GMAT Math
Percent change
(FV -IV) / IV *100
Graphing Inequalities
-Get into "y=" form (y >mx+b or y < mx + b) Greater than/less than = Dashed line Greater than/less than or equal to = Solid Line
parallel lines
1. Never cross 2. Same slope and different Y-intercepts
sum-of-digits method
1. Sum of all the numbers which can be formed by using the n digits without repetition is: (n-1)!*(sum of the digits)*(111.....n times). 2. Sum of all the numbers which can be formed by using the n digits (repetition being allowed) is: n^(n−1)*(sum of the digits)*(111.....n times). Example: Find the sum of all the four-digit numbers formed using the digits 2,3,4 and 5 without repetition. 3! *(2+3+4+5) * 1111 = 93,324
triangle inscribed in a circle
If 3 points are on a circle, then the triangle is a RIGHT triangle and the hypotenuse measures the diameter.
If Z is divisible by both X and Y, then Z must also be divisible by the ____ of X and Y
LCM Example: if integer z is divisible by 3 & 4, then z must also be divisible by 12 * Note: this DOES NOT mean z is divisible by multiples of the LCM
the LCM of two positive integers x and y cannot be ______ than either of them
Less
2 is the only even ____ number
Prime
If a person/letter must sit on the left or right of another person/letter
Take the total number of arrangements and divide by 2 to get number of arrangements.
Different prime factors of n!
To find the different prime factors of n!, just take all the prime numbers less than n. Example: The different prime factors of 6! is 2,3,5
"at least" combination problems
Typically involve the addition of outcomes.
Creating Codes (Mississipi Rule)
Use the permutation formula for indistingushable items P = N!/(r1!)*(r2!)*...(rn!) N = total # of objects to be arranged r = the frequency of each indistinguishable object.
Catch up rate questions
We typically determine how long it will take for the faster car to catch the slower car. The time is a function of the difference in speeds of the 2 objects. * The distance will be equal when they catch up.
Remainder after division by 10^n
When a whole number is divided by 10, the remainder will be the units digit of the numerator.
perpendicular bisector
a line, ray, or segment that intersects the line segment at its midpoint and forms a right angle
Sum of Consecutive Integers
(1) Average the first and last term to find the middle of the set (2) Count the number of terms (3) Multiply the middle term by the number of terms to find the set Formula: N*(first+last)/2 Ex: Sum of even integers from 100 to 300 1. Average: 100 + 300 / 2 = 200 2. Count: (Highest # divisible by the given number+ Lowest # divisible by the given number)/ Given number )+ 1 --> (300 + 100)/2 + 1 = 201 3.Multiply: 201*200 = 20,200
# of multiples in a range
(Last multiple of x in range - first multiple of x in range / multiple )+ 1
Arrangements in a circle
(n-1)!
Reflection over origin
(x,y) -> (-x,-y)
Reflection over y-axis
(x,y) -> (-x,y)
Reflect over y = -x
(x,y) -> (-y,-x)
Reflect over x = a
(x,y) -> (2a - x, y)
Reflection over point (a , b)
(x,y) -> (2a-x, 2b-y)
Reflection over x-axis
(x,y) -> (x,-y)
Reflect over y = b
(x,y) -> (x,2b-y)
Reflect over y = x
(x,y) -> (y,x)
midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
Special case of choosing "At least 1" from a group (combinations)
- Apply mutually exclusive and collective exhaustive. Total # ways = # ways to include at least 1 item (this is what we are looking for)+ # ways that DOES NOT include at least 1 item
Divisibility Rules
0 - no number 1- any number 2 - even number 3 - if sum adds up to a multiple of 3 4 - if last two digits is a multiple of 4 5 - if unit digit ends in a 5 or 0 6 - if the number is divisible by 2 & 3 7 - Just do DIVISION 8 - if the number is even, divide the last 3 digits by 8 9 - if sum is divisible by 9 10 - if unit digit is a 0 11 - sum of (odd numbered place) - (even numbered) is divisible by 11 --> 2915 = (9 + 5) - (2 + 1) = 11/11 --> yes 12 - if number is divisible by 3 & 4
Rate relationships
1) Distance is directly proportional to Rate and Time 2) Rate is inversely proportional to Time and Directly proportional to Distance 3) Time is inversely proportional to Rate and Directly proportional to Distance.
