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Words 15488

Pages 62

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.

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.

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.

Zero is an even number.

Addition / Subtraction: even +/- even = even; even +/- odd = odd; odd +/- odd = even.

Multiplication: even * even = even; even * odd = even; odd * odd = odd.

Division of two integers can result into an even/odd integer or a fraction.

IRRATIONAL NUMBERS

Fractions (also known as rational numbers) can be written as terminating (ending) or repeating decimals (such as 0.5, 0.76, or 0.333333....). On the other hand, all those numbers that can be written as non-terminating, non-repeating decimals are non-rational, so they are called the "irrationals". Examples would be ("the square root of two") or the number pi (~3.14159..., from geometry). The rationals and the irrationals are two totally separate number types: there is no overlap.

Putting these two major classifications, the rationals and the irrationals, together in one set gives you the "real" numbers.

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.

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

Prime Numbers

A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise a number is called acomposite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1. 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.

• The first twenty-six prime numbers are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

• Note: only positive numbers can be primes.

• There are infinitely many prime numbers.

• The only even prime number is 2, since any larger even number is divisible by 2. Also 2 is the smallest prime.

• 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.

• Any nonzero natural number can be factored into primes, written as a product of primes or powers of primes. Moreover, thisfactorization is unique except for a possible reordering of the factors.

• 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. For instance integer with three unique prime factors , , and can be expressed as , where , , and are powers of , , and , respectively and are .

Example: .

• Verifying the primality (checking whether the number is a prime) of a given number can be done by trial division, that is to say dividing by all integer numbers smaller than , thereby checking whether is a multiple of .

Example: Verifying the primality of : is little less than , from integers from to , is divisible by , hence is not prime.

• If is a positive integer greater than 1, then there is always a prime number with.

Factors

A divisor of an integer , also called a factor of , is an integer which evenly divides without leaving a remainder. 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.

• Every integer is a divisor of itself.

• Every integer is a divisor of 0, except, by convention, 0 itself.

• 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.

• 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.

• 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.

Example: 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:

• 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 .

• If is a factor of and is a factor of , then is a factor of .

• If is a factor of and is a factor of , then or .

• If is a factor of , and , then a is a factor of .

• If is a prime number and is a factor of then is a factor of or is a factor of .

Finding the Number of Factors of an Integer

First make prime factorization of an integer , where , , and are prime factors of and , , and are their powers.

The number of factors of will be expressed by the formula . NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450:

Total number of factors of 450 including 1 and 450 itself is factors.

Finding the Sum of the Factors of an Integer

First make prime factorization of an integer , where , , and are prime factors of and , , and are their powers.

The sum of factors of will be expressed by the formula:

Example: Finding the sum of all factors of 450:

The sum of all factors of 450 is

Greatest Common Factor (Divisior) - GCF (GCD)

The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

To find the GCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors).

• 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. 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.

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

Perfect Square

A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is an perfect square.

There are some tips about the perfect square:

• The number of distinct factors of a perfect square is ALWAYS ODD.

• The sum of distinct factors of a perfect square is ALWAYS ODD.

• 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.

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.

4 - If the last two digits form a number divisible by 4, the number is also.

5 - If the last digit is a 5 or a 0, the number is divisible by 5.

6 - If the number is divisible by both 3 and 2, it is also divisible by 6.

7 - Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

8 - If the last three digits of a number are divisible by 8, then so is the whole number.

9 - If the sum of the digits is divisible by 9, so is the number.

10 - If the number ends in 0, it is divisible by 10.

11 - If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11.

Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11.

12 - If the number is divisible by both 3 and 4, it is also divisible by 12.

25 - Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25.

Factorials

Factorial of a positive integer , denoted by , is the product of all positive integers less than or equal to n. For instance .

• Note: 0!=1.

• Note: factorial of negative numbers is undefined.

Trailing zeros:

Trailing zeros 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.

125000 has 3 trailing zeros;

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 .

It's easier if you look at an example:

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!

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 .

The formula is: ... till

What is the power of 2 in 25!?

Finding the power of non-prime in n!:

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!.

Find the power of 2:

=

Find the power of 3:

=

Find the power of 5:

=

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!.

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.

• 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 .

• If n is odd, the sum of consecutive integers is always divisible by n. Given , we have consecutive integers. 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. Given , we have consecutive integers. The sum of 9+10+11+12=42, therefore, is not divisible by 4.

• The product of consecutive integers is always divisible by .

Given consecutive integers: . The product of 3*4*5*6 is 360, which is divisible by 4!=24.

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. The set of integers is an example of evenly spaced set. Set of consecutive integers is also an example of evenly spaced set.

• If the first term is and the common difference of successive members is , then the term of the sequence is given by:

• In any evenly spaced set the arithmetic mean (average) is equal to the median and can be calculated by the formula , where is the first term and is the last term. Given the set , .

• The sum of the elements in any evenly spaced set is given by:

, the mean multiplied by the number of terms. OR,

• Special cases:

Sum of n first positive integers:

Sum of n first positive odd numbers: , where is the last, term and given by: . Given first odd positive integers, then their sum equals to .

Sum of n first positive even numbers: , where is the last, term and given by: . Given first positive even integers, then their sum equals to .

• If the evenly spaced set contains odd number of elements, the mean is the middle term, so the sum is middle term multiplied by number of terms. 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. 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.

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). 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. The number on bottom of the fraction is called denominator. In the fraction, , 9 is the numerator and 7 is denominator.

• Fractions that have a value between 0 and 1 are called proper fraction. The numerator is always smaller than the denominator. is a proper fraction.

• Fractions that are greater than 1 are called improper fraction. Improper fraction can also be written as a mixed number. is improper fraction.

• An integer combined with a proper fraction is called mixed number. is a mixed number. This can also be written as an improper fraction:

Converting Improper Fractions

• Converting Improper Fractions to Mixed Fractions:

1. Divide the numerator by the denominator

2. Write down the whole number answer

3. Then write down any remainder above the denominator

Example #1: Convert to a mixed fraction.

Solution: Divide with a remainder of . Write down the and then write down the remainder above the denominator , like this:

• Converting Mixed Fractions to Improper Fractions:

1. Multiply the whole number part by the fraction's denominator

2. Add that to the numerator

3. Then write the result on top of the denominator

Example #2: Convert to an improper fraction.

