Combo with GMAT Math 1 and 19 others
what's the general probability formula?
# of desired or successful outcomes / Total # of positive outcomes
Key Note: 2^ -x < .01 is = 2^x > .01^ -1
..
Be prepared to use proportions in Data Sufficiency
...
Fewer Factors, More Multiples.
...
If you add or subtract two multiples N, you get another multiple of N
...
1/6 =
.1667
1/5
.2
1/4
.25
1/3
.333
1/2
.5
xy/99 =
.xyxy
0/1 = ?
0
Number Properties
0 and 1 are not prime #'s
Number Properties
0 is an even number
It takes an elevator four seconds to go up one floor. How many floors will the elevator rise in two minutes?
1 floor/4 seconds or .24 floors/second (NOT 4 seconds per floor --> this is a frequent error)
30-60-90 =
1,2,sqrt3 - x, xsqrt 2, 2x
Lines Cut by a Transversal
1. Interior angles formed by intersecting lines add up to 360 degrees
Area of Trapezoid
1/2 (long base+short base) * height
Area of a triangle ________?
1/2 * B * H
What is the formula to calculate the area of a triangle with the following vertices L(1,3), M(5,1), and N(3,5)?
1/2 [X1(Y2-Y3) + X2(Y3-Y1) + X3(Y1-Y2)]
11^2
121
2^5
32
2^9
512
75^2
5625
800/3 =? Another way to format -
8/3 * 100
2500/3 = ?
800 r. 100/3
3^4
81
9^2
81
Question: How many multiples of 3 are within range of 244-811?
811/3 = 270 r.1 244/3 = 81 r.1 270-81=189 + 1= 190 or 811-244= 567/3 = 189 +1
3^2
9
When the #9 is raise to an odd power, ex. ^3- the units digit will equal ?
9
if there are 9 baseball teams - what is the total number of combinations for the top 4 spots?
9x8x7x6/4x3x2x1
(2 x 5)^3 =
= (10)^3
Sum of the number of consecutive integers =
= (Last - First + 1)
Sum of the number of consecutive multiples =
= (Last - First)/(Increment) + 1
9 raised to an even power, the units digit is ______?
= 1
9 raised to an odd power, the units digit is ______?
= 9
y percent more than z
= z + z (y/100)
y percent less than z
= z - z(y/100)
8k8 + k88 1,6p6 If k and p represent non-zero digits within the integers above, what is p? 6 7 8 9 17
A
Area of Trapezoid
A = (sum of bases)(height)/2 A = {[(b1 + b2)/2](height)}/2
Simple Interest
A = P(1 + r)n A = amount accumulated P = principal r = annual rate of interest n = number of years
Area of a Rhombus
A = bh OR A = [(d1)(d2)]/2
A Common Digits Problem
A Common Digits Problem BA => 47 or 83 +AB +74 +38 CDC 121 121
The greatest possible distance within a rectangular solid is =
A diagonal touching opposite corners equation = sqrt of(L^2 + W^2 + H^2)
Function Graphs
A functions can be visualized by graphing it in the coordinate plane. The input variable is considered the domain of the function, or the x-coordinate. The corresponding output is considered the range of the function, or the y-coordinate Create input & out put tables and plug in the x coordinate to find the y coordinate and vice versa
Gerund
A gerund is the -ing form of a verb acting as a noun, in any of the "noun roles" possible in a sentence. A gerund can be the subject, the direct object, or the object of a prepositional phrase.
DS: Rephrase
A good data sufficiency strategy is to rephrase the information in a question: z + z < z? => z < 0? (...or 0 < z < 1)
What is a multiple?
A multiple of an integer is formed by multiplying that integer by any integer, so 8, 16, 24 and 32 are some of the multiples of 8.
Prime Numbers
A prime number is a positive integer that has exactly two different positive divisors: 1 and itself. • 1 is NOT prime • 2 is both the smallest prime and the only even prime
Pythagoreans' Theorem (*only with right triangles) -
A^2 + B^2 = C^2
DS: Yes/No Question Frequency
About 1/3 of DS questions are "Yes/No" questions.
And means what in probability?
And means multiply the probabilities. You'll wind up w/ a smaller number, which indicates a lower probability.
Similar triangles
Angles are equal; areas proportional
A rabbit on a controlled diet is fed daily 300 grams of a mixture of two foods, food X and food Y. Food X contains 10 percent protein and food Y contains 15 percent protein. If the rabbit's diet provides exactly 38 grams of protein daily, how many grams of food X are in the mixture?
Ans:0.10X + 0.15(300 - X) = 0.38 .... 140
Does the integer k have a factor p such that 1<p<k? (1) k>4! (2) 13! + 2 <= k < 13! +13
Answer B: 1 - if, for example, k=29 then p could not have a factor of p.... 2. for each number of k from 13! +2 to 13! + 13 there is a factor p such that 1<p<k; Sufficient
How many distinct prime divisors does a positive integer N have? 1. 2N has one prime divisor 2. 3N has one prime divisor
Answer: C 1. N has no prime divisor or N has only digit 2 as a prime divisor/factor.... 2, 4, 8 2. Same as 1
Set Problem: Each of 25 people is enrolled in history, math, or both. If 20 are enrolled in history and 18 are enrolled in math, how many are enrolled in both?
Answer: create a Venn diagram with one circle for history, one for math, and an overlapping space. Overlap = n History only = 20 - n Math only = 18 - n n + (20 - n) + (18 - n) = 25 38 - n = 25 n = 13 people in both history and math
Prime Factor?
Any number that is divisible by 1 and itself
Circles - Arc Length
Arc length can be found by determining what fraction the arc is of the entire circumfrence (To find length, first find the circumference of the circle).
Vertices of a triangle have coordinates (1,0), (4,0) and (0.A). Is the area of the triangle bigger than 15? 1. A<3 2. The triangle is right
B stmt 2: if the given triangle is right angle then one side which forms the hypotenuse is 5 so the other two sides will be less than 4. so definitely area of triangle has to be less than 15
No matter how many 2's (or more specifically the same prime) are on 1 side of an equation; all those two's can NEVER -
Combine with one another in such a way to form a multiple of another prime ex. 2's on 1 side & the number 15, or 3^5 on another side
P(E)P(F)
Combined Events: E and F
P(E) + P(F) - P(E and F)
Combined Events: E or F
1 - P(E)
Combined Events: Not E = P(not E) = ?
2nd Rule of Probability: Complementary events
Complementary Events: The probability of an event occurring plus the probability of the event not occurring = 1. P(E) = 1 - P(not E)
Principal (1 + interest/number times compounded)^(t)(n)
Compound interest formula
A basketball coach will select the members of a five-player team from among 9 players, including John and Peter. If the five players are chosen at random, what is the probability that the coach chooses a team that includes both John and Peter? 1/9 1/6 2/9 5/18 1/3
D Teams with John and Peter = (1)(1)(7!)/((4!)(3!)) Total = (9!)/(5!3!)
If x is a positive integer, is x^3 - 3x^2 + 2x divisible by 4? (1) x = 4y + 4, y = integer (2) x = 2z + 2, z = integer
D. Both 1 and 2 are sufficient
Small standard deviation
Data points are clustered
Large standard deviation
Data points are spread apart
Add or subtract 2 odd numbers, and the result is...
EVEN
Consecutive Integers
Even: 2n, 2n + 2, 2n + 4 Odd: 2n + 1, 2n + 3, 2n + 5
Compound Interest Example: If $10,000 is invested at 8% annual interest, compounded semiannually, what is the balance after 1 year?
Final balance = Principal x (1 + r/n)^(yn) Final = 10,000 x (1 + .08/2)^(1)(2) = 10,000 x (1.04)^2 = $10,816
Calculate out the first DS questions to make sure they are correct. It is important to start out the section strong.
First DS questions...
Factors
For "N" to be a factor of "X", N must contain ALL prime factors that X does. Must beware of overlapping factors. All primes must be unique.
the total amount before any deductions
Gross
multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs
How do you multiply roots together.
What is the absolute value?
How far away a number is from 0
Absolute Value?
How far away is the number from 0 on the number line?
Rate Problem
How far we have to go/ how fast we are getting there
14 liters
How many liters of a solution that is 10% alcohol by volume must be added to 2 liters of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume?
0.15n + 0.08(5) = 0.1(n+5)
How many liters of a solution that is 15% salt must be added to 5 liters of a solution that is 8% salt so that the resulting mixture is 10% salt?
about 1/2 of the time
How often do you have to look at both statements combined?
Check each prime number up to the approximate square root of the number. If you haven't found a number less than or equal to the square root of the number, then the number is prime.
How to check for a prime number.
Sum of digits is multiple of 3, last two digits multiple of 4.
How to check whether a number is a multiple of 12.
Sum of digits is multiple of 3
How to check whether a number is a multiple of 3.
Last two digits are multiple of 4 or the number can be divided by 2 twice.
How to check whether a number is a multiple of 4.
Number is a multiple of 3 and 2
How to check whether a number is a multiple of 6
$11, 025
If $10,000 is invested at 10% annual interest, compounded semi-annually, what is the balance after 1 year?
Combos & Equations Strategy
If a DS question has an equation with multiple variables in the question stem, it is probaly a Combo is disguise - In this situation isolate the wanted variable and create the simplest combo you can ** Do not make assumptions about sufficiency based on the number of variables/equations. Sometimes one statement can provide sufficient information about two variables, and sometimes two statements together are still not enough to find the value of one variable**
Factor foundation Rule
If a is a factor of b and b is a factor of c, the a is a factor of c
Probability and Geometry
If a point is chosen at random within a space with an area, volume, or length of Y and a space with a respective area, volume, or length of X lies within Y, the probability of choosing a random point within Y is the area, volume, or length of X divided by the area, volume, or length of Y.
What are distinct numbers?
If two numbers are distinct, they cannot be equal.
Percent Change
If two precent changes are the same in a profit like relationship, then the third change will be equal to the other two.
Reciprocals of Inequalities
If x < y then: - 1 / x > 1 / y (when both x and y are positive. Flip the inequality) - 1 / x > 1 / y (when x and y are both negative. Flip the inequality) - 1/ x < 1/y (when x is negative and y is positive. Do not flip the inequality) If you dont know the signe of x or y, you cannot take reciporcals ** if you know the signs flip the inequality, unless they have different signs**
Absolute Value
If you have a value inside of absolute value signs (isolate value on one side)..drop the absolute value and set up two equations, on positive and one negative (ie. IzI=4, z=4 and -z=4 .. solve)
D or E
If you have to guess in a problem, which ones should you guess? Especially if you have to plug numbers.