8 major types of rate*distance problesm
1) Elementary 2) Average rate 3) Converging Rate 4) Diverging Rate 5) Round Trip Rate 6) Catch up 7) Relative motion 8) if/then questions
perpindicular lines
1) LINES CROSSING AT RIGHT ANGLES. 2) Any pair of perpendicular lines create 4 right angles 3) the slopes of 2 perpendicular lines are negative reciprocals (slopes multiply to -1)
How to find the number of factors in a particular number
1) Prime factorize 2) Add 1 to the value of each exponent, then multiply 3) Result is the total # of factors
5 common reflections
1) Reflection over x-axis 2) reflection over y-axis 3) Reflection over origin 4) Reflection over point 5) Line reflection
You can define a line by knowing a point and:
1) the slope of the line, or the line that is parallel or perpendicular to the line. 2) A second point on the line (y-intercept, x-intercept, or any other point)
The smallest circle will be between __ points on the circumference
2
finding an area of a triangle in a coordinate plane with 3 given vertices
2 options: 1) Find out if it is right triangle 2) Draw a larger shape around the triangle, such as a rectangle and subtract the areas.
Catch up and PASS questions
2 variations: 1) The faster objects starts at the same location. and must catch up and pass the slower object and reach some distance beyond the slower object. 2) The faster object starts at a location behind the slower objects starting point and must either catch up with with the slower object or must catch up and pass the slower object by some distance. * VERY SIMILAR To catch up questions, EXCEPT the distance that the objects travel are not necessarily equal. * We set up the FASTER objects distance equal to the slower objects distance PLUS (+) the additional distance the faster object has traveled.
Number of games played for n team to win
2^n - 1 = number of losers
Remainders exhibit pattern
3^123/4 has a reminder pattern of 3-1-3-1
Dependent combinations
A & B are dependent if the # of ways in which event B could be selected depends on which specific way event A is selected.
The information needed to define a line
A second point on the Lin, we can then calculate the slope, y-intercept, and x Interco.
We can _____ ,_____ ,_____ reminders
Add, subtract, and multiply *Note: Correct any excess remainders
exponential growth
An initial value grows by a constant factor during each growth period Initial Value * (growth factor) ^ (growth period)
Terminating decimals
Any denominator whos prime factors contain only 2,5, or both.
anchor method
Apply when encoutering arrangements with restrictions. - Begin by "anchoring" the restricted items in their designated positions prior to handling the other items that must be arranged.
When distance is same, the average speed is
Average speed = 2*(rate1 * rate2) / (rate1 + rate2)
Some items can never be together in the same subgroup (combinations)
Consider using your knowledge of collective exhaustive and mutually exclusive Total # of ways in which the scenario can occcur =# of ways in which A can occur + # of ways in which B can occur Example: 3 people are to be selected from a pool of 5 people. If 2 of the people can never be together in the club, how many different ways can the club be formed? Total # of ways = # of ways in which they are part of the same club + # of ways they are NOT part of the same club (This is what we are looking for)
the number of integers between two evenly spaced integers inclusive
Count: (Highest # divisible by the given number+ Lowest # divisible by the given number)/ Given number )+ 1
Round trip questions
Distances are equal in both directions, but the time can differ from the time it takes to return. Example: if total time = 10hours, then... one leg= t 2nd lef = 10 - t.