Solution: Multiply the whole number by the denominator: . Add the numerator to that: . Then write that down above the denominator, like this:

Reciprocal

Reciprocal for a number , denoted by or , is a number which when multiplied by yields . The reciprocal of a fraction is. To get the reciprocal of a number, divide 1 by the number. For example reciprocal of is , reciprocal of is .

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.

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.

• Dividing fractions: Change the divisor into its reciprocal and then multiply.

Example #1:

Example #2: Given , take the reciprocal of . The reciprocal is . Now multiply: .

Decimal Representation

The decimals has ten as its base. Decimals can be terminating (ending) (such as 0.78, 0.2) or repeating (recuring) decimals (such as 0.333333....).

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. For example: is a terminating decimal , as (denominator) equals to . Fraction is also a terminating decimal, as and denominator .

Converting Decimals to Fractions

• To convert a terminating decimal to fraction:

1. Calculate the total numbers after decimal point

2. Remove the decimal point from the number

3. Put 1 under the denominator and annex it with "0" as many as the total in step 1

4. Reduce the fraction to its lowest terms

Example: Convert to a fraction.

1: Total number after decimal point is 2.

2 and 3: .

4: Reducing it to lowest terms:

• To convert a recurring decimal to fraction:

1. Separate the recurring number from the decimal fraction

2. Annex denominator with "9" as many times as the length of the recurring number

3. Reduce the fraction to its lowest terms

Example #1: Convert to a fraction.

1: The recurring number is .

2: , the number is of length so we have added two nines.

3: Reducing it to lowest terms: .

• To convert a mixed-recurring decimal to fraction:

1. Write down the number consisting with non-repeating digits and repeating digits.

2. Subtract non-repeating number from above.

3. 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.

Example #2: Convert to a fraction.

1. The number consisting with non-repeating digits and repeating digits is 2512;

2. Subtract 25 (non-repeating number) from above: 2512-25=2487;

3. 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.

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. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:

5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.

5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.

5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Ratios and Proportions

Given that , where a, b, c and d are non-zero real numbers, we can deduce other proportions by simple Algebra. These results are often referred to by the names mentioned along each of the properties obtained.

- invertendo

- alternendo

- componendo

- dividendo

- componendo & dividendo

EXPONENTS

Exponents are a "shortcut" method of showing a number that was multiplied by itself several times. 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.

Exponents one and zero: Any nonzero number to the power of 0 is 1.

For example: and

• Note: the case of 0^0 is not tested on the GMAT.

Any number to the power 1 is itself.

Powers of zero:

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.

Powers of one: The integer powers of one are one.

Negative powers:

Powers of minus one:

If n is an even integer, then .

If n is an odd integer, then .

Operations involving the same exponents:

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:

Exponential Equations:

When solving equations with even exponents, we must consider both positive and negative possibilities for the solutions.

For instance , the two possible solutions are and .

When solving equations with odd exponents, we'll have only one solution.

For instance for , solution is and for , solution is .

Exponents and divisibility: is ALWAYS divisible by . is divisible by if is even.

is divisible by if is odd, and not divisible by a+b if n is even.

LAST DIGIT OF A PRODUCT

Last digits of a product of integers are last digits of the product of last digits of these integers.

For instance last 2 digits of 845*9512*408*613 would be the last 2 digits of 45*12*8*13=540*104=40*4=160=60

Example: The last digit of 85945*89*58307=5*9*7=45*7=35=5?

LAST DIGIT OF A POWER

Determining the last digit of :

1. Last digit of is the same as that of ;

2. Determine the cyclicity number of ;

3. Find the remainder when divided by the cyclisity;

4. 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.

• Integers ending with 2, 3, 7 and 8 have a cyclicity of 4.

• Integers ending with 4 (eg. ) have a cyclisity of 2. 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. When n is odd will end with 9 and when n is even will end with 1.

Example: What is the last digit of ?

Solution: Last digit of is the same as that of . Now we should determine the cyclisity of :

1. 7^1=7 (last digit is 7)

2. 7^2=9 (last digit is 9)

3. 7^3=3 (last digit is 3)

4. 7^4=1 (last digit is 1)

5. 7^5=7 (last digit is 7 again!)

...

So, the cyclisity of 7 is 4.

Now divide 39 (power) by 4 (cyclisity), remainder is 3.So, the last digit of is the same as that of the last digit of , is the same as that of the last digit of , which is .

ROOTS

Roots (or radicals) are the "opposite" operation of applying exponents. For instance x^2=16 and square root of 16=4.

General rules:

• and .

•

•

•

•

• , 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.

That is, , NOT +5 or -5. In contrast, the equation has TWO solutions, +5 and -5. Even roots have only a positive value on the GMAT.

• Odd roots will have the same sign as the base of the root. For example, and .

• For GMAT it's good to memorize following values:

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". Since a percent is an amount per 100, percents can be represented as fractions with a denominator of 100. For example, 25% means 25 per 100, 25/100 and 350% means 350 per 100, 350/100.

• A percent can be represented as a decimal. The following relationship characterizes how percents and decimals interact. Percent Form / 100 = Decimal Form

For example: What is 2% represented as a decimal?

Percent Form / 100 = Decimal Form: 2%/100=0.02

• Percent change

General formula for percent increase or decrease, (percent change):

Example: 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. By what percent did the ratio of royalties to sales decrease from the first $10 million in sales to the next $100 million in sales?

Solution: Percent decrease can be calculated by the formula above:

, so the royalties decreased by 60%.

• Simple Interest

Simple interest = principal * interest rate * time

Example: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months?

Solution: $15,000*0.1*9/12 = $1125

• Compound Interest

, where C = the number of times compounded annually. If C=1, meaning that interest is compounded once a year, then the formula will be: , where time is number of years.

Example: If $20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 year?

Solution:

ORDER OF OPERATIONS - PEMDAS

Perform the operations inside a Parenthesis first (absolute value signs also fall into this category), then Exponents, thenMultiplication and Division, from left to right, then Addition and Subtraction, from left to right - PEMDAS.

Special cases:

• 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. But means while ; a factorial in an exponent applies to the exponent, while a factorial not in the exponent applies to the entire power.

• If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus: and not

Definition

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". Since a percent is an amount per 100, percents can be represented as fractions with a denominator of 100. For example, 25% means 25 per 100, 25/100 and 350% means 350 per 100, 350/100.