Factor Out and Simplify
Immediately try factoring/simplifying when possible. Example: Is 2x/6 + 24/6 an integer? => (2x + 24)/6 => x/3 + 4
What are consecutive integers?
Integers listed in order of increasing size without any integers in between
Simple Interest Example: If $12,000 is invested at 6% simple annual interest, how much interest is earned after 9 months?
Interest = (12,000)(.06)(9/12) = $540
Diagnoles of rhombus
Intersect at midpoints and are perpendicular bisectors
| A union B| = |A| + |B| - |A intersect B|
Intersecting Sets
Consecutive Interger DS Problem
Is k^2 off? 1) K - 1 is divisible by 2 2) The sum of consecutive integers is divisible by K Statemement (1) tells us that K must be odd so K^2 will be odd Statement (2) tesll us that the sum of K consecutive integers is divisible by K. Therefore, this sum divided by K is an integer. Moreover, the sume of K consecutive integers divided by K is the avg of that set of K integers. As a result, statement (2) tells us that the avg of the K consecutive intergers is and integer. Therefore, must be ODD.
Inequalities : Multiplication
Is mn < 10 1) m < 2 2) n < 5 Tempting to multiply inequalities together. However negative values would creaete a positve number > 10. Could only multiply if the question stated that both m and n were POSITIVE
Joey runs a race 30 seconds faster than Tommy
Joey --> t - 30 Tommy --> t
Kelly received 1/3 more votes than mike
Kelly received M + 1/3(M)
Lie vs. Lay
Lie - To recline or be located. PT lay PPT lain Lay - to put, place or put down. PT Laid Laid
Trial Problems
Look at the probability of NOT OCCURRING. P(Event Not Occurring) = 1 - P(Event Occurring)
1. Start by writing each number as a product of primes. 2. Write so that each new prime factor begins in the same place. 3. Lowest common multiple is found by multiplying all factors in either list.
Lowest Common Multiple 60: 2 x 2 x 3 x 5 72: 2 x 2 x 2 x 3 x 3 LCM: 2 x 2 x 2 x 3 x 3 x 5
Ratio of men to women in a room is 3 : 4. If there are 56 people in the room, how many of the people are men?
M men/W women = 3/4 and M + W = 56 OR 3x + 4x = 56 x = 8 M = 3 * 8 = 24 NEVER HAVE TWO UNKNOWN MULTIPLIERS IN A PROBLEM
Fractions - Multiplication
Mulitply tops and bottoms, CANCELLING COMMON FACTORS FIRST. Can even break fractions down in to their units (ie. 20/10 = (5*4)/(5*2)...can cancel the 5's)
Quick Division & Multiplication
Multiple by 2 & Divide by 10. AB=2A*(1/2B). Example: 3.5*22=2*3.5*(22/2)=7*11=77
3, 6, 9, 12
Multiples of 3
Is 0 positive or negative?
Neither
the amount after deductions
Net
Coordinate Plane
Note that sometimes the GMAT will only give you one point on the line, along with the y-intercept. This is the same thing as giving you two points on the line because the y-intercept is a point! (ie. a y-intercept of 4 is the same a (4,0)
Total = mean x (number of terms) Number deleted = (original total) - (new total) Number added = (new total) - (original total)
Number added or deleted
Sum of 2 and any other prime is...
ODD
DS: Both Together
Only about half the time do you have to look at both statements in combination.
n! / (n - r)!
Permutations: Order Matters
What is the arithmetic order of operation?
Please excuse my dear aunt sally *Parenthesis, exponents, multiplication, division, addition, subtraction
Fractions - Square's
Proper fractions produce smaller numbers (ie (1/3)^2 = (1/9)
Even integer. Neither positive nor negative. Multiple of every number. Not a factor of any number.
Properties of 0
Purchase Price vs. Market Value
Purchase Price = price purchased for by wholesaler Market Value = price sold for by retailer (after markup)
To find the distance between two points on a plane, use?
Pythagorean Theorem sqrt of( (x1 - x2)^2 + (y1 - y2)^2 )
When adding & subtracting exponents -
Realize you can't necessarily combine
View separately
Remember not to fall in the trap
If p is an integer, is p/18 an integer? (1) 5p/18 is an integer. (2) 6P/18 is an integer.
Rephrase to ask are there two 3's and and a 2 in the prime box of p?
A good data sufficiency strategy is to rephrase the information in a question.
Rephrase.
(x-n(n)y-n)
Set Problems formula
45 - 45 - 90 Triangles
Side Ratio: x : x : x√2 (hypotnuse) Half of a sq, rect, Quad is 45-45-90 triangle
30 - 60 - 90 Triangles
Side ratio: x (short) : x√3 (middle) : 2x (hypotnuse) Half of equilaterall tri = 30-60-90
*Key Note to Remember: When dealing with negatives, -9 is ________ than -1.
Smaller than; think number line
Problems Involving Either/Or
Some GMAT word problems involve groups with distinct "either/or" categories (male/female, blue collar/white collar, etc.). The key is to organize the information into a grid with the totals.
Organize into a grid.
Some GMAT word problems involve groups with distinct "either/or" categories (male/female, blue collar/white collar, etc.) The key is to do what with the information?
Formulas with unspecified amounts
Some formula problems are tricky becaue they never give you real values, they only tell you how the value of a variable has changed. (ie, Cost is expressed by the formula tb^4. If b is doubled, the new cost is what percent of the orginal cost? Simply replace b with 2b. New cose = t(2b)^4 = 16tb^4 16tb^4 = 16 * tb^2 = 16 * orginal cost...equiv to 1600% For any formula that does not specify amounts, apply te changes the quest describes directly to the orginal expression
Terminating or Non Terminating Decimals
Steps for finding no. of either zero digits or non zero digits: Make 10 pairs and multiple individual left overs.
GCF
Take the lowest common power of the prime factorization of both numbers (the GCF of two numbers cannot be larger than the difference between the two numbers.
If the inflation index between '89 and '70 is 3.56 and price of Model K mixer was 102.40 more in 1989 then 1970. What is the price of Model K in 1970?
The 1989 price (3.56x) minus the 1970 price (x) = 2.56x. 2.56x = 102.40.
What is the Surface Area?
The SUM of the areas of ALL of the faces
Remainders contd...
The decimal of a quiotent is the remanider and must be an integer. Moreover, the decmial part of the quiotent can be multipled by the divisor to get the intger remainder. Ex: 7/5 = 1.4.... (.4)*5=2.0 (2 is the integer version of the remainder)!
What is the arithmetic mean (average) of a evenly spaced set?
The median or middle number (or average of middle number).
Of all quadrilaterals with a given perimeter which one has the largest area and which one has the minimun perimeter?
The square
Variables in Answer Choices
With variables in answer choices pick numbers
always try to factor
always try to factor
Probaility - Domino Effect
dont forget to analyze events by considering whether one event affects subsequent events
When dividing two number and result contains a remainder, such remainder is always smaller than denominator.
e.g. 5/10 -- remainder = 5 (10>5) 25/10 -- remainder - 5 ( 5<10)
Ratio's give us proportional relationship's but can not _________?
give us actual values
A number is divisible by 2 if -
its even
When given sqrt(x^2) * sqrt(y^2), the presence of sqrt means it will be positive (+) sqrt of a number -
meaning sqrt of(x^2) is essentially = |x| So, if |x| * |y| also = x*y
Combinations (Order Does Not Matter)
nCr = n! / (r! (n - r)!) Where n is the total number of items in the set and r is the number of chosen items.
perpendicular lines have what kind of reciprocal slope?
perpendicular lines have negative reciprocal slopes
Unsquaring and inequality: x < y and x&y > 0
sqrt(x) < sqrt (y)
Fraction in the answer choices
when they are fractional parts or ratios of unsuspecified amounts
x^2 + x^3 =
x^2(1+x)
A number is divisible by 7 if -
you can Double the last digit, then subtract that number from the rest of the number; If after all that, the number can still be / by 7 - then your original # can be divided by 7 ex. 259 --> 9*2 = 18 --> 25 - 18 = 7; so 259 divisible by 7
The sum of consecutive integers =
(The Number of Items in the Set) X "Middle Number" Or Median
X(X-2) also =
(X-0) * (X-2)
Interior Angles
(n-2) x 180 = Sum of Interior Angles of a polygon
Balancing Method for Mixtures/Dilutions
(percent/price difference between weaker solution and desired solution) x (amt weaker solution) = (percent/price difference between stronger solution and desired solution) x (amt stronger solution)
Quadratics
(x + y)^2 = x^2 + 2xy + y^2 (x - y)^2 = x^2 - 2xy + y^2 (x+y)(x-y)=x^2 -y^2 When you see an equation in factored form in a question, immediately UNFACTOR it; vice versa.
Quadratic equations with one solution - also called perfect square. Solve: X^2 +8x + 16=0
(x+4) (x+4) = 0 (x+4)^2= 0 The only solution for x is -4
(x^2 - y^2) = difference of squares = ??
(x+y)*(x-y)
Key Note: Know the pattern of the difference of squares -
(x-y)*(x+y) = (x^2 - y^2) If y=1, or -1, then will also look like - x^2 - 1
Midpoint of a line (equation) =
(x1 + x2/2 , y1 + y2/2)
Exponents
(x^r)(y^r)=(xy)^r (3^3)(4^3)=12^3 = 1728
12 is divisible by 3 can be said in many different ways...
* 12 is a multiple of 3 * 12/3 is an integer *12 = 3n, where n is an integer *12 can be shared among 3 people so that each person has the same number of items *3 is a divisor of 12, or 3 is a factor of 12 *12/3 yields a remainder of 0 *3 "goes into" 12 evenly
If the problem states or assumes that a number is an integer, you may need to use prime factorization to solve the problem.
* determining whether one number is divisible by another number * Determining the greatest common factor of two numbers * Reducing fractions * Finding the least common multiple of two (or more) numbers * Simplifying square roots * Determining the exponent on one side of an equation with integrer constraints
16.6%
1/6 = what %
1/8 = 3/8 = 7/8 = 1/6 = 5/6 =
1/8 = .125 3/8 = .375 7/8 =87.5 1/6 = .167 5/6 = 83
Useful Percentages to Know
1/8 = 12.5% 1/6 = 16.6% 2/3 = 66.6% 5/6 = 83.3%
12.5%
1/8 = what %
Taking the reciprocals and flipping the sign: x < y and x > 0; y > 0 then you know....
1/x > 1/y
10^2
100
When an altitude is drawn down an isosceles triangle, it creates ____?