Odd + odd Common terms will result in ____ term Opposite terms will result in ___ term
Even Odd
linear growth formula
F(n) = K*n + p Where F is the final value after the growth has occured K is the growth constant n is the number of growth periods that have occurred. P is the original value prior to the growth
Permutation Formula for indistiguishable items
For a permutation problem that consists of identical items, use: P = N!/(r1!)*(r2!)*...(rn!) N = total # of objects to be arranged r = the frequency of each indistinguishable object.
Divisibily rule
If integers a and b are both multiples of some integer k>1 (divisible by k), then their sum and difference will also be a multiple of k (divisible by k):Example: a=6 and b=9, both divisible by 3 --->a+b=15 and a−b=−3, again both divisible by 3. If out of integers a and b, one is a multiple of some integer k>1 and another is not, then their sum and difference will NOT be a multiple of k (divisible by k):Example: a=6, divisible by 3 and b=5, not divisible by 3 ---> a+b=11 and a−b=1, neither is divisible by 3. If integers a and b both are NOT multiples of some integer k>1 (divisible by k), then their sum and difference may or may not be a multiple of k (divisible by kk) :Example: a=5 and b=4, neither is divisible by 3 ---> a+b=9, is divisible by 3 and a−b=1, is not divisible by 3;OR:a=6 and b=3, neither is divisible by 5 ---> a+b=9 and a−b=3, neither is divisible by 5;OR: a=2 andb=2, neither is divisible by 4 ---> a+b=4 and a−b=0, both are divisible by 4.
Property of consecutive integers
If n is odd, the sum of n consecutive integers is always divisible by n. Given {9,10,11}, we have n=3= odd consecutive integers. The sum is 9+10+11=30, which is divisible by 3. if n is even, the sum of n consecutive integers is not divisible by n. Given {4,5,6,7}, we have n = 4 = even consecutive integers. The sum is 4 + 5 + 6 + 7 = 22, which is not divisible by 4.
When some items must be together, Link those items together
Link together any items that can be considered one unit. - When x items must be together in an arrangement of y items, treat those x items together as one unit. Now, there are Y - x + 1 items that can be arranged (y - x + 1)!. In addition, the x items can be arranged in x! ways so the total ways to arrange things is x! * (y-x+1)!
Evenly Spaced Sets
Mean = median = (first+last)/2
When a distance is not given in rate questions, you can create your ___
Own
Every even number greater than 2 can be written as the sum of two ____ numbers.
Prime
Prime factorization of a perfect cube
Prime factorization will only contain multiples of 3.
Line reflection
Rules are same Simply reflect the endpoints of the line segment, draw a line segment connecting them, and this new line segment is the reflection.
Converging rate and their variations
Set up a matrix, when they meet the total distance will be the sum of the individual distances of each object d1+ d2 = total Distance 4 Variations: 1) Both leave at the same time 2) Objects leave at different times 3) One object travels faster(or slower) than the other objects 4) One object is RELATIVELY faster(or slower) than the other object.
Average of n consecutive integers or evenly spaced set
The average of consecutive integers is the median if n = odd, then the average = middle term = (n + 1)/2 if n = even, then the average = median = average of the 2 middle numbers = average of (n/2) & (n+2)/2 positions
Collectively Exhaustive Events
The collection of all possible outcomes in an experiment. If 2 events A & B occur in a scenario and are collectively exhaustive and mutually exclusive, it must be true that #total ways = # of ways in which A can occur + # of ways in which B can occur.
greatest common divisor
The greatest number that divides into two or more numbers with no remainder.
if/then rate questions
They provide us info about a scenario that actually happened, and information about a scenario that would have happened if the actual set of. conditions have been met. "if [object] had traveled [some rate], it would [saved/added] t hours to its time. * TO approach the question, define the actual rate and time of object, and then define hypothetical rate & time of object, and then substitute.
Some items CANNOT be next to each other (permutations)
Use collectively exhaustive and mutually exclusive concept. 1) determine the total # of arrangements 2) determine the total # of arrangements with the items together 3) # of arrangements in which the items are NOT together comes from the formula Total # = Total # together + Total # not together.