• A percent can be represented as a decimal. The following relationship characterizes how percents and decimals interact. Percent Form / 100 = Decimal Form

For example: What is 2% represented as a decimal?

Percent Form / 100 = Decimal Form: 2%/100=0.02

Percent change

General formula for percent increase or decrease, (percent change):

Example: 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. By what percent did the ratio of royalties to sales decrease from the first $10 million in sales to the next $100 million in sales?

Solution: Percent decrease can be calculated by the formula above:

, so the royalties decreased by 60%.

Simple Interest

Simple interest = principal * interest rate * time, where "principal" is the starting amount and "rate" is the interest rate at which the money grows per a given period of time (note: express the rate as a decimal in the formula). Time must be expressed in the same units used for time in the Rate.

Example: If $15,000 is invested at 10% simple annual interest, how much interest is earned after 9 months?

Solution: $15,000*0.1*9/12 = $1125

Compound Interest

, where C = the number of times compounded annually.

If C=1, meaning that interest is compounded once a year, then the formula will be:

, where time is number of years.

Example: If $20,000 is invested at 12% annual interest, compounded quarterly, what is the balance after 2 year?

Solution:

Percentile

If someone's grade is in percentile of the grades, this means that of people out of has the grades less than this person.

Example: Lena’s grade was in the 80th percentile out of 120 grades in her class. In another class of 200 students there were 24 grades higher than Lena’s. If nobody had Lena’s grade, then Lena was what percentile of the two classes combined?

Solution:

Being in 80th percentile out of 120 grades means Lena outscored classmates.

In another class she would outscored students.

So, in combined classes she outscored . As there are total of students, so Lena is in , or in 85th percentile.

Official GMAC Books:

The Official Guide, 12th Edition: PS #10; PS #17; PS #19; PS #47; PS #55; PS #60; PS #64; PS #78; PS #92; PS #94; PS #109; PS #111; PS #115; PS #124; PS #128; PS #131; PS #151; PS #156; PS #166; PS #187; PS #193; PS #200; PS #202; PS #220; DS #2; DS #7; DS #21; DS #37; DS #48; DS #55; DS #61; DS #63; DS #78; DS #88; DS #92; DS #120; DS #138; DS #142; DS #143;

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. They are most often used for two purposes :

(a) In Algebraic Equations : To represent unknown quantities in known relationships. For eg : "Mary's age is 10 more than twice that of Jim's", we can represent the unknown "Mary's age" by x and "Jim's age" by y and then the known relationship is

(b) In Algebraic Identities : 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. So the variable acts like a true placeholder, which may be replaced by any number.

Basic rules of manipulation

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

Eg.

When switching terms from one side to the other in an algebraic expression * becomes / and vice versa.

Eg.

you can add/subtract/multiply/divide both sides by the same amount. Eg. you can take to the exponent or bring from the exponent as long as the base is the same.

Eg 1.

Eg 2.

It is important to note that all the operations above are possible not just with constants but also with variables themselves. So you can "add x" or "multiply with y" on both sides while maintaining the expression. But what you need to be very careful about is when dividing both sides by a variable. When you divide both sides by a variable (or do operations like "canceling x on both sides") you implicitly assume that the variable cannot be equal to 0, as division by 0 is undefined. This is a concept shows up very often on GMAT questions.

Degree of an expression

The degree of an algebraic expression is defined as the highest power of the variables present in the expression.

Degree 1 : Linear

Degree 2 : Quadratic

Degree 3 : Cubic

Degree 4 : 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. Let us consider some special cases of linear equations :

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. 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. The general form is :

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. 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 :

Case 1 : No solution --> Then there is no solution for bigger set

Case 2 : Unique solution --> Substitute in other equations to see if the solution works for all others

Case 3 : Infinite solutions --> Out of the 2 equations you picked, replace any one with an un-picked equation and repeat.

More than 2 variables

This is not a case that will be encountered often on the GMAT. But in general for n variables you will need at least n equations to get a unique solution. Sometimes you can assign unique values to a subset of variables using less than n equations using a small trick. For example consider the equations :

In this case you can treat as a single variable to get :

These can be solved to get x=0 and 2y+5z=20

There is a common misconception that you need n equations to solve n variables. This is not true.

Solving equations of degree 2 : 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 :

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. And the solutions are f,g

Eg.

The sum is -11 and the product is 30. So numbers are -5,-6

Solving equations 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.

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. The basic idea is that if you can write an equation in the form :

where each of A,B,C are algebraic expressions. Once this is done, the solution is obtained by equating each of A,B,C to 0 one by one.

Eg.

So the solution is x=0,-5,-6

Reducing to lower degree

This is useful sometimes when it is easy to see that a simple variable substitution can reduce the degree.

Eg.

Here let

So the solution is y=1,2 or x^3=1,2 or x=1,cube_root(3)

Other tricks

Sometimes we are given conditions such as the variables being integers which make the solutions much easier to find. When we know that the solutions are integral, often times solutions are easy to find using just brute force.

Eg. and we know a,b are integers such that a<b

We can solve this by testing values of a and checking if we can find b a=1 b=root(115) Not integer a=2 b=root(112) Not integer a=3 b=root(107) Not integer a=4 b=root(100)=10 a=5 b=root(91) Not integer a=6 b=root(80) Not integer a=7 b=root(67) Not integer a=8 b=root(52)<a

So the answer is (4,10)

Algebraic Identities

These can be very useful in simplifying & solving a lot of questions :

Triangle A closed figure consisting of three line segments linked end-to-end. A 3-sided polygon.

Vertex The vertex (plural: vertices) is a corner of the triangle. Every triangle has three vertices.

Base The base of a triangle can be any one of the three sides, usually the one drawn at the bottom.

• You can pick any side you like to be the base.

• Commonly used as a reference side for calculating the area of the triangle.

• In an isosceles triangle, the base is usually taken to be the unequal side.

Altitude The altitude of a triangle is the perpendicular from the base to the opposite vertex. (The base may need to be extended).

• Since there are three possible bases, there are also three possible altitudes.

• The three altitudes intersect at a single point, called the orthocenter of the triangle.

Median The median of a triangle is a line from a vertex to the midpoint of the opposite side.

• The three medians intersect at a single point, called the centroid of the triangle.