2 mirrored triangles that are congruent
Combinatorics
2!=2, 3!=6, 4!=24, 5!=120, 6!=720, 7!=5040, 8!=40320 nc2=nc(n-2)=n(n-1)/2 When picking balls or jelly beans take worst case scenario.
What are the first 10 prime factors?
2, 3, 5, 7, 11, 13, 17, 19, 23 and 29
Prime Numbers up to 100. 25 in number
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
14 Lowest Primes
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43,47
List all prime numbers under 30
2,3,5,7,11,13,17,19,23,29
√5
2.236
(216)^(1/3) = ...
216 = 3 x 3 x 3 x 2 x 2 x 2 = 6^3
15^2
225
6^2
36
19^2
361
pythagorean triples
3:4:5...5:12:13,....7:24:25
3^ (3/2) expands differently than 3^ (2/3), what are both ways to expand?
3^ (3/2) = 3^ (2/2 + 1/2) = 3^1 * 3^ (1/2) = 3*sqrt(3) 3^ (2/3) = cube root(3^2)
Powers of 3
3^1=3 3^2=9 3^3=27 3^4=81
12^3
3^3 x 4^3 = ?
the probability of event A AND event B occurring is the probability of event A times the probability of event B, given that A has already occurred.
3rd Rule of Probability: Conditional Probability
2^2
4
Both cube root(8^2) & (cube root(8) ^2) =
4
What are the types of word problem's?
4) Work/Rate - equation: Work = Rate * Time Rate = (1 / Time it takes to complete entire job); if 2/3 of a job -then (1/T * 2/3 = 2/3T *Rates are additive
Illustrate this rule: For any set of consecutive intergers wiht an ODD number of items, the sum of all the intergers is ALWAYS a multiple of the number of items
4+5+6+7+8 = 30 (divisble by 5
Volume of sphere
4/3 * π *r^3
20^2
400
45-45-90 Triangle
45-45-90 x (shorter legs), x(sqrt 2) (hypotenuse)
x, x, x(sq. rt 2)
45-45-90 triangle basic lengths of sides
21^2
461
7^2
49
(4^3)^2
4^(3x2) = 4^6
Powers of 4
4^1=4 4^2=16 4^3=64
The probability of event A OR B occurring is the probability of event A occurring plus the probability of event B occurring minus the probability of both events occurring. P(A or B) = P(A) +P(B) - P(A and B)
4th rule of Probability
3Sqrt5=
5 1/3
83.3%
5/6 = what %
5^3 = 6^3 = 7^3 = 8^3 = 9^3 =
5^3 = 125 6^3 = 216 7^3 = 343 8^3 =512 9^3 =729
Consecutive Inters Contd
6. The average of an odd number of consecutive intergers will always be and integer 7. The average of an even number of consecutive intergers will NEVER be an integer 8. The product of K consecutive integers ias always divisible by K factorial (K!) 9. For any set of consecutive integers with an ODD number of items, the sum of all integers is ALWAYS a multiple of the number of items 10. For any set of consecutive integers with an EVEN number of items, the sum of all the items is NEVER and multiple of the number of items 11. For any odd number of consecutive integers, the sum of those integers is divisible by the number of integers
What number is 150% greater than 60?
60 (1 + 150/100) = 60*2.5 = 150
25^2
625
2^6
64
4^3
64
8^2
64
6^2 X 6^3=
6^5
If x is a positive integer, is x! + (x + 1) a prime number? (1) x < 10 (2) x is even
7!+ 6 --> divisible by 1 to 7 7! + 11 --> prime 7! + 13 --> prime 6! + 11 --> prime
Give two examples of the following: The product of k intergers is always divisible by k!
7,8,9 = 24 (divisible by 3) 2,3,4,5 = 120 (divisible by 4)
Is 27 a factor of 72?
72 Prime Factors = 2, 2, 2, 3, 3; 27 Prime Factors = 3, 3, 3. 72 has only two 3's and 27 has three 3's; therefore, 27 is not a factor of 72.
Inscribed Angle, Minor Arc
An inscribed angle = two chords that have a vertex on the circle Inscribed angle with one chord as diameter = 35 degrees Minor arc = 2 x inscribed angle = 70 degrees
Rate x Time = Distance (rt = d)
For a fixed distance, the average speed is inversely related to the amount of time required to make the trip. Ex: Since Mieko's average speed was 3/4 of Chan's, her time was 4/3 as long. (3/4)r(4/3)t = d
Compound Interest
For Compound Interest: Divide interest by # of times compounded in 1 year to find interest for the compound period.
347
Number of integers from A to B inclusive = B - A + 1 How many consecutive integers are there from 73 through 419, inclusive?
Add or subtract an odd with an even, and the result is...
ODD
When you multiply integers, and none of the integers are even, the result is...
ODD
ODD+EVEN
ODD +
ODD*ODD
ODD*
Integer Constraints
Occasionally, GMAT algebra problems contain integer constraints. In such a case, there might be many possible solutions among all numbers bu only one integer solution. ie 2y - x = 2xy. If x and y are integers, which of the following could equal y First solve for x in terms of y, so that you can test values of y in the answer choices (x = 2y / 2y+1) Next, simply find which integer value of y generates and interger value for x
Recursive Sequences
Occasionally, a sequence will be defined recursively. A recursive sequence defines each term relative to other terms When a sequence is defined recursively, the question will have to give you the value of at lease one of the terms. Those values can be used to find the value of the desired term.
Odd * Odd
Odd
Odd +- Even
Odd
Odd +- Even =
Odd
Odd x Odd =
Odd
Odd & Even multiplied -
Odd * Odd = Odd Odd * Even = Even Even * Even = Even
Odds and Evens
Odd + Odd = Even Even + Even = Even Odd + Even = Odd Odd × Odd = Odd Even × Even = Even Odd × Even = Even
Odd & Even added/subtracted -
Odd +/- Odd = Even Even +/- Even = Even Odd +/- Even = Odd
odd numbers only have odd factors
Odd Factors
Remainder 1
Odd Integers (5/2 --> R = 1; 3/2 --> R =1)
Any multiplication involving an even number creates an even product.
Odd and Even rule.
Factors of Odd Numbers
Odd numbers have only odd factors
Maximum Area - Quadrilateral
Of all quadrilaterals with a given perimether, the square has the largest area
Minimum Perimeter - Quadrilateral
Of all quadrilaterals with given area, the square has the minimum perimeter
DS: Sufficiency in Yes/No Questions
On "Yes/No" DS questions, if a statement answers the question conclusively in the affirmative or in the negative, then IT IS SUFFICIENT.
Critical Reasoning - assumption question
On GMAT assumption questions, choices that RESTATE or CONTRADICT stated evidence are automatically incorrect.
DS: Hard Questions
On harder DS questions, answer choices tend to be more sufficient than they might appear. • DON'T CHOOSE (E) if you have to guess. • Pick between (A) or (C), if you can eliminate (B). • Historically, (A) is slightly more common as the right answer.
1. Don't choose E if you have to guess. 2. Pick between A or C if you can eliminate B. 3. Historically, A is slightly more common as the right answer.
On harder DS questions, choices tend to be more sufficient than they appear...
Squaring Fractions
When positive fractions between 0 and 1 are squared, they get smaller. Ex: (1/2)^2 = 1/4
When should you use the 1 - X probability trick
When the success contains multiple probabilities - especially, if the wording contain phrases such as "AT LEAST" and "AT MOST" then consider finding the probability that success does not happen.
Add/SUB
When we add or subtract two numbers the only way the answer will be even is if BOTH numbers or even or odd, otherwise, result will be odd
Circles
When you know one thing you can find everything: 1. Circumfrence = ∏*D or (2∏r) 2. Diameter = 2r 3. Area = ∏r^2
All sets of consecutive integers are sets of...
consecutive multiples.
all factorials except 0! and 1! are
even because they have 2 as a factor
All sets of consecutive multiples are ...
evenly spaced sets.
Don't confuse part/part & part/whole when dealing with ratio's
ex. Ratio of men/women is 1/4(part/part); percentage of men is 1/5(part/whole) or 20%
Key to comparing proportions is to have the same parts in the same location -
ex. ratio of apples to oranges 3:4, You have 9 apples 3/4 = 9/x
If you are given a question that asks to find the greatest value of 5 answers, you can __________
answer by comparing 2 answers together at a time
1/11 =
approx. .0909
For an evenly spaced set, the sum of the elements in the set equals
arithmetic mean (average) X the number of elements
combinations
arrangements, permuations = groups
Remember: If the remainder is 23 of 123/x, then X must _______
be greater than 23, or you would be able to fit another multiple in there. x can = 25 But if x= 20, then 123/20 would yield 6 r.3 vs. 5 r.23
Given (c-b)/(a-b) = d; b=?
c-b = da - db; but to see problem better, then c-b = ad - bd so would just b - bd = c - ad --> b(1-d) = c - ad b = c - ad/ (1-d)
x^a * x^b = x^(a+b)
c^3 * c^5 = c^8
Always Try to Factor!
ex: x^3 − 2x^2 + x = −5(x − 1)^2 x(x^2 − 2x + 1) = −5(x − 1)^2 x(x − 1)2 + 5(x − 1)^2 = 0 (x + 5)(x − 1)^2 = 0 x = −5, 1
Permutations (Order Does Matter)
nPr = n! / (n - r)! Where n is the total number of items in the set and r is the number of chosen items.
Purchase price
price purchased for by wholesaler
market value
price sold for by retailer (after markup)
s/t = 64.12...what is the remainder?
s = 64t + .12t
Perimeter of a triangle ________?
side A + side B + side C
Area of an equilateral triangle is _______?
side^2 * sqrt(3)/4
LCM- Least Common Multiple -
smallest non-zero number that is a multiple of 2 or more number's(Prime factorization, all unique primes to the highest power multiplied together)
s Sq. rt (x^r)
x^r/s = ?
Multiplying two inequalities: x < y; z < w and x,y,z,w > 0 then you know....
xz < yw (you can multiple two inequalities but NOT divide them)
Slope of a Line
y = mx + b m = slope = (difference in y coordinates)/(difference in x coordinates) = (y2 - y1)/(x2-x1)
is 0 a multiple of all integers?
yes
is 1 a factor of all integers?
yes
A number is divisible by 6 if -
you can combine rules for #2 & #3
To perform in a certain youth dance group, girls must be between 4 feet tall and 4 feet 6 inches, inclusive. If x represents a girl's height, in inches, write an absolute value equation for the heights of girls who are eligible to perform.
|x - 51| <= 3.
