GMAT - Quant Basics

Reciprocal of 1 and -1

1 and -1

Subtracting a fraction from a whole number. Complete example: 6-(2/3)=

5(3/3) - (2/3)

Adding/subtracting fractions. Complete the example: (a/b)-(c/d)

((ad)-(cb))/(bd)

Equivalent fractions exist if: (a/b)=(c/d) if…

ad=cb

different denominator addition. Finish this formula: (a/b)+(c/d)=

((ad)+(cb))/(bd)

Comparing Fraction Sizes. Complete: (a/b)>(c/d) if

(ad)>cb

properties of a number between 0 and 1

x² < x < sqrt(x)

Common quadratic equation 1: (x+y)²

x² + 2xy + y²

Common quadratic equation 2: (x-y)²

x² - 2xy + y²

Common quadratic equation 3: (x+y)(x-y)=

x² -y²

First quadratic identity example : x² - 9

(x-3)(x+3)

Second quadratic identity example 4x² - 100

(2x - 10) (2x+ 10)

Third quadratic identity example: 3^(30) - 2^(30)

(3^15)² - (2^15)² = (3^15 + 2^15)(3^15-2^15)

Number divisible by 3 if…

the sum of all its digits are divisible by 3

Number divisible by 4 if…

the last two digits of a number are a number that is divisible by 4. for example 244, because 44 is divisible by 4. Also, 00 is divisible by 4. Such as 100 is divisible by 4.

Number divisible by 6 if…

if the number is divisible by both 2 and 3

Number divisble by 8 if…

the last 3 digits of the number are divisible by 8. Meaning, when dividing the last 3 digits, it comes out as an integer. Remember 000 is divisible by 8.

A number is divisible by 9 if..

The sum of all the digits are divisible by 9

Number is divisible by 11 if…

the sum of the odd-number place digits minus the sum of the even-number digits place is divisible by 11

How many problem solving questions are on the test?

21

When solving DS questions, we use any given information and the information from two statements to determine whether we can _____ answer the question

Definitively

Answer choice A for DS questions

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Answer choice B for DS questions

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Answer choice C for DS questions

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Answer choice D for DS questions

EACH statement ALONE is sufficient to answer the question asked.

Answer choice E for DS questions

Statements (1) and (2) TOGETHER are NOT sufficient

What are the two groups of answer choices we can group together for DS questions?

AD and BCE

If statement 1 is not sufficient, which answer choice group do you eliminate?

AD, and BCE remaining

If statement 1 is sufficient, which answer choices do you eliminate?

B,C, E

Is creating a common denominator the only way to compare fractions? And how would you know which is bigger?

No, changing to a common numerator allows for this, then the number with the largest denominator is the smallest fraction.

Multiplying numbers with decimals. Complete the example: 15.534 × 2.8 =

15534×28= 434952, then move the decimal as many points as the two original numbers combined (in this case 4)

when dividing decimals, what do you do? Complete the example:
10.36 / 2.8

Make the divisor (2.8) a whole number (28) and move the dividend decimal the same number of times (103.6), then divide.

When comparing decimals, how do you know which is largest?

Compare at the tenths place first, then as you go, which ever number is larger.

What is the first step of adding percentages together?

Convert them into fractions and then add and change to a percentage

Write 3.4 and .34 as fractions

34/10 and 34/100

Convert the following to decimals (rounded to the thousandths) 1/2 , 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10

.5, 0.333, 0.25, 0.2, 0.167, 0.143, 0.125, 0.111, 0.1

when converting 5/7 to a decimal, what are my steps?

1/7 = 0.143, and then multiply by 5, so 0.715

Definition of a perfect square. Give the example with 144

Is a number where all of its prime factors have even exponents. 144= 2^4 × 3²

Definition of a perfect cube, Give the example with 27.

A perfect cube, other than 1 or 0, is a number such that its prime factors have exponents that are divisible by 3. 27 = 3³

Perfect squares to memorize

0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225

perfect cubes to memorize

0,1,8,27,64,125,216,343,512,729,1000

non-perfect square roots to memorize

sqrt(2)=1.4, sqrt (3) = 1.7, sqrt (5)= 2.2 (approximately)

principal square root definition

the non negative square root of a number. For example 2 is the principal square root of 4, not -2

For all nonnegative numbers, sqrt(a) * sqrt(b)=

sqrt(ab)

What should I do if I see an operation such as .8977×112,568,219?

Round to 1×100,000,000 or whatever the answer choices look closest to.

What should I do first with answering a math question?

1. read the stem and look at the answer choices.

2. think about how you may answer it, is there any way to eliminate answers?

What do we do when we’re presented with a large integer number raised to the second power?

if the answer choices all have unique ones digit, then we can square just the ones digit to get the result.

What to do when presented with a non integer number raised to the second power?

Square the digit furthest to the right

Perfect squares have to end in which numbers?