• Each median divides the triangle into two smaller triangles which have the same area.

• Because there are three vertices, there are of course three possible medians.

• No matter what shape the triangle, all three always intersect at a single point. This point is called the centroid of the triangle.

• The three medians divide the triangle into six smaller triangles of equal area.

• The centroid (point where they meet) is the center of gravity of the triangle

• Two-thirds of the length of each median is between the vertex and the centroid, while one-third is between the centroid and the midpoint of the opposite side.

• , where , and are the sides of the triangle and is the side of the triangle whose midpoint is the extreme point of median .

Area The number of square units it takes to exactly fill the interior of a triangle.

Usually called "half of base times height", the area of a triangle is given by the formula below.

•

Other formula:

•

•

Where is the length of the base, and the other sides; is the length of the corresponding altitude; is the Radius of circumscribed circle; is the radius of inscribed circle; P is the perimeter

• Heron's or Hero's Formula for calculating the area where are the three sides of the triangle and which is the semi perimeter of the triangle.

Perimeter The distance around the triangle. The sum of its sides.

• For a given perimeter equilateral triangle has the largest area.

• For a given area equilateral triangle has the smallest perimeter.

Relationship of the Sides of a Triangle

• The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.

Interior angles The three angles on the inside of the triangle at each vertex.

• The interior angles of a triangle always add up to 180°

• Because the interior angles always add to 180°, every angle must be less than 180°

• The bisectors of the three interior angles meet at a point, called the incenter, which is the center of the incircle of the triangle.

Exterior angles The angle between a side of a triangle and the extension of an adjacent side.

• An exterior angle of a triangle is equal to the sum of the opposite interior angles.

• If the equivalent angle is taken at each vertex, the exterior angles always add to 360° In fact, this is true for any convex polygon, not just triangles.

Midsegment of a Triangle A line segment joining the midpoints of two sides of a triangle

• A triangle has 3 possible midsegments.

• The midsegment is always parallel to the third side of the triangle.

• The midsegment is always half the length of the third side.

• A triangle has three possible midsegments, depending on which pair of sides is initially joined.

Relationship of sides to interior angles in a triangle

• The shortest side is always opposite the smallest interior angle

• The longest side is always opposite the largest interior angle

Angle bisector An angle bisector divides the angle into two angles with equal measures.

• An angle only has one bisector.

• Each point of an angle bisector is equidistant from the sides of the angle.

• The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment DC is equal to the ratio of the length of side AB to the length of side AC:

• The incenter is the point where the angle bisectors intersect. The incenter is also the center of the triangle's incircle - the largest circle that will fit inside the triangle.

Similar Triangles Triangles in which the three angles are identical.

• It is only necessary to determine that two sets of angles are identical in order to conclude that two triangles are similar; the third set will be identical because all of the angles of a triangle always sum to 180º.

• In similar triangles, the sides of the triangles are in some proportion to one another. For example, a triangle with lengths 3, 4, and 5 has the same angle measures as a triangle with lengths 6, 8, and 10. The two triangles are similar, and all of the sides of the larger triangle are twice the size of the corresponding legs on the smaller triangle.

• If two similar triangles have sides in the ratio , then their areas are in the ratio

Congruence of triangles Two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in size.

1. SAS (Side-Angle-Side): If two pairs of sides of two triangles are equal in length, and the included angles are equal in measurement, then the triangles are congruent.

2. SSS (Side-Side-Side): If three pairs of sides of two triangles are equal in length, then the triangles are congruent.

3. ASA (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent.

So, knowing SAS or ASA is sufficient to determine unknown angles or sides.

NOTE IMPORTANT EXCEPTION:

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also known as ASS, or Angle-Side-Side) does not always prove congruence, even when the equal angles are opposite equal sides.

Specifically, SSA does not prove congruence when the angle is acute and the opposite side is shorter than the known adjacent side but longer than the sine of the angle times the adjacent side. This is the ambiguous case. In all other cases with corresponding equalities, SSA proves congruence.

The SSA condition proves congruence if the angle is obtuse or right. In the case of the right angle (also known as the HL (Hypotenuse-Leg) condition or the RHS (Right-angle-Hypotenuse-Side) condition), we can calculate the third side and fall back on SSS.

To establish congruence, it is also necessary to check that the equal angles are opposite equal sides.

So, knowing two sides and non-included angle is NOT sufficient to calculate unknown side and angles.

Angle-Angle-Angle

AAA (Angle-Angle-Angle) says nothing about the size of the two triangles and hence proves only similarity and not congruence.

So, knowing three angles is NOT sufficient to determine lengths of the sides.

Scalene triangle all sides and angles are different from one another

• All properties mentioned above can be applied to the scalene triangle, if not mentioned the special cases (equilateral, etc)

Equilateral triangle all sides have the same length.

• An equilateral triangle is also a regular polygon with all angles measuring 60°.

• The area is

• The perimeter is

• The radius of the circumscribed circle is

• The radius of the inscribed circle is

• And the altitude is (Where is the length of a side.)

• For any point P within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.

• For a given perimeter equilateral triangle has the largest area.

• For a given area equilateral triangle has the smallest perimeter.

• With an equilateral triangle, the radius of the incircle is exactly half the radius of the circumcircle.

Isosceles triangle two sides are equal in length.

• An isosceles triangle also has two angles of the same measure; namely, the angles opposite to the two sides of the same length.

• For an isosceles triangle with given length of equal sides right triangle (included angle) has the largest area.

• To find the base given the leg and altitude, use the formula:

• To find the leg length given the base and altitude, use the formula:

• To find the altitude given the base and leg, use the formula: (Where: L is the length of a leg; A is the altitude; B is the length of the base)

Right triangle A triangle where one of its interior angles is a right angle (90 degrees)

• Hypotenuse: the side opposite the right angle. This will always be the longest side of a right triangle.

• The two sides that are not the hypotenuse. They are the two sides making up the right angle itself.

• Theorem by Pythagoras defines the relationship between the three sides of a right triangle: , where is the length of the hypotenuse and , are the lengths of the the other two sides.

• In a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices

• A right triangle can also be isosceles if the two sides that include the right angle are equal in length (AB and AC in the figure above)

• Right triangle with a given hypotenuse has the largest area when it's an isosceles triangle.