Area of a circle
π * r^2
Length of diagonal for square
√2 * side
Height in Equil Triangle
√3/2 * side
Fractions within fractions
Always work out from the deepest level
An odd number divided by any other interger cannot produce what?
An even interger
0 is an even, odd, both, or neither?
An even, but neither positive nor negative
Squaring inequalities: x > y and x > 0; y > 0 then you know....
x^2 < y^2
x/y < y/x and x and y > 0 then you know...
x^2 < y^2
(x * y)^3 expands out to equal?
x^3 * y^3 = ex. (4 * 3)^3 = 4^3 * 3^3 = 12^3 =1728
Basic probability formula
# of outcomes wanted/ total possible outcomes
Percent Increase
% Increase of ... New % Mult Org Value by.. 10% 110% 1.1 20% 120% 1.2 or 6/5 25% 125% 1.25 or 5/4 50% 150% 1.5 or 3/2 100% 200% 2
Percent Decrease
% decrease of ... New % Mult Org Value by.. 10% 90% 0.9 20% 80% .8 or 4/5 25% 75% .75 or 3/4 50% 50% .5 or 1/2 75% 25% .25 or 1/4
What's the area of rhombus?
( Diagonal1 * Diagonal2) / 2
Simple Probability
(# of favorable outcomes) / (# of possible outcomes)
Balancing Method for Mixtures/Dilutions: Example: How many liters of a solution that is 10% alcohol by volume must be added to 2 liters of a solution that is 50% alcohol by volume to create a solution that is 15% alcohol by volume?
(% diff b/w weaker solution and desired solution) x (amt weaker solution) = (% diff b/w stronger solution and desired solution) x (amt stronger solution) n(15-10) = (50-15)(2) 5n = 70 n = 14 L of 10% solution
While on a straight road, Car X and Car Y are traveling at different constant rates. If car X is now 1 mile ahead of Car Y, how many minutes from now will Car X be 2 miles ahead of Car Y? (1) Car X is traveling 50 mph; Car Y is traveling 40 mph. (2) 3 minutes ago Car X was 1/2 mile ahead of Car Y.
(1) Shrinking distance problem: 50-40 = 10 mph*t=1 (2) We are told that 3 minutes ago Car X was 1/2 mile ahead of Car Y. Therefore, Car X took 3 minutes to get 1/2 mile ahead, or r3=1/2 --> r = 6 mph 6t = 1 mile --> t = 1/6 of an hour or 6 minutes THEREFORE, D. Growing distance you subtract the rates. Word translations.
1/16
(1/4)^2
How many multiples of 7 are there between 100 and 150?
(147 - 105)/7+1 = 7
If given a = b^2 * c; and told to multiply b (x)2, be sure to ________
(2*b) then ^2, square; to get 4b
Factoring Exponents
(5^k)−(5^k−1) (5^k)-(1/5)(5^k) (5^k)(1 - 1/5) (4/5)(5^k)
(2 + 5)^3=
(7)^3 (NOT 2^3 + 5^3)
If Kelly received 1/3 more votes than Mike in a student election, which of the following could have been the total number of votes cast for the two candidates. (A) 12 ; (B) 13; (C) 14; (D) 15; (E) 16
(C) 14
For an evenly spaced set, the mean and median of the set are equal to the average of the...
(First + Last) /2
What are factors and multiples?
*A number is a FACTOR of another numbers if it can be divided evenly into that number *A number x is considered to be a multiple of another number y, if y times an integer = x *Example - 15 is a multiple of 3 (3x5)
List divisibility rules
*Div by 2 if units digit can be evenly div by 2 *Div by 3 if sum of digits can evenly be div by 3 *Div by 4 if sum of last 2 digits can be div by 4 *Div by 5 if final digit is 0 or 5 *Div by 6 if div by 2 and 3 *Division by zero is undefined *Zero in numerator is fine; it equals zero
All evenly spaced sets are fully defined if the following are known...
*The smallest (first) or largest (last) number in the set *The increment (always 1 for consecutive integers) *The number of items in the set.
What are digits?
*There are 10 digits 0-9 *ALL integers are made up of digits *The digits can be different units in decimals; ones, tens, tenths, hundredths
How to set up people leaving room problem -
- 160 people originally in room & 15% are women(=24) - A group leaves, 30% are women - Of people in room, 10% are now women, how many left originally? 24 - (30/100 * X) = 10/100 *(160-x) = 24 - 3x/10 = 16 - x/10 => 8 = 2x/10 ; x = 40
Perfect Squares, Cubes, etc..
- All perfect squares have an odd total number of factors - The prime factorization of a perfect suqare contains only even powers of primes (Also true that any number whose prime factorization contains only even powers of primes must be a prime number) - If a numbers prime factorization contains any odd powers of primes, then the number is not a perfect square
Word Problems contd..
- If you multiply two quanties with units multi the units as well - Likewise, units cancel in the same way as numbers and variables do
Heavy Division Shortcut
- Move the decimals in the same direction and round to whole numbers Q: what is 1,530,794 / (31.49 x 10^4) 1. Set up in fraction form 2. Rewrite eliminating powers of 10 ( 1,530,794 / 314,900) 3. Goal is to get a single digit to the left of the decimal in the denominator. In this prob must move decimal point back 5 places. Must do same to numerator (15.30794 / 3.14900) 4. Foucus on whole number parts of numerator and denominator and solve (15 / 3 = ~5) 5. if not precise enough keep one more decimal place and do long division
For any set of consecutive integers with an EVEN number of items, the sum of all the integers is NEVER a multiple of the number of items.
...
For any set of consecutive integers with an ODD number of items, the sum of all the integers is ALWAYS a multiple of the number of items.
...
In general, the divisor (y in this case) will always be greater than the remainder. To illustrate this concept, let's look at a few examples: 15/4 gives 3 remainder 3 (the divisor 4 is greater than the remainder 3) 25/3 gives 8 remainder 1 (the divisor 3 is greater than the remainder 1) 46/7 gives 6 remainder 4 (the divisor 7 is greater than the remainder 4)
...
Interest = Principal x (1 + rate/accrual per year)^(accruals per year x years)
...
Know exponent chart dealing with units digits
...
The product of k consecutive integers is always divisible by k!.
...
The sum of n consecutive integers id divisible by n if n is odd, but it is not divisible by n if n is even.
...
Two data sufficient statements always provide TRUE information
...
X-intercept is found by = -b/m
...
Yes/No data sufficiency problem, YES is a sufficient answer and NO is a sufficient answer. MAYBE is not a sufficient answer.
...
You can separate or combine the product or quotient of two roots
...
a^x + a^x + a^x = 3a^x
...
((.000064)^1/3)^1/2
.02
1/12
.0833
1/10
.1
1/10 =
.10
1/9
.111
1/9 =
.111
1/8
.125
1/8 =
.125
1/7
.14
1/7 =
.143
1/6
.166
Fundmental Counting Principle
1. If you must make a number of separate decisions, then MULTIPLY the numbers of ways to make each individual decision to find the number of ways to make all decisions 2. For problems in which certain choices are restricted and/or affect other choices, choose the most restricted options first
Factorial of Zero
0! = 1
1
0! = ?
1^2
1
2^0
1
2^0=
1
3^1
1
When the #9 is raised to an even power, ex. ^2- the units digit will equal ?
1
"Y percent less than" can be translated as what?
1 - (Y/100)
Boomtown urban planners expect the city's population to increase by 10% per year over the next two years. If that projection were to come true, the population two years from now would be exactly double the population of one year ago. Which of the following is closest to the percent population increase in Boomtown over the last year? 20% 40% 50% 65% 75%
1 Year Before Now --> 100 1 Year After --> 110 2 Years After --> 121
Simple Factorials
1! = 1 2! = 2 3! = 6 4! = 24 5! =120 6! = 720 The number of ways of putting n distinct objects in order, if there are no restrictions is, N!
19 / 5 = 3 r. 4
1) 19 - 4 = 15/ 5= 3 2) 5 - 4 = 1; 1 + 19 = 20/ 5 =4
What are the types of word problem's?
1) Conversion - use railroad technique 2) Weighted Avg. - remember to stick with totals/totals comparison's; can also use ratio of distances from mean, flip-flopped 3) Mixture - solve using 2 equations, or combination of 2 equations
Name the 2 ways to solve ratio problems?
1) The multiplier 2) The proportion (4/5 = 20/x)
What are the combinations of info needed to determine a line equation?
1) any 2 points on a line 2) any 1 point and a slope 3) any 1 point and a perpendicular slope 4) any 1 point and a slope parallel to line
The 3rd side of a triangle is always _________ than the __________ of the other 2 sides.
1) greater & 2)difference or, 1) less & 2) sum
Divisibility & Addition/Subtraction
1) if you add a multiple of N to a non multiple of N, the restult is a non multiple of N 2) If you add two non multiples o fN, the result couldbe either a multiple of N or a non multiple of N
plugging in checklist
1) plug in a normal # for your variable, if not satisfied, try another. 2) plug in a weird number, 0,1, 1/4 etc, repeat
Illustrate this rule: Illustrate this rule: For any set of consecutive intergers wiht an EVEN number of items, the sum of all the intergers is NEVER a multiple of the number of items
1+2+3+4 = 10 (Not divisible by 4)
Percents
1. "what percent" = x / 100 2. "% of" means multiply (ie. 30% of 200 = .30 * 200)
GCF & LCM
1. (GCF of m and n) * (LCM of m and n) = m * N 2. The GCF of m and n cannot be larger than the difference between m and n 3. Consecutive multiples of n have a GCF of n (for this reason, the GCF of any two consecutive integers is 1, because both integers are multiples of 1 and the numbers are 1 unit apart)
Area of Equilateral triangle
1. (s^2√3) / 4 2. can also be split into two 30-60-90 triangles
Perfect Squares
1. All perfect squares have an ODD number of total factors 2. The prime factorization of a perfect sq contains only even powers of primes 3. Same logic applied for primes of perfect cubes and other "perfect" powers (ex: all the powers of primes are multiples of 3 in the factorization of a perfect cube)
Circles - Inscribed Angle / Triangles
1. An inscribed angle is equal to half of the arc it intercepts, in degrees 2. If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle
Trapezoid
1. Area = Height * (b1 + b2) / 2
Testing Odd & Even Cases
1. Ask yourself: how do any given constraints limit the possible scenarios 2. Create a table to keep track of the allowable scenarios, given any constraints in the problem. (If unsure how to limit the scenarios, write out all possibilities and then eliminate the ones that dont fit the constraints.)