0,1,4,5, 6, 9

In PEMDAS, what order does absolute value signs and radicals come in as?

Parenthesis

if n is a positive integer, n! includes what?

the product of all integers between 1 and n

0! and 1! are equal to what?

1 and 1

If you a solve a system of equations where you produce k=0 where k is a nonzero number, what is this called?

it has zero solutions

A system of equations have infinite solutions if what?

The two equations are identical

if a system of equations comes out to 0=0, then how many solutions does it have?

It has infinitely many solutions

if a system of equations comes out that the variable terms are equal, but the constant terms are no, what is it called?

It has zero solutions

If solving a system of equations with fractions, what should I do?

consider multiplying the entire equation by the LCM of the denominators to cancel it out

What are the roots or the solutions to the factored quadratic (x+1)(x+8)

-8,-1

The discriminant is the number under the radical in the quadratic equation. If it is positive, zero, or negative, how many roots are there in the equation?

positive =2 roots; zero= 1 root; negative=0 roots

Given the quadratic formula ax² + bx +c, what is the sum of the two solutions? Given it has exactly two solutions.

-b/a

Given the quadratic formula ax² + bx +c, what is the product of the two solutions? Given it has exactly two solutions

c/a

how do you solve x²=100x

move 100x over and factor to solve

(x+y)³ =?

x³+3x²y+3xy²+y³

(x-y)³

x³-3x²y+3xy²-y³

(a/b)²

A²/B²

A * (b/c) simplified

(a*b)/c

The sum or difference will be an ____ number only if both of the numbers are even or both are odd

even

Odd times and even number is

even

Odd x odd is

odd

The product of an even number and any number is always

even

Even number/odd number is

even number

Odd number is divisible by an odd number

odd number

If finding an algebraic expression is even or odd with variable X, it must only include what types of operations?

addition, subtraction and/or multiplication

If finding an algebraic expression is even or odd with variable X,we can determine how the expression behaves when x is even by replacing x with _____

zero

If finding an algebraic expression is even or odd with variable X,we can determine how the expression behaves when x is odd by replacing x with _____

one

two even numbers added or subtracted will be what? Two odd numbers added or subtracted will be what?

even number, even number

and even and an odd number added together will be what?

odd number

even/even =

even

even/odd=

can be even or a non-integer

odd/odd

can be either odd or a non-integer

odd/even

is always a non-integer

x is a multiple of y if and only if (?) is an integer?

x/y

What are the first 25 prime numbers?

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,84,89,97

How do you find the total number of factors of a number?

Find the prime factorization, add 1 to the value of each exponent, then multiply these results.

how to determine how many positive odd factors N has

1) find prime factorization 2) from the prime factorization of N, remove the power with base 2 (if there is one) 3) add 1 to each remaining exponent and multiply the resulting sums.

What number is a factor of all integers

1

Definition of Unique prime factors

number of factors that differ from each other. Ex: 6= 2×3 (2 unique prime factors). Ex 12= 2×2×3 (2 unique prime factors)

When does the number of unique prime factors not change?

when that number is raised to a positive integer exponent. Prime factorization of 18 is 3² 2 (2 unique prime factors). 18³= 5832. Prime factorization of 5832 is 3^6 2³. 5832 has same unique prime factors of 18:3 and 2

Steps to find the LCM of two integers

1. Prime factorize the two integers

2. Pick out the higher exponents of the repeated prime factors on both sides

3. Bring down all extra factors as well

4. multiply them together

If finding the LCM of a set of positive integers, how many numbers in the set have to share a prime factor for the prime factor to be considered repeating?

at least two

If two positive integers, x and y, share no prime factors, the LCM is what? And otherwise, what?

xy, otherwise the LCM is some number less than xy.

Steps to finding the GCF

1. Prime factorize each number

2. Identify repeated prime factors among the numbers

3. of the repeated prime factors among the numbers, take only those with the smallest exponent (if there are none, the GCF is 1)

4. multiply together the numbers that you found in step 3; the product is the gcf

LCM will always be greater than or equal to what?

the largest number in the set

The GCF will always be less than or equal to what?

the smallest number in the set

If we know that positive integer y divides evenly into positive integer x, the LCM of x and y is what?

x

If we know that positive integer y divides evenly into positive integer x, the GCM of x and y is what?

y

What is the relationship (equation) between the two integers, the LCM and GCF?

x*y= LCM (x,y) * GCF (x,y)

If two or more entities return to a common starting position at various frequencies, how do you find the shortest amount of time it takes for all of the entities to return to the same starting point?

the LCM

positive integer x is divisible by positive integer y if an only if what?

the prime factorization of x contains the prime factorization of y. Example: 15 =3×5, 60=2×2×3×5, so 60 is divisible by 15

If x and y are positive integers and x/y is an integer, then x/(any factor of y) is what?

also an integer