• A right triangle can never be equilateral, since the hypotenuse (the side opposite the right angle) is always longer than the other two sides.

• Any triangle whose sides are in the ratio 3:4:5 is a right triangle. Such triangles that have their sides in the ratio of whole numbers are called Pythagorean Triples. There are an infinite number of them, and this is just the smallest. If you multiply the sides by any number, the result will still be a right triangle whose sides are in the ratio 3:4:5. For example 6, 8, and 10.

• A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer. There are 16 primitive Pythagorean triples with c ≤ 100:

(3, 4, 5) (5, 12, 13) (7, 24, 25) (8, 15, 17) (9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97).

• A right triangle where the angles are 30°, 60°, and 90°.

This is one of the 'standard' triangles you should be able recognize on sight. A fact you should commit to memory is: The sides are always in the ratio .

Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°).

• A right triangle where the angles are 45°, 45°, and 90°.

This is one of the 'standard' triangles you should be able recognize on sight. A fact you should also commit to memory is: The sides are always in the ratio . With the being the hypotenuse (longest side). This can be derived from Pythagoras' Theorem. Because the base angles are the same (both 45°) the two legs are equal and so the triangle is also isosceles.

• Area of a 45-45-90 triangle. As you see from the figure above, two 45-45-90 triangles together make a square, so the area of one of them is half the area of the square. As a formula . Where S is the length of either short side.

• Right triangle inscribed in circle:

• If M is the midpoint of the hypotenuse, then . One can also say that point B is located on the circle with diameter . Conversely, if B is any point of the circle with diameter (except A or C themselves) then angle B in triangle ABC is a right angle.

• A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.

• Circle inscribed in right triangle:

Note that in picture above the right angle is C.

• Given a right triangle, draw the altitude from the right angle.

Then the triangles ABC, CHB and CHA are similar. Perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle.

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.

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. 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:

Example #1

Q: Find the distance between the point A (3,-1) and B (-1,2)

Solution: Substituting values in the equation we'll get

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:

• 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.

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.

Every straight line in the plane can represented by a first degree equation with two variables.

There are several approaches commonly used in coordinate geometry. It does not matter whether we are talking about a line, ray or line segment. In all cases any of the below methods will provide enough information to define the line exactly.

1. General form.

The general form of the equation of a straight line is

Where , and are arbitrary constants. This form includes all other forms as special cases. For an equation in this form the slope is and the y intercept is .

2. Point-intercept form.

Where: is the slope of the line; is the y-intercept of the line; is the independent variable of the function .

3. Using two points

In figure below, a line is defined by the two points A and B. 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.

The equation of a straight line passing through points and is:

Example #1

Q: Find the equation of a line passing through the points A (17,4) and B (9,9).

Solution: 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. 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.

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). With these two facts we can establish a unique line.

The equation of a straight line that passes through a point with a slope m is:

Example #2

Q: Find the equation of a line passing through the point A (14,23) and the slope 2.

Solution: Substituting the values in equation we'll get -->

4. Intercept form.

The equation of a straight line whose x and y intercepts are a and b, respectively, is:

Example #3

Q: Find the equation of a line whose x intercept is 5 and y intercept is 2.

Solution: Substituting the values in equation we'll get --> OR if we want to write the equation in the slope-intercept form:

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.

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.

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. 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 .

SLOPE DIRECTION

The slope of a line can be positive, negative, zero or undefined.

Positive slope

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

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. 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. 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. The line above is exactly vertical, so it has no defined slope.

SLOPE AND QUADRANTS:

1. If the slope of a line is negative, the line WILL intersect quadrants II and IV. X and Y intersects of the line with negative slope have the same sign. Therefore if X and Y intersects are positive, the line intersects quadrant I; if negative, quadrant III.

2. If the slope of line is positive, line WILL intersect quadrants I and III. Y and X intersects of the line with positive slope have opposite signs. Therefore if X intersect is negative, line intersects the quadrant II too, if positive quadrant IV.

3. Every line (but the one crosses origin OR parallel to X or Y axis OR X and Y axis themselves) crosses three quadrants. Only the line which crosses origin OR is parallel to either of axis crosses only two quadrants.

4. If a line is horizontal it has a slope of , is parallel to X-axis and crosses quadrant I and II if the Y intersect is positive OR quadrants III and IV, if the Y intersect is negative. Equation of such line is y=b, where b is y intersect.

5. If a line is vertical, the slope is not defined, line is parallel to Y-axis and crosses quadrant I and IV, if the X intersect is positive and quadrant II and III, if the X intersect is negative. Equation of such line is , where a is x-intercept.

6. For a line that crosses two points and , slope

7. If the slope is 1 the angle formed by the line is degrees.

8. Given a point and slope, equation of a line can be found. 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. All points on the line will have the same x-coordinate.

A vertical line has no slope. Or put another way, for a vertical line the slope is undefined.

The equation of a vertical line is:

Where: x is the coordinate of any point on the line; a is where the line crosses the x-axis (x intercept). Notice that the equation is independent of y. Any point on the vertical line satisfies the equation.

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

A horizontal line has a slope of zero.

The equation of a horizontal line is:

Where: x is the coordinate of any point on the line; b is where the line crosses the y-axis (y intercept). Notice that the equation is independent of x. Any point on the horizontal line satisfies the equation.

Parallel lines

Parallel lines have the same slope.

The slope can be found using any method that is convenient to you:

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:

Distance between two parallel lines and can be found by the formula:

Example #1

Q:There are two lines. One line is defined by two points at (5,5) and (25,15). The other is defined by an equation in slope-intercept form form y = 0.52x - 2.5. Are two lines parallel?

Solution:

For the top line, the slope is found using the coordinates of the two points that define the line.

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. 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. Therefore, the lines are not parallel.

Example #2

Q: Define a line through a point C parallel to a line passes through the points A and B.

Solution: We first find the slope of the line AB using the same method as in the example above.

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:

-->

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 .

The two lines and are perpendicular if .

The equation of a line passing through the point ) and perpendicular to line is:

Example #1:

Q: Are the two lines below perpendicular?

Solution:

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.

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:

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. However, if you looked carefully at the diagram, you might have noticed that point C is a little too far to the left for the lines to be perpendicular.

Example # 2.

Q: Define a line passing through the point E and perpendicular to a line passing through the points C and D on the graph above.