Consecutive Integers
1. Evenly Spaced Integers - the mean and the median are equal the avg of the first and last terms 2. The sum of the elements in the set are equal to the mean * the number of items in the set 3. Counting Intergers: Last term - First term + 1 4. Counting Intergers for consecutive multiples: (Last - First) / Increment + 1 ( ie. All EVEN intergers between 12 and 24 = (24-12) / 2 + 1 5. To find the sum of consecutive intergers simply find the number of terms and multiply by the median or avg in the set
How do you solve this: What is the sum of all the intergers from 20 to 100 inclusive?
1. Find the median --> (100 + 20 )/2 = 60 2. Count the number of terms (100-20) + 1 = 81 3. Multiply median by # of terms = 60*81 = 480
data sufficiency and equations
1. For a data sufficiency statement to be sufficient to solve for the variables, there must be as many quotations as there are variables. A single equation with two variables cannot be solved, but two distinct equations with the same two variables can be solved, using simultaneous equations, 2. Just because there is only one variable doesn't mean an equation has just one solution and equation with a variable raised to and even power may have more than one solution. In equation with a variable raised to an odd power will have only one solution 3. But sometimes, it's possible to get an answer even if there is only a single equation with two variables if the problem asks for an expression that contains both variables.
Ratios
1. If you see that the ratio of sharts to dolphins is 3 to 13..write as a proportion ( ie. 3/13) and write each quanity in terms of an unknown multiplier (ie. Sharks = 3x, Dolphins = 13x) 2. If you have two parts that make a whole and have a ratio of 3 to 4...write "Part : Part : Whole" ratio as 3 : 4 : 7 and use unknow multipler as needed (ie. Lefties = 3x, Righities = 4x so people = 7x)
Lines
1. Parallel lines have equal slopes 2. perpendicular lines have negative reciprocal slopes 3. If two lines in a plane do not intersect, then the lines are parallel (no pair of numbers (x,y) satifies both equations at the same time 4. Two equations may arepresent the same line. In this case , infinitely many points satifiy the equations (usually same equation, just disgushied) 5. at the point of intersection (x,y) makes both equations true.
Quadrilaterals
1. Parallelogram - Opposite sides and opposite angles are equal 2. Rectangle - all angels are 90 deg and opposite sides are equal 3.Square - all angles 90. All sides equal (both rectangle and rhombus) 4. Rhombus - All sides are equal. Opposite angles are equal 5. Trapezoid - One pair of opposite sides is parallel
Triangles - Right Triangles
1. Paythagorean Theorem: a^2 + b^2 = c^2 2. Triplets: (3 - 4 - 5 ), (5 - 12 - 13), (8 - 15 - 17), (or any triple or double of the 3)
Check for Prime
1. Pick a number n. 2. Start with the least prime number, 2. See if 2 is a factor of your number. If it is, your number is not prime. 3. If 2 is not a factor, check to see if the next prime, 3, is a factor. If it is, your number is not prime. 4. Keep trying the next prime number until you reach one that is a factor (in which case n is not prime), or you reach a prime number that is *equal to or greater than the square root of n.* 5. If you have not found a number less than or equal to the square root of n, you can be sure that your number is prime. Ex: the number n=19 has a square root of ~4.35. Test 2, 3, 4 --> you know 19 is prime because none of them are factors, and any other factor would be greater than sqrt(19).
Fractions to Powers
1. Rasing a proper fraction to a power causes that fraction to move closer to zero on a number line 2. Rasing any negative number to an odd power will make the result negative 3. Any positive proper fraction rasied to a power greater than 1 will result in a number smaller than the org. fraction 4. Any positive fraction rasied to a power between 0 and 1 will result in a number larger than the original
3 Dimensions : Surface Area
1. Rectangle - Find the area of each face (3) and double. Then add. 2. 6e^2 (e=side/edge)
3 Dimensions : Volume
1. Rectangle - L*W*H 2. e^3 (e= side/edge)
Geometry Word Problems
1. Redraw, fill in, label target 2. Spot relationships & write equations 3. Slove for what you can 4. Make inferences
Cylinders & Surface Area
1. SA = 2(∏r^2) + 2∏rh (the only info you need to find the surface area is (1) the radius of the cyliner and (2) the height of the cylinder) 2. Vol = ∏r^2H
Prime Factorization: Greatest Common Factor (GCF)
1. Start by writing each number as product of its prime factors. 2. Write so that each new prime factor begins in same place. 3. Greatest Common Factor (GCF) is found by multiplying all factors appearing on BOTH lists. 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 GCF = 2 x 2 x 3 = 12
Prime Factorization: Lowest Common Multiple (LCM)
1. Start by writing each number as product of its prime factors. 2. Write so that each new prime factor begins in same place. 3. Lowest common multiple found by multiplying all factors in EITHER list. 60 = 2 x 2 x 3 x 5 72 = 2 x 2 x 2 x 3 x 3 LCM = 2 x 2 x 2 x 3 x 3 x 5 = 360
Triangles
1. Sum of any two sides will always be greater than thrid 2. 3rd side must be less than the sum of the other two sides but greater than the difference of the other two sides 3. Sum of all angles = 180 4. Longest side is opposite the largest angle, smallest side opposite smallest angle 5. same sides = same angles 6. Area = (1/2)B * height
Circles - Sector
1. The boundaries of a sectro of a circle are formed by the arc and two radii. If you know the length of the radius and the central (or inscrived) angle, you can find the perimeter 2. You can find the area of a sector by determining the fraction of the entire area that the sector occupies. To determine the fraction, look at the central angle that defines the sector.
Diagonals of Other Polygons
1. The diagonal of a square = d=s√2 (s= side of the square) 2. diagonal of a cube = d=s√3 3. To find the diagonal of a rectangle you must know either the length and the width OR one dimension and the proportion of one to the other 4. To find the diagonal of a rectangular solid, use the pythagorean theorem twice
Similar Triangles
1. Tirangles are defined as similar if all their corresponding angles are equal and their corresponding sides are in proportion 2. If two right triangles have one other angle in common, they are similar triangles 3. If two similar triangles have corresponding side lengths in ratio a:b, then their areas will be in ratio a^2 : b^2 4. For similar solids with corresponding sides in ratio a : b, their volumes will be in ratio aV3 : b^3 5. Anytime two triangles each have a right angle and also share an additional right angle (or span of 90 degrees), they will be similar
Conbinations - Indistinguishable Items
1. To count possible GROUPS/Teams, divide the total factorial by two factorials: one for the chosen group and one for those not chosen 2. n! / k! (n-k)! n = Total number of items k = number chosen ** all that matters is who is in and who is out
Last Digit Shortcut
1. To find the digit of a product or a sum of integers, only pay attention to the units digits of the numbers you are working with. Drop any other digits 2. Simply multiply the last digits of each of the products
Polygons and Area
1. Triangle - 1/2Base x Height or (Base x Height) / 2 2. Rectangle - Length x Width 3. Trapezoid - (base 1 + base 2) x Height / 2 4. Parallelogram - Base x Height 5. Rhombus - (Diagonal x Diagonal) / 2 * If you forget the area formula for a particular shape, simply cut the shape into rectangles and right triangles, and then find the areas of these individual pieces*
Maximum Area - Triangle / Parallelogram
1. if you are given two sides of a triangle or parallelogram, you can maximize the area by placing those two sides perpendicular to each other (making them the base & height)
Finding the number of factors of an interger
1. make a prime factorazation n=a^p * b^q * c^r ; where a,b,c are prime factors # of factors= (p+1)(q+1)(r+1)
Cordinate Plane
1. y= Mx + B (M=slope, B = y-intercept) 2. Slope: (y2 - Y1) / (X2 - X1)
2^(1/2)
1.41
√2
1.414
3^(1/2)
1.73
√3
1.732
DS: Strategy
1.Focus on the question stem—thinking about the information needed to answer the question. 2. Look at each stem separately. 3.If neither statements was sufficient alone, look at both statements in combination. 4.Half of the Data Sufficiency (DS) answers on the GMAT come down to step 3.
Area of a Triangle
1/2 base * height
Powers of 10
10^1=10 10^2=100 10^3=1,000
Roots contd...
11. √a * √b = √ab 12.√a/√b=√a/b 13. to simplfy √ factor our sqs (ex: √50=√25*2=√25*√2=5√2 14. add or sub under √ factor out a sq factor from the sum or difference (EX: √4^14+4^16=√4^14(1+4^20=√4^14*√1+16=4^7√17 15. 3√x=x^1/3
Square of Numbers
11=121, 12=144, 13=169, 14=196, 15=225, 16=256, 17=289, 18=324, 19=361, 20=400, 21=441, 22=484, 23=529, 24=576, 25=625
The population of locusts in a certain swarm doubles every two hours. If 4 hours ago there were 1,000 locusts in the swarm, in approximately how many hours will the swarm population exceed 250,000 locusts? 6 8 10 12 14
12. Remember 4 hours ago, then 2 hours ago, then now...
35^2
1225
5^3
125
2^7
128
12^2
144
2^4
16
4^2
16
13^2
169
14^2
196
Squares
1^2=1 2^2=4 3^2=9 4^2=16 5^2=25 6^2=36 7^2=49 8^2=64 9^2=81 10^2=100 11^2=121 12^2=144 13^2=169 14^2=196 15^2=225 16^2=256 17^2=289 18^2=324 19^2=361 20^2=400 30^2=900
Cubes
1^3=1 2^3=8 3^3=27 4^3=64 5^3=125 6^3=216 7^3=343
The probability of event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes.
1st Rule of Probability: Basic Rule is what?
Dimension's of the 45-45-90?
1x, 1x, sqrt(2)x *Also note that its 1/2 a square
Dimension's of the 30-60-90?
1x, sqrt(3)x, 2x 1x = opposite 30 degree sqrt(3)x = opposite 60 degree 2x = opposite 90 degree
2^1
2
2^1=
2
If you see the sum of two primes that is odd, what must one of the numbers be?
2
The sum of any two primes is even unless one of the primes is?
2
What is the only even prime number?
2
Circumference
2 * π * radius
Smallest Prime Number =?
