probability, and counting note: But it isn't the right answer. It would be right if we specified exactly each position for events in the sequence.


GMAT sets, probability, and counting note: So, we need to take into account that there are more than one outcomes. HHHTTTTT and HHTTTTTH are different mutually exclusive outcomes but they both have 3 heads and 5 tails.


GMAT sets, probability, and counting note: Therefore, we need to include all combinations of heads and tails. In our general question, probability of occurring event k times in n-time sequence could be expressed as:.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. The total number of possible committees is.


GMAT sets, probability, and counting note: The number of possible committee that includes both Bob and Rachel is. 2) reversal combinatorial approach:.


GMAT sets, probability, and counting note: Instead of counting probability of occurrence of certain event, sometimes it is better to calculate the probability of the opposite and then use formula p = 1 - q. The number of possible committee that does not includes both Bob and Rachel is: where, - the number of committees formed from 6 other people.


GMAT sets, probability, and counting note: The number of committees formed from Rob or Rachel and one out of 6 other people. The probability of choosing Bob or Rachel as a first person in committee is 2/8.


GMAT sets, probability, and counting note: The probability of choosing Rachel or Bob as a second person when first person is already chosen is 1/7. The probability that the committee includes both Bob and Rachel is.


GMAT sets, probability, and counting note: 4) reversal probability approach:. We can choose any first person.


GMAT sets, probability, and counting note: Then, if we have Rachel or Bob as first choice, we can choose any other person out of 6 people. If we have neither Rachel nor Bob as first choice, we can choose any person out of remaining 7 people.


GMAT sets, probability, and counting note: Given that there are 5 married couples.


GMAT sets, probability, and counting note: Separate total outcomes from desired outcomes and count carefully. We choose 1 couple out of 5 couples.


GMAT sets, probability, and counting note: We chose one person out of remaining 8 people. The total number of combinations to choose 3 people out of 10 people.


GMAT sets, probability, and counting note: 3) probability approach: 1st person: - we choose any person out of 10. 2nd person: - we choose any person out of 8 = 10-2(one couple from previous choice) 3rd person: - we choose any person out of 6 = 10-4(two couples from previous choices).


GMAT sets, probability, and counting note: Probability tree Sometimes, at 700+ level you may see complex probability problems that include conditions or restrictions. For such problems it could be helpful to draw a probability tree that include all possible outcomes and their probabilities.


GMAT sets, probability, and counting note: Julia and Brian play a game in which Julia takes a ball and if it is green, she wins. If the first ball is not green, she takes the second ball (without replacing first) and she wins if the two balls are white or if the first ball is gray and the second ball is white.