Solution: The point E is on the y-axis and so is the y-intercept of the desired line. 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:

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. 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 another point on the line using the equation and then draw the line through that point and the intercept.

Intersection of two straight lines

The point of intersection of two non-parallel lines can be found from the equations of the two lines.

To find the intersection of two straight lines:

1. 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. 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.

This gives us the x and y coordinates of the intersection.

Example #1

Q: Find the point of intersection of two lines that have the following equations (in slope-intercept form):

Solution: 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:

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:

We now have both x and y, so the intersection point is (10, 27)

Example #2

Q: Find the point of intersection of two lines that have the following equations (in slope-intercept form): and (A vertical line)

Solution: When one of the lines is vertical, it has no defined slope. 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.

Equation for a line

Set x equal to 12

So the intersection point is at (12,33).

Note: 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.

The distance from a point to a line is given by the formula:

When the line is horizontal the formula transforms to:

Where: is the y-coordinate of the given point P; is the y-coordinate of any point on the given vertical line L. | | 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:

Where: is the x-coordinate of the given point P; is the x-coordinate of any point on the given vertical line L. | | the vertical bars mean "absolute value" - make it positive even if it calculates to a negative.

When the given point is origin, then the distance between origin and line ax+by+c=0 is given by the formula:

Circle on a plane

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that:

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram above, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x-a and y-b.

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.

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:

.

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 . Remember this statement can ALWAYS be expressed as:

.

Parabola

A parabola is the graph associated with a quadratic function, i.e. a function of the form .

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.

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.

x-intercepts: 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:

If discriminant is positive parabola has two intercepts with x-axis;

If discriminant is negative parabola has no intercepts with x-axis;

If discriminant is zero parabola has one intercept with x-axis (tangent point).

y-intercept: Given , the y-intercept is , as y intercept means the value of y when x=0.

Vertex: The vertex represents the maximum (or minimum) value of the function, and is very important in calculus.

The vertex of the parabola is located at point .

Note: typically just is calculated and plugged in for x to find y.

Official GMAC Books:

The Official Guide, 12th Edition: DT #39; PS #9; PS #25; PS #39; PS #88; PS #194; PS #205; PS #210; PS #212; PS #229; DS #69; DS #75; DS #93; DS #94; DS #108; DS #121; DS #149; DS #164;

The Official Guide, Quantitative 2th Edition: PS #21; PS #85; PS #102; PS #123; DS #22;

The Official Guide, 11th Edition: DT #39; PS #7; PS #23; PS #36; PS #89; PS #199; PS #222; PS #227; PS #229; PS #248; DS #15; DS #78; DS #85; DS #124;

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).

A number expressing the probability (q) that a specific event will not occur:

Examples

Coin

There are two equally possible outcomes when we toss a coin: a head (H) or tail (T). Therefore, the probability of getting head is 50% or and the probability of getting tail is 50% or .

All possibilities: {H,T}

Dice

There are 6 equally possible outcomes when we roll a die. The probability of getting any number out of 1-6 is .

All possibilities: {1,2,3,4,5,6}

Marbles, Balls, Cards...

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?

The number of all marbles: 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. 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 ). 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. For n independent events the probability is the product of all probabilities of independent events:

p = p1 * p2 * ... * pn-1 * pn

or

P(A and B) = P(A) * P(B) - A and B denote independent events

Example #1

Q:There is a coin and a die. After one flip and one toss, what is the probability of getting heads and a "4"?

Solution: Tossing a coin and rolling a die are independent events. The probability of getting heads is and probability of getting a "4" is . Therefore, the probability of getting heads and a "4" is:

Example #2

Q: 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?

Solution: The probability of rain is 0.2; therefore probability of sunshine is q = 1 - 0.2 = 0.8. This yields that the probability of rain on the first day and sunshine on the second day is:

P = 0.2 * 0.8 = 0.16

Example #3

Q:There are two sets of integers: {1,3,6,7,8} and {3,5,2}. 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?

Solution: 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 . 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 . Finally, the probability of of getting two odd integers is:

Mutually exclusive events

Shakespeare's phrase "To be, or not to be: that is the question" is an example of two mutually exclusive events.

Two events are mutually exclusive if they cannot occur at the same time. For n mutually exclusive events the probability is the sum of all probabilities of events:

p = p1 + p2 + ... + pn-1 + pn

or

P(A or B) = P(A) + P(B) - A and B denotes mutually exclusive events

Example #1

Q: If Jessica rolls a die, what is the probability of getting at least a "3"?

Solution: There are 4 outcomes that satisfy our condition (at least 3): {3, 4, 5, 6}. The probability of each outcome is 1/6. The probability of getting at least a "3" is:

Combination of independent and mutually exclusive events

Many probability problems contain combination of both independent and mutually exclusive events. 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:

Q: 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: All events are independent. So, we can say that:

(1)

But it isn't the right answer. It would be right if we specified exactly each position for events in the sequence. So, we need to take into account that there are more than one outcomes. Let's consider our example with a coin where "H" stands for Heads and "T" stands for Tails:

HHHTTTTT and HHTTTTTH are different mutually exclusive outcomes but they both have 3 heads and 5 tails. 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:

(2)

In the example with a coin, right answer is

Example #1

Q.:If the probability of raining on any given day in Atlanta is 40 percent, what is the probability of raining on exactly 2 days in a 7-day period?

Solution: We are not interested in the exact sequence of event and thus apply formula #2:

A few ways to approach a probability problem

There are a few typical ways that you can use for solving probability questions. Let's consider example, how it is possible to apply different approaches:

Example #1

Q: There are 8 employees including Bob and Rachel. If 2 employees are to be randomly chosen to form a committee, what is the probability that the committee includes both Bob and Rachel?

Solution:

1) combinatorial approach: The total number of possible committees is . The number of possible committee that includes both Bob and Rachel is .

2) reversal combinatorial approach: 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 total number of possible committees is . The number of possible committee that does not includes both Bob and Rachel is: where, - the number of committees formed from 6 other people. - the number of committees formed from Rob or Rachel and one out of 6 other people.

3) probability approach: The probability of choosing Bob or Rachel as a first person in committee is 2/8. 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.

4) reversal probability approach: We can choose any first person. 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. The probability that the committee includes both Bob and Rachel is.

Example #2

Q: Given that there are 5 married couples. If we select only 3 people out of the 10, what is the probability that none of them are married to each other?