2 - And is the only even prime
Number Prop
2 is the only EVEN prime
3^5
243
5^2
25
16^2
256
2^8
256
3^3
27
17^2
289
Quick Cubes
2=8, 3=27, 4=64, 5=125, 6=216, 7=343, 8=512, 9=729
If given (2F - 5) / 2, this can be broken down into:
2F/2 - 5/2, or F - 5/2
Powers of 2
2^1=2 2^2=4 2^3=8 2^4=16 2^5=32 2^6=64 2^7=128 2^8=256 2^9=512 2^10=1,024
a^x * b^x = (ab)^x
2^4 * 3^4 = 6^4
Odd
2n+1, 2n+3, 2n+5
Even
2n, 2n+2, 2n+4
The probability of an event occurring plus the probability of the event not occurring = 1
2nd Rule of Probability: P(E) = 1 - P(not E)
Right Triangle Frequent Combos
3 4 5 and 6 8 10 and 5 12 13
Asked if every possible value answer's the question the exact same way:
3 or a 5 both lead to yes?
Right Angle Triangle Combinations
3,4,5 (multiples too) || 5,12,13 (X2 multiple) || 7,24,25 || 8,15,17
Special Right Triangles by Side:
3-4-5 5-12-13
Common Right Triangles
3-4-5 or 6-8-10 or 9-12-15 5-12-13
Percent Example: 15 is 3/5 percent of what number?
3/5 percent = 3/500 15 = (3/500) x whole whole = 2500
If a coin is tossen 3 times, what is the probability that it will turn up heads exactly twice
3/8 - you can use the counting tree method to figured it out
Special Right Triangles by Angle:
30-60-90 45-45-90
30-60-90 Triangle
30-60-90 x (shorter leg), x(sqrt 3) (longer leg), 2x (hypotenuse)
x(sq. rt 3), x, 2x
30-60-90 triangle basic lengths of sides
18^2
324
Units Digits
7= 7,9,3,1 & 3=3,9,7,1
2^3
8
On data sufficiency questions, after 1st statement, the Answer's are either -
A or D if statement is sufficient or B,C, or E if statement is not sufficient
Any number raised to an even exponent you know the result is -
A positive (+) or zero But you can Not automatically know the sign of the number or variable being raised; could be (+) or (-)
What are prime numbers?
A positive integer that can be divided evenly only by 2 numbers; itself and one. *Two is the smallest and ONLY even prime number. One and zero are NOT primes. *ALL prime numbers are positive.
Car A is chasing Car B. How long does it take for Car A to catch up to Car B?
A's distance = a mph x t B's distance = b mph x t A - B distance = a - b mph x t
Car A and Car B start driving towards each other at the same time. Eventually they crash into each other. Or, Car A and B start driving away from each other at the same time.
A's distance = a mph x t hours B's distance = b mph x t hours A + B distance = a + b mph x t hours (ADD) (ADD) (SAME)
k,m, t are > 0. k/6 + m/4 = t/12. Do t and 12 have common factors > 1? (1) k is a multiple of 3 (2) m is a multiple of 3
A. 2k + 3m = t (1) 2k + 3m = t --> 2k has the factors 2 and 3 --> 3m has a factor of 3 and something else. Therefore, there is a GCF greater than 1. (2) 2k + 3m = t --> 3m already has 3 as a factor --> 2k does not tell us anything
If x>y, x<6, and y>-3, what is the largest prime number that could be equal to x+y?
A=11 --> there is no integer constraint
Transalte: Jane bought twice as many apples as bananas
A=2B
DS: Strategy
AD or BCE: If you can determine that choice (A) is correct in your DS question, then you know that the ultimate answer must be either (A) or (D). If you can determine that choice (A) is not correct in your DS question, then you know that the ultimate answer must be (B), (C), or (E). Think of this as the before/after Christ distinction!!
If the length of side AB is 17, is triangle ABC a right triangle? (1) The length of side BC is 144. (2) The length of side AC is 145.
According to the Pythagorean Theorem, in a right triangle a2 + b2 = c2. (1) INSUFFICIENT: With only two sides of the triangle, it is impossible to determine whether a2 + b2 = c2. (2) INSUFFICIENT: With only two sides of the triangle, it is impossible to determine whether a2 + b2 = c2. (1) AND (2) SUFFICIENT: With all three side lengths, we can determine if a2 + b2 = c2. It turns out that 172 + 1442 = 1452, so this is a right triangle. However, even if it were not a right triangle, this formula would still be sufficient, so it is unnecessary to finish the calculation. The correct answer is C.
A number's unique divisor's are?
All combination's of a number's prime factorization
Consecutive Multiples?
All the values in the set are multiples of the increment {12, 16, 20, 24}
Work rate= given # of jobs/ given amount of time
Always express work rates as jobs per unit time, not as time/job
Simplify the Base of Exponential Expression
Always try to simplify the base. • If 27^n = 9^4 • then (3^3)^n = (3^2)^4 => n = 8/3
If w, x, y and z are integers and w + x = y, is y divisble by z? (1) w and x have a remainder of 1 when divided by z (2) z = 2
C
Squaring Inequalities
As w/ reciprocals, cannot square both sides of an inequality unless you know the signs of both sides of the inequality. - If both signs are negative flip the inequality sign when squaring (ie. if x < -3, left side (x) must be negative). However if given x > -3, x could be positive or negative - If both sides are positive dont flip sign when squaring - if one side is positive and one side is negative, then cannot square
What are the associative and distributive laws?
Associative - when adding or multiplying you can group or regroup numbers the way you like *Example - 2+3+4 is the same as 3+2+4 Distributive - when multiplying you can regroup numbers *Example - a(b+c) is the same as ab + ac
Isosceles triangle means ?
At least 2 sides & 2 angles are the same
Average Rate
Average A per B = (Total A)/(Total B) Average Speed = (Total Distance)/(Total Time)
(total A) / (total B)
Average Rate: Average A per B
(total distance) / (total time)
Average Rate: Average speed
If n is a non-negative integer such that 12^n is a divisor of 3,176,793, what is the value of n^12 - 12^n ? - 11 - 1 0 1 11
B
What is the median value of the set R, if for every term in the set, Rn = Rn-1 + 3? (1) The first term of set R is 15. (2) The mean of set R is 36.
B
What is the mean of four consecutive even integers a,b,c,d? a is the smallest of the four integers. 1. a+d = b + c 2. b+c =d - a
B Ste 2: as they are consecutive the condition b+c = d-a is equal to a+2 + a+4 = a+6 - a -> a=0
x/y < 1 and y > 0 then you know...
B/c y is positive you can multiple y: x < 1(y) --> x < y
What's the area of a Parallelogram?
Base * Height
1. Focus on the question stem - thinking about the information needed to answer the question. 2. Look at each stem separately. 3. If neither statement was sufficient alone, look at both statements together. 4. Half of the DS answers on GMAT come down to step 3.
Basic DS Strategy
1st Rule of Probability: Likelihood of A
Basic rule: The probability of event A occurring is the number of outcomes that result in A divided by the total number of possible outcomes.
Percent Increase vs. Percent Of
Be careful about percent increase vs. percent of. % increase = (amount of change) / (original amount) % of = portion / whole
A number between 0 & 1 squared -
Becomes smaller
Immediately write out the DS problem type (value, range, yes/no) on your scratch pad.
Before you begin strategy.
Common Factors
Break down both numbers to their prime factors to see what factors they have in common. Multiply all combinations of shared prime factors to find all common factors.
Greatest Common Factor
Break down both numbers to their prime factors to see what factors they have in common. Multiply all occurrences of prime factors that are shared. Ex: 54 = 3 x 3 x 3 x 2, 72 = 2 x 2 x 2 x 3 x 3, GCF = 3 x 3 x 2 = 18.
What is one way to speed up multiplying and dividing numbers?
By always reducing one number to the nearest "10" ex. 23 * 18 = 1) 20 * 18=360 2) take 360 and add to 3 * 18 = 54 get 360 + 54 = 414 * When dividing focus on "10's" & "100's"
If n is an integer and n^3 is between 1 and 100, inclusive, what is the value of n? (1) n = 2k + 1, where k is an integer (2) n is a prime number
C.
If r + s > 2t, is r > t ? (1) t > s (2) r > s
COMBINE INEQUALITIES!!!!!!! Answer is D. Try it. Do it now. Now.
DS: First Data Sufficiency Question
Calculate out the first DS questions to make sure they are correct. It is important to start out the section strong.
Inequalities
Cannot multiply or divide an inequality by a variable, unless you kno wthe sign of the number that the variable stands for
Car A and B are driving directly toward each other.
Car A --> a Car B --> b shrinking distance b/w a + b; If Car A is going 30 mph and Car B is going 40 mph, then the distance b/w them is shrinking at 70 mph.
Car A is chasing Car B and falling behind
Car A --> a mph Car B --> b mph growing distance between is a - b
The % change in something =
Changed amount - Original amount/ Original amount * 100
(n-1)!
Circular permutation: The number of ways to arrange n distinct objects along a fixed circle is?
Find simple interest then look for the answer that is a little bigger
Compound interest rule
3rd Rule of Probability: Conditional Probability
Conditional Probability: The probability of event A AND event B occurring is the probability of event A times the probability of event B, given that A has already occurred. P(A and B) = P(A) × P(B|A)
Work Problems
Consider work done in one hour (jobs/hour) Inverse of the time it takes everyone working together = Sum of the inverses of the times it would take each person working individually. For example, if worker A and B are doing a job, their combined rate of work is (1/A) + (1/B) = (1/T)
Cindy has her eye on a sundress but thinks it is too expensive. It goes on sale for 15% less than the original price. Before Cindy can buy the dress, however, the store raises the new price by 25%. If the dress cost $68 after it went on sale for 15% off, what is the difference between the original price and the final price? $0.00 $1.00 $3.40 $5.00 $6.80
D
X, A, B and are positive integers. When X is divided by A, the remainder is B. If when X is divided by B, the remainder is A - 2, which of the following must be true? (A) A is even (B) X + B is divisible by A (C) X - 1 is divisible by A (D) B = A - 1 (E) A + 2 = B+1
D When dividing two number and result contains a remainder, such remainder is always smaller than denominator. e.g. 5/10 -- remainder = 5 (10>5) 25/10 -- remainder - 5 ( 5<10)
If the outcome of one event affects the outcome of the other event.
Dependent events: When are two events said to be dependent events?