Fundamental Counting Principle
Uses multiplication of the number of ways each event in an experiment can occur to find the number of possible outcomes in a sample space. m ways to perform one tasks and n ways to perform another task, if the tasks are indendedent, there are m*n possible ways to perform both taks.
Diverging Rate question
When 2 objects start at the same points, and move away from each other, the distance they cover is a function of the sum of their speeds. R1*t + R2*t = Total Distance
Combinations with restrictions
When certain items must be selected, determine how many items remains in the main pool and how many spots remain to be filled in the subgroup. When certain items MUST NOT be selected, we eliminate them from the main pool.
Remainder after division by 5
When integers with the same unit digit are divided by 5, the remainder will always be the same Example: 9/5 = R4, 19/5 = R4, 29/5 = R4.... 59/5 = R4
Vowels
a, e, i, o, u
N! is divisible by.....
all of the integers from 1-n and any factor combinations from 1-n.
average of consecutive numbers
average of smallest + largest. (S + L)/2
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²]
if x is divisible by y, then x is ALSO divisible by all the _____ of ____
factors, y
Variation 2: two objects leave at different times, add time to the object leaving _____ because the total travel time will be greater for that object and use a matrix.
first
In converging or diverging questions, you may want to focus on the _____ shrinking or expanding
gap * If the objects are moving in opposite directions, then then add the rates. (converge or diverge) * if the objects are moving in the same direction, then subtract the rates (catch up)
Determining the number of leading zeroes in the fraction 1/x
if x is an integer with k digits, then it will have k - 1 leading zeros, unless it is a perfect power of 10, then it will have k-2 leading zeros.
The product on any set of n consecutive integers is dvisible by ___
n!
Sum of the first n integers
n(n+1)/2
Number of diagonals in a polygon
n(n−3)/2 Where n is the number of sides
Combination Formula
nCr = n!/r!(n-r)! Or Box and fill method example: Choosing 3 from a pool of 7 - each box represents a decision that is to be made 7 * 6 * 5 / 3
Permutation Formula
nPr = n!/(n-r)! Or Box and Fill but DO NOT divide the boxes by the factorial of the number of the boxes.
Co-prime numbers
numbers that have no factors in common with another, usually consecutive numbers. Example: Does 40! + 1 have 11, 19, or 37 as prime factors? 40! and 40!+1 are consecutive integers. Hence, no common factors. => 11, 19, and 37 are factors of 40!. So, these can't be a factor of 40!+1.
Colinear
points that lie on the same line
Two consecutive integers will never share the same _____ factors
prime
Variation 3: One object travels faster than another object, then let the slower objects speed be r and the faster object speed be _____. Use a _____ form and have the SAME units
r + difference in speed, matrix
Variation 1: Both objects leave at the same time, you will represent the travel time of each object using the ____ variable and use a matrix
same
Calculating an unknown number of items in a group (combinations)
set up an Box and Fill equation with x being variable: x* (x-1)/2! = # of ways
Catch up and pass formula
time = delta(distance)/delta(rate) change in distance is simply the extra distance that the faster object must travel over that of the smaller object.
Average Rate
total distance/total time * DO NOT add a set of rates together and then divide by 2.
LCM provides us with all the ____ _____ factors of some set of positive integers
unique, prime
Patterns in Unit digits
when a positive digit is raised to consecutive powers, a pattern will arise in the units digit. Note: all integers greater than 9 follow the same unit digit pattern as the powers of its unit digit. 3
LCM* GCF = ___
x*y
Converting a decimal remainder to an integer
x/y = n.d remainder =y*d
if x in the range 0 < x < 1, then...
x^2 < x < sqrt(x) < 1
Number of integers from x to y inclusive
y - x + 1
slope-intercept form
y=mx+b, where m is the slope and b is the y-intercept of the line. * The equation must be in slope-intercept form prior to answering questions.
sqrt(2) value
~1.3