Solution:

1) combinatorial approach: - we choose 3 couples out of 5 couples. - we chose one person out of a couple. - we have 3 couple and we choose one person out of each couple. - the total number of combinations to choose 3 people out of 10 people.

2) reversal combinatorial approach: In this example reversal approach is a bit shorter and faster. - we choose 1 couple out of 5 couples. - we chose one person out of remaining 8 people. - the total number of combinations to choose 3 people out of 10 people.

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).

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.

Example #1

Q: 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. What is the probability of Julia winning if the jar contains 1 gray, 2 white and 4 green balls?

Solution: Let's draw all possible outcomes and calculate all probabilities.

Now, It is pretty obvious that the probability of Julia's win is:

Tips and Tricks: Symmetry

Symmetry sometimes lets you solve seemingly complex probability problem in a few seconds. Let's consider an example:

Example #1

Q: There are 5 chairs. Bob and Rachel want to sit such that Bob is always left to Rachel. How many ways it can be done ?

Solution: Because of symmetry, the number of ways that Bob is left to Rachel is exactly 1/2 of all possible ways:

Enumeration

Enumeration is a method of counting all possible ways to arrange elements. Although it is the simplest method, it is often the fastest method to solve hard GMAT problems and is a pivotal principle for any other combinatorial method. In fact, combination and permutation is shortcuts for enumeration. The main idea of enumeration is writing down all possible ways and then count them. Let's consider a few examples:

Example #1

Q:. There are three marbles: 1 blue, 1 gray and 1 green. In how many ways is it possible to arrange marbles in a row?

Solution: Let's write out all possible ways:

Answer is 6.

In general, the number of ways to arrange n different objects in a row

Example #2

Q:. There are three marbles: 1 blue, 1 gray and 1 green. In how many ways is it possible to arrange marbles in a row if blue and green marbles have to be next to each other?

Solution: Let's write out all possible ways to arrange marbles in a row and then find only arrangements that satisfy question's condition:

Answer is 4.

Example #3

Q:. There are three marbles: 1 blue, 1 gray and 1 green. In how many ways is it possible to arrange marbles in a row if gray marble have to be left to blue marble?

Solution: Let's write out all possible ways to arrange marbles in a row and then find only arrangements that satisfy question's condition:

Answer is 3.

Arrangements of n different objects

Enumeration is a great way to count a small number of arrangements. But when the total number of arrangements is large, enumeration can't be very useful, especially taking into account GMAT time restriction. Fortunately, there are some methods that can speed up counting of all arrangements.

The number of arrangements of n different objects in a row is a typical problem that can be solve this way:

1. How many objects we can put at 1st place? n.

2. How many objects we can put at 2nd place? n - 1. We can't put the object that already placed at 1st place.

.....

n. How nany objects we can put at n-th place? 1. Only one object remains.

Therefore, the total number of arrangements of n different objects in a row is

Combination

A combination is an unordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted as:

knowing how to find the number of arrangements of n distinct objects we can easily find formula for combination:

1. The total number of arrangements of n distinct objects is n!

2. Now we have to exclude all arrangements of k objects (k!) and remaining (n-k) objects ((n-k)!) as the order of chosen k objects and remained (n-k) objects doesn't matter.

Permutation

A permutation is an ordered collection of k objects taken from a set of n distinct objects. The number of ways how we can choose k objects out of n distinct objects is denoted as:

knowing how to find the number of arrangements of n distinct objects we can easily find formula for combination:

1. The total number of arrangements of n distinct objects is n!

2. Now we have to exclude all arrangements of remaining (n-k) objects ((n-k)!) as the order of remained (n-k) objects doesn't matter.

If we exclude order of chosen objects from permutation formula, we will get combination formula:

Circular arrangements

Let's say we have 6 distinct objects, how many relatively different arrangements do we have if those objects should be placed in a circle.

The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have:

Tips and Tricks

Any problem in Combinatorics is a counting problem. Therefore, a key to solution is a way how to count the number of arrangements. It sounds obvious but a lot of people begin approaching to a problem with thoughts like "Should I apply C- or P- formula here?". Don't fall in this trap: define how you are going to count arrangements first, realize that your way is right and you don't miss something important, and only then use C- or P- formula if you need them.

Arithmetic Progressions

Definition

It is a special type of sequence in which the difference between successive terms is constant.

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 :

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

Examples

All odd positive integers : {1,3,5,7,...}

All positive multiples of 23 : {23,46,69,92,...}

All negative reals with decimal part 0.1 : {-0.1,-1.1,-2.1,-3.1,...}

Geometric Progressions

Definition

It is a special type of sequence in which the ratio of consequetive terms is constant

General Term is the ith term is the common ratio 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 geometric mean of the other two

For all i,j > k >= 1 :

Summation

The sum of an infinite GP will be finite if absolute value of r < 1

The general sum of a n term GP with common ratio r is given by

If an infinite GP is summable (|r|<1) then the sum is

Examples

All positive powers of 2 : {1,2,4,8,...}

All positive odd and negative even numbers : {1,-2,3,-4,...}

All negative powers of 4 : {1/4,1/16,1/64,1/256,...}

Harmonic Progressions

Definition

It is a special type of sequence in which if you take the inverse of every term, this new sequence forms an AP

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 :

Or in other words :

APs, GPs, HPs : Linkage

Each progression provides us a definition of "mean" :

Arithmetic Mean : OR

Geometric Mean : OR

Harmonic Mean : OR

For all non-negative real numbers : AM >= GM >= HM

In particular for 2 numbers : AM * HM = GM * GM

Example :

Let a=50 and b=2, then the AM = (50+2)*0.5 = 26 ; the GM = sqrt(50*2) = 10 ; 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

For Example : Consider the AP with {1,3,5,7,9,11,...}, so a_n=1+2*(n-1)=2n-1

Pick out the subsequence of terms

New sequence is {9,19,29,...} which is an AP with and

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

For Example : Consider the GP with {1,2,4,8,16,32,...}, so b_n=2^(n-1)

Pick out the subsequence of terms

New sequence is {4,16,64,...} which is a GP with and

The special sequence in which each term is the sum of previous two terms is known as the fibonacci sequence. It is neither an AP nor a GP. The first two terms are 1. {1,1,2,3,5,8,13,...}

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 even. In either case this is also equal to the mean of the first and last terms

Some examples

Example 1

A coin is tossed repeatedly till the result is a tails, what is the probability that the total number of tosses is less than or equal to 5 ?