In an exponential (or geometric) sequence, the ratio between successive terms is always the same. Provide formula(s)
Direct Formula Sn=xk^n Recursive Formula Sn=kSn-1 S1=xk
Common Function Types - Direct Proportionality
Direct Proportionality - two quantities always change by the same facor and in the same direction. y = kx (where x is the input and y is the output value. K is the constant) Can also be expressed by y / x =k
If K is the constant difference between successive terms, and x is some other constant, then provide formulas:
Direct formula Sn= kn + x Recursive formula Sn=sn-1 + k S1=k+x
Distance Formula
Distance = Rate x Time *Multiply a rate by the denominators unit and youll get the numerators unit* - always put time in the demoninator
Evenly Divisible Problem: Determine the number of integers less than 5000 that are evenly divisible by 15
Divide 4999 by 15 => 333 integers OR => 5000/15 =333.something, so round DOWN to integer 333
Determining # Integers within a Range of 1 - X that are Evenly Divisible by a Number N
Divide X by N and round down to the nearest integer. Ex: How many numbers less than 13 are divisible by 3? 13/3 = 4.33 --> 4 Proof: 3, 6, 9, 12
How many of the three-digit numbers are divisible by 7? (A) 105 (B) 106 (C) 127 (D) 128 (E) 142
Divide all of the three-digit numbers 999-100+1=900 (Don't forget to add 1 to get the number of all the 3-digit numbers) by 7, which is 128.57, and then round it off to 128.
DS: Common Trap
Do NOT use the information in one statement as an assumption in the second statement. • Statements are not necessarily related. • View separately!
Intersecting Sets
Draw Venn Diagram for sets A and B with overlap representing A intersect B |A union B| = |A| + |B| - |A intersect B|
EVEN+EVEN
EVEN
ODD+ODD
EVEN
Sum of any two primes above 2 is ...
EVEN
When you multiply integers, and any of the integers are even, the result is...
EVEN
ODD*EVEN
EVEN *
EVEN*EVEN
EVEN2
Even +- Even
Even
Even +- Even =
Even
Odd * Even
Even
Odd +- Odd =
Even
Odd X Even =
Even
Odd+- Odd =
Even
Even x Even =
Even (and divisible by 4)
Even + Even
Even (divisible by 4)
Remainder 0
Even integers
What are odd and even numbers?
Even numbers are integers that can be divided by 2; Odd numbers are integers that cannot be divided by 2. *ZERO IS EVEN Rules: *even x even = even *odd x odd = odd *even x odd = even *even + even = even *odd + odd = even *even + odd = odd
Note the difference parentheses make in dealing with exponents -
Ex. (x^3)^2 = x^6 & x^(3)^2 = x^9 ..the exponent is actually raised
Exterior Angles in Triangles
Exterior angle d is equal to the sum of the two remote interior angles a and b. d = a + b
What is a factor?
Factor is a positive integer that divides evenly into an integer (1,2,4 and 8 are all factors (also called divisors of 8).
Circles - Sectors
Figure out the fraction of the circle that the sector represents and you can figrure out everything: 1. (Central angle / 360) 2. (Sector area / circle area) 3. (arc length / circumfrence)
1. If you have as many distinct linear equations as you have variables, you can answer ANY question about the system. 2. If you are only asked to solve for part of the system, you don't necessarily need all n equations. 3. If you are asked to solve for a relationship instead of the value of variables, you don't necessarily need all n equations.
For a system with n variables..?
DS: Equations
For a system with n variables: • If you have as many distinct linear equations as you have variables, you can answer ANY question about the system. • If you are only asked to solve for part of the system, you don't necessarily need all n equations. • If you are asked to solve for a relationship instead of the value of variables, you don't necessarily need all n equations.
Quadratic Formula
For any quad equation of the form ax^2 + bx + c = 0, the solutions for x are given by x = (-b +/- √b^2 - 4ac )/ 2a Not imp to memorize formula, but expression (√b^2 - 4ac) is call the discriminat and conveys how many solutions the equation has. If the discriminat = 0, there will be on solution If discriminat is > 0 there will be two solutions If discriminat is < 0 there will be no solutions
Combined Events
For events E and F: • not E = P(not E) = 1 - P(E) • E or F = P(E or F) = P(E) + P(F) - P(E and F) • E and F = P(E and F) = P(E)P(F)
Skip statements that you do not understand. Eliminate as much as possible.
For hard questions do the following..
Squence Problems - Alt Method
For simple linear sequences, in which the same number is added to any term to yield the next term, can use following method: "If each number in a sequence is three more than the previous number, and the 6th number is 32. What is the 100th number?" From the 6th number to the 100th number there are 94 "jumps" (100-6) of 3. Since 94*3=282, there is an increase of 282 from the sixth term to the one hundreth term: 32+282=314
Group 1 + Group 2 + Neither - Both = Total
Formula for Mixed Group problems (involving Both/Neither)
(sum of bases)(height) / 2
Formula for area of a Trapezoid
GCF & LCM
GCF : lowest power of SHARED prime factors LCM: Highest factor of ALL prime factors
Disguised Positives & Negative Questions
Generally Speaking, whenever you see inequalities with zero on either side of the inequality, you should consider testing positive/negative cases to help solve the problem
Gross Profit = Selling Price - Cost
Gross Profit formula
Gross vs. Net
Gross is the total amount before any deductions are made. Net is the amount after deductions are made.
Gross Profit
Gross profit = Selling Price - Cost
Group Problems Involving Both/Neither
Group1 + Group2 + Neither - Both = Total
DS: How Often will Problems be Both Insufficient?
Half the time statements (A) and (B) are both insufficient.
Sum of digits is multiple of 9
How to check whether number is multiple of 9
Find all prime factors
How to find all divisors of a number
y2 - y1 / x2 - x1
How to find the slope.
Which of the following would result in a remainder of 7 when divided by 12? I. 151 II. 3,443 III. 5,995 I only I and II I and III II and III I, II and III
I. TRUE: 151 - 7 = 144. 44 is divisible by 4, and the sum of the digits of 144 is 9, which is divisible by 3. II. FALSE: 3,443 - 7 = 3,436. Because 36 is divisible by 4, the number 3,436 must also be divisible by 4. However, the sum of digits of 3,436 is 16, and 16 is not divisible by 3. III. TRUE: 5,995 - 7 = 5,988. 88 is divisible by 4, and the sum of the digits of 5,988 is 30, which is divisible by 3. The correct answer is C.
Disguised Postive & Negative Example
If (a-b) / c < 0, is a>b ? 1) c> 0 2) a+b < 0 The fact that (a-b)/ c < 0 tells you that a-b and c have different signs. Thus one expression is positve and the other is negative. Stmt 1: tells you that c is negative, therefore a- b must be positive. Sufficent Stmt 2: Says that sum of a and b is negative. doesnt tell you weather a is lager than b. Insufficent.
Stick Separator Method (same gifts or items)
If 16 oranges are distributed among 4 children such that each gets at least 3 oranges, the number of ways of distributing them is a. 30 b. 210 c. 15 d. 35 e. 40 Number of grouping simply becomes number of ways to arrange remaining 4 oranges and 3 separators between themselves = (4 + 3)!/(4!*3!) = 35
Factor Foundation Rule
If a is a factor b, and b is a factor of c, then a is a factor of c. Or, any integer is divisible by all of its factors and it is also divisible by all all the factors of its factors.
What is a remainder?
If a number cannot be evenly divided by another number, the number left over at the end of the division is called the remainder.
P(event NOT occurring) = 1 - P(event occurring)
Trial Problems: look at the probability of NOT OCCURRING
Non-terminating Decimals
If in a fraction, the denominator is factor of a number other than 2 or 5, the decimal will be non terminating. Also if denominator gives 10 multiple after deducting one, the denominator can be 9,99,999 and hence non terminating
Wes works at a science lab that conducts experiments on bacteria. The population of the bacteria multiplies at a constant rate, and his job is to notate the population of a certain group of bacteria each hour. At 1 p.m. on a certain day, he noted that the population was 2,000 and then he left the lab. He returned in time to take a reading at 4 p.m., by which point the population had grown to 250,000. Now he has to fill in the missing data for 2 p.m. and 3 p.m. What was the population at 3 p.m.? 50,000 62,500 65,000 86,666 125,000
If we decide to find a constant multiple by the hour, then we can say that the population was multiplied by a certain number three times from 1 p.m. to 4 p.m.: once from 1 to 2 p.m., again from 2 to 3 p.m., and finally from 3 to 4 p.m. Let's call the constant multiple x. 2,000(x)(x)(x) = 250,000 2,000(x3) = 250,000 x3 = 250,000/2,000 = 125 x = 5 Therefore, the population gets five times bigger each hour.At 3 p.m., there were 2,000(5)(5) = 50,000 bacteria. The correct answer is A.
DS: What is Being Asked?
In Data Sufficiency questions, you are usually being asked 1 of 3 things: 1. A specific value 2. A range of numbers 3. Yes/No Immediately write out the DS problem type (value, range, yes/no) on your scratch paper before you begin a DS problem.
Geometry
In a square. If length of the diagonal is sqrt(52). Hence the length of each side of the square = sqrt(26)
Critical Reasoning - assumption question
In an assumption question, the key thing to realize is that you are not looking for the answer choice that does the most for the stimulus when added - you are looking for the answer choice that hurts the most when taken away. The correct answer to an assumption question is something that is "necessary" to the argument. For example, taking the GMAT is necessary for acceptance into most MBA programs. If you take the GMAT away, then the conclusion - being accepted into a MBA program - is harmed. An assumption is not something that has to really strengthen the conclusion when added. For example, just taking the GMAT does not guarantee admission into an MBA program.
at least 3 steps
In general, difficult questions require how many steps to solve?
2 steps
In general, medium questions require how many steps to solve?
Combining Inequalities: Add Em Up
In order to add, must make sue inequality signs are facing the same direction Ex: is a + 2b < c + 2d 1) a < c 2) d > b 1. Need to line up the inequalities so that they are all facing the same direction a < c b < d 2. Can take the sume of the two inequalites to prove the result. You will need to add the second inequality twice: a + 2b < c + 2d
To find the number of distinct permutations of a set of items with indistinguishable ("repeat") items, divide the factorial of the items in the set by the product of the factorials of the number of indistinguishable elements.
Indistinguishable events how to find the number of permutations
Minor arc = 2(inscribed angle)
Inscribed Angle, Minor Arc
John and Mary were each paid x dollars in advance to do a certain job together. J worked on the job for 10 hours and Mary worked for 2 hours less than John. If Mary gave John y dollars of her payment so that they would have received the same hourly wage, what was the dollar amount, in terms of y, that John was paid in advance?
J --> 10 hours M --> 8 hours J --> x+y M--> x -y Hourly Wage = $/hr (x+y)/10=(x-y)/8 --> 9y
Advanced data sufficiency traps
Joe blogs favorite answer or advanced data sufficiency questions is usually choice he statements 1 and 2 together are not sufficient. His second favorite choice answer is C both statements together are sufficient but neither statement alone is sufficient. Joe likes choice C because he assumes that a difficult question me a lot of information. Because these are Joe blogs first choice answers you have to be wary of them when you are answering difficult data sufficiency questions. On medium or difficult questions that appear easy open avoid the Joe blogs answer
Compound Functions
Key to solving compound functions is to work from the inside out (ie "what is f(g(3)). Start with g(3), then plug into f.