Solution

P(<=5 tosses) = P(1 toss)+...+P(5 tosses) = P(T)+P(HT)+P(HHT)+P(HHHT)+P(HHHHT)

We know that P(H)=P(T)=0.5

So Probability = 0.5 + 0.5^2 + ... + 0.5^5

This is just a finite GP, with first term = 0.5, n=5 and ratio = 0.5. Hence :

Probability =

Example 2

In an arithmetic progression a1,a2,...,a22,a23, the common difference is non-zero, how many terms are greater than 24 ?

(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. 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. Sufficient

Answer is B

Example 3

For positive integers a,b (a<b) arrange in ascending order the quantities a, b, sqrt(ab), avg(a,b), 2ab/(a+b)

Solution

Using the inequality AM>=GM>=HM, the solution is : a <= 2ab/(a+b) <= Sqrt(ab) <= Avg(a,b) <= b

Example 4

For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) *(1/2^k). 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.

Solution

The sequence given has first term 1/2 and each subsequent term can be obtained by multiplying with -1/2. So it is a GP. We can use the GP summation formula

1023/1024 is very close to 1, so this sum is very close to 1/3

Answer is d

Example 5

The sum of the fourth and twelfth term of an arithmetic progression is 20. What is the sum of the first 15 terms of the arithmetic progression?

A. 300

B. 120

C. 150

D. 170

E. 270

Solution

Now we need the sum of first 15 terms, which is given by :

Answer is (c)

Scope

The GMAT often tests on the knowledge of the geometries of 3-D objects such cylinders, cones, cubes & spheres. The purpose of this document is to summarize some of the important ideas and formulae and act as a useful cheat sheet for such questions

Cube

A cube is the 3-D generalisation of a square, and is characterized by the length of the side, . Important results include :

Volume =

Surface Area =

Diagnol Length =

Cuboid

A cube is the 3-D generalisation of a rectangle, and is characterized by the length of its sides, . Important results include :

Volume =

Surface Area =

Diagnol Length =

Cylinder

A cylinder is a 3-D object formed by rotating a rectangular sheet along one of its sides. It is characterized by the radius of the base, , and the height, . Important results include :

Volume =

Outer surface area w/o bases =

Outer surface area including bases =

Cone

A cone is a 3-D object obtained by rotating a right angled triangle around one of its sides. It is charcterized by the radius of its base, , and the height, . The hypotenuse of the triangle formed by the height and the radius (running along the diagnol side of the cone), is known as it lateral height, . Important results include :

Volume =

Outer surface area w/o base =

Outer surface area including base =

Sphere

A sphere is a 3-D generalisation of a circle. It is characterised by its radius, . Important results include :

Volume =

Surface Area=

A hemisphere is a sphere cut in half and is also characterised by its radius . Important results include :

Volume =

Surface Area w/o base =

Surface Area with base =

Some simple configurations

These may appear in various forms on the GMAT, and are good practice to derive on one's own :

Sphere inscribed in cube of side : Radius of sphere is

Cube inscribed in sphere of radius : Side of cube is

Cylinder inscribed in cube of side : Radius of cylinder is ; Height

Cone inscribed in cube of side : Radius of cone is ; Height

Cylinder of radius in sphere of radius () : Height of cylinder is

Examples

Example 1 : A certain right circular cylinder has a radius of 5 inches. There is oil filled in this cylinder to the height of 9 inches. If the oil is poured completely into a second right cylinder, then it will fill the second cylinder to a height of 4 inches. What is the radius of the second cylinder, in inches?

A. 6

B. 6.5

C. 7

D. 7.5

E. 8

Solution : The volume of the liquid is constant.

Initial volume =

New volume =

Answer is (d)

Example 2 : A spherical balloon has a volume of 972 cubic cm, what is the surface area of the balloon in sq cm?

A) 324

B) 729

C) 243

D) 324

E) 729

Solution :

Answer (d)

Example 3 : A cube of side 5cm is painted on all its side. If it is sliced into 1 cubic centimer cubes, how many 1 cubic centimeter cubes will have exactly one of their sides painted?

A. 9

B. 61

C. 98

D. 54

E. 64

Solution : Notice that the new cubes will be each of side 1Cm. So on any face of the old cube there will be 5x5=25 of the smaller cubes. Of these, any smaller cube on the edge of the face will have 2 faces painted (one for every face shared with the bigger cube). The number of cubes that have exacly one face painted are all except the ones on the edges. Number on the edges are 16, so 9 per face.

There are 6 faces, hence 6*9=54 smaller cubes with just one face painted.

Answer is (d)

Example 4 : What is the surface area of the cuboid C ?

(1) The length of the diagnol of C is 5

(2) The sum of the sides of C is 10

Solution : Let the sides of cuboid C be

We know that the surface area is given be

(1) : Diagnol = . Not sufficient to know the area

(2) : Sum of sides = . Not sufficient to know the area

(1+2) : Note the identity

Now we clearly have enough information.

Sufficient

Answer is (c)

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...The Story of Stuff: A Review The Story of Stuff is a 20-minute film aimed at raising awareness on the dangers of excessive human consumption. Presented by Annie Leonard – an activist for consumerism and environmental sustainability (Story Of Stuff Project, 2013) – this video provides a simple yet well researched explanation on the ‘behind-the-scenes’ management of the life-cycle of common items consumed daily in the United States. From mining and manufacturing to distribution, consumption and waste, this presentation takes viewers on a journey to explore many missing fragments that are hardly ever reported in the system (Leonard, 2009). Watched by over 12 million people worldwide, and translated into over 15 languages (Roosevelt, 2010), Leonard has received both applauds and criticisms for her unapologetic condemnation of humans (especially Americans) – for their excessive consumption and wasteful habits – and her rather conspicuous allegation of the devious relationship between the government and private corporations. One significant issue that Leonard (2009) presented in the video was that, due to human activity, natural resources around the world are rapidly diminishing. Natural resources such as minerals and trees are constantly mined and felled to satisfy men’s insatiable appetite for more goods. The presentation revealed that in the last decade, one-third of the world’s resource base has depleted. Furthermore, it is said that only four......

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