Dividing Rules: Given M/n = X r. of Y Then: if you subtract Y from M (M-Y); the result is then divisible by n evenly
M-Y = Result/n = integer
If there is an evenly spaced set...
MEAN = MEDIAN!!!!!
The GOAL in exponent equations is to ______?
Make each side of the equation contain the same prime numbers
OR means what in probability?
Means add the probabilities. You'll wind up w/ a larger #
1. Find total number of possible outcomes. 2. Find the number of desired outcomes.
Multiple event probability
Consecutive Numbers Product
Multiplication of K consecutive integers is always divisible by K
When multiplying a number by (x)5, it is often easier to ________?
Multiply by 10, and then divide by 2
When multiplying a number by (x)9, it is often easier to do ________?
Multiply by 10, then subtract that number away ex. 48 * 9= 48 * 10= 480 (-) 48= 432
Successive % Change
Multiply org value by the "new percents" for BOTH percent changes (ie. $50 + 10% then +20% = $50(11/10)(6/5) = $66
Numbers Added or Deleted
Number added: (new sum) - (original sum) Number deleted: (original sum) - (new sum) Example: The average of 5 numbers is 2. After onenumber is deleted, the new average is -3. What number was deleted? CORRECT: Original sum: 5 x 2 = 10 New sum: 4 x (-3) = - 12 Number deleted = 10 - (- 12) = 22
Interest Problem: If $10,000 is invested at 10% annual interest, compounded semi-annually, what is the balance after 1 year?
P = 10,000 r = .10 y = 1 n = 2 FV = P (1 + r/n)^ny FV = 10,000 (1 + .1/2)^(2)(1) FV = 10,000 (1.1025)^2 = 10,000 (1.1025) = $11,025
P is X percent of Q
P = x/100*q
FDP's equation
Part = Fraction * Whole
The % of something =
Part/Whole * 100
p/100 = is/of
Percent Formula
(amount of change) / (original amount)
Percent increase = ?
Perpendicular Bisectors
Perpendicular bisector has the negative reciprocal slop of the line segment it bisects 1. find the slope 2. find the slope of the perpendicular bisector 3. Find the midpoint of the line ((x1+x2)/2, (y1+y2)/2) 4. set eq equal to eachother to find out where they intersect
Positive & Negative multiplied -
Pos * Pos = Pos Pos * Neg = Neg Neg * Neg = Pos
Positive & Negative added -
Pos + Pos = Pos Pos + Neg = Which even has largest absolute value Neg + Neg = Neg
What are positive and negative numbers?
Positive numbers are to the right of zero, Negative numbers are to the left of zero. Rules: *pos x pos = pos *pos x neg = neg *neg x neg = pos *pos + neg = subtraction of neg number *neg + neg = neg
A number raised to an odd exponent can be -
Positive, Negative or zero And the sign of the variable will be the same as the solution
1. Start by writing each number as product of primes. 2. Write so that each new prime factor begins in the same place. 3. Greatest Common Factor is found by multiplying all factors appearing in BOTH lists
Prime Factorization to find Greatest Common Factor
Immediately try factoring/simplifying when possible
What to do with equations that have fractions
If a point is chosen at random within a space with an area, volume, or length of Y and a space with a respective area, volume, or length of X lies within Y, the probability of choosing a random point within Y is the area, volume, or length of X divided by the area, volume, or length of Y.
Probability and Geometry.
1. A and B < A or B 2. A or B > Individual probabilities of A, B 3. P(A and B) = P(A) x P(B) <-- "fewer options" 4. P(A or B) = P(A) + P(B) <-- "more options"
Probability of multiple events rules.
GMAT Volume Trick
Q: How many books with a vol of 100 in, can be packed into a crate with a vol of 5,000 in? A: It is tempting to answer "50 books" (since 50x100=5,000). Howeverm this is incorrect bc we dont know the exact dimensions of each book! Without knowing the exact shape of all of the books we cannot tell whether they would all fit into the crate
Remainders
Q: If a/b yields a remainder of 5, c/d yields a remainder of 8, and a,b,c and d are all integers, what is the smallest possible value for b+d? A: Since the remainder must be smaller than the divisor, 5 must be smaller than 5 b. b must be an integer, so be is at least 6. Similarly, 8 must be smaller than d and d must be an interger so d must be atleast 9.Therefore, smallest possible value for b+d is 6+9=15 ** Remainder cannot be greater than divsor or negative, must adjust for excess by taking out the divisor from the remainder or adding the divisor to the remainder (when is negative)
Discriminant
Quadratic equation: ax^2+ bx + c = 0 Dicriminant = b^2 - 4ac If discriminiant > 0, there are two roots If discriminant = 0, there is one roots If discriminant < 0, there are no (real) roots
-b +- sq. rt(b^2 - 4ac) / 2a
Quadratic formula
Why can't an odd÷even produce an interger?
Quotient can't be divided by 2
Question: How many 10's go into 40!?
Rewrite, to think of as how many (2*5) go into 40! - Because there are fewer 5's, this will be your basis that dictates the # of 10's. - multiples of 5 = 5-10-15-20-25-30-35-40 Looks to be 8 5's, but '25' is actually 2 5's = so 9
Probability of Multiple Events
Rules: • A *and* B < A *or* B • A *or* B > Individual probabilities of A, B • P(A and B) = P(A) x P(B) ← "fewer options" • P(A or B) = P(A) + P(B) ← "more options"
To determine similar triangles, what measures can be used?
Side-Side-Side Angle-Angle-Angle Side-Angle-Side
A = P(1 + r) ^n
Simple Interest Formula (remember this is the total amount of money in the bank after the interest is earned)
principle (interest rate - in decimal form) (time - in years)
Simple Interest formula (remember this is only the interest earned, not the total amount of money present in the bank after interest earned)
Simple Interest
Simple interest = (principal)(interest rate)(time)
(# of favorable outcomes) / (# of possible outcomes)
Simple probability
Using Conjugates to Rationalize Denominators
Simplifying a denominator that contains ths sum or difference of a square root and another terms is more diff: 4 / 3-√2 To simply problem need to use the conjugate of the denominator: for a+√b, the conjugate is a - √b for a-√b, the conjugate is a+√b Simply change the sign of the sq root term. This eliminates the sq root from the denom
For a fixed distance, the average speed is inversely related to the amount of time required to make the trip.
Since Mieko's average speed was 3/4 of Chan's, her time was 4/3 as long.
Exterior angle d is equal to the sum of the two remote interior angles a and b
Triangle abc with d on the outside with a line. What does d = ?
A necklace is made by stringing N individual beads together in the repeating patter red, green, white, blue and yellow. If the necklace design begins with a red bead and ends with a white bead, then N could equal? 16 32 41 54 68
Since the pattern repeats every five beads.. RGWBY 5, 15, 20 is a yellow bead. Remainder 1 = Red, 2 = Green = 2, White = 3, Blue = 4, Yellow = 0. Therefore, which answer choice has a remainder of 3, since it ends on W.
Soap, Alcohol, Water: 2, 50, 100 Soap to Alcohol is doubled? Soap to Water is halved? What is the new ratio?
Soap to Alcohol = 2*2 to 50 = 4 to 50 Soap to Water = 1/2*2 to 100 = 1 to 100 Soap, Alcohol, Water: 2, 50, 100 New Soap to Alcohol: 4, 50 New Soap to Water: 1 100 Soap, Alcohol, Water: 4, 50, 400
Approximations of Common Square Roots
Square root of 2 = 1.4 Square root of 3 = 1.7 Square root of 5 = 2.25
Standard Deviation of n Numbers
Standard deviation measures the "spread" of the data points vs the mean. Higher SD = Higher Variation 1. Find arithmetic mean. 2. Find differences between mean and each of the n numbers. 3. Square each of the differences. 4. Find average of squared differences. 5. Take non-negative square root of this average. *Probably won't need to calculate this!
Sue left her office at the same time as Tara left her. They met some time later.
Sue --> t Tara --> t
Sue left the office 1 hour after Tara, but they met on the road.
Sue --> t - 1 Tara --> t
Sue and Tara left at the same time, but Sue arrived home 1 hour before Tara did.
Sue --> t -1 Tara --> t
If the statement answers Yes OR No, the statement is sufficient. MAYBE is when the statement is insufficient.
Sufficiency in "Y/N" question types
Sum of Consecutive Numbers
Sum = (average)x(number of terms)
sum = (average)(number of terms)
Sum of consecutive numbers
Sum of Angles in a Regular Polygon
Sum of interior angles in a polygon with n sides =180(n - 2)
22
The average of 5 numbers is 2. After one number is deleted, the new average is -3. What number was deleted?
Average of Consecutive Numbers
The average of a set of evenly spaced consecutive numbers is the average of the smallest and largest numbers in the set. Average Set = (Smallest + Largest)/2
The average of a set of evenly spaced consecutive numbers is the average of the smallest and largest numbers in the set.
The average of consecutive numbers
If (x # y) represents the remainder that results when the positive integer x is divided by the positive integer y, what is the sum of all the possible values of y such that (16 # y) = 1? 8 9 16 23 24
The definition given tells us that when x is divided by y a remainder of (x # y) results. Consequently, when 16 is divided by y a remainder of (16 # y) results. Since (16 # y) = 1, we can conclude that when 16 is divided by y a remainder of 1 results. Therefore, in determining the possible values of y, we must find all the integers that will divide into 16 and leave a remainder of 1. These integers are 3 , 5, and 15. The sum of these integers is 23. The correct answer is D.
Function Graphs and Quadratics
The most importatnt questions about a parabola are: 1. How many times does the parabola touch the x-axis 2. if the parabola does touch the x-axis, where does it touc To find the number of times it touches, use the discriminat: (b^2 - 4AC) 1. If > 0, parabola crosses the x-axis twice and has 2 x-intercepts 2. If = 0, the parabola has one root of the quadratic equation, touches the x-axis once and has just one x-intercept 3. If <0, no roots for the quad eq, parabola never touches x-axis
Multiplication Principle
The number of ways independent events can occur together can be determined by multiplying together the number of possible outcomes for each event.
the number of ways independent events can occur together can be determined by multiplying together the number of possible outcomes for each event.
The number of ways independent events can occur together.
Circular Permutations
The number of ways to arrange n distinct objects along a fixed circle is: (n - 1)!
Inequalities
The only numbers that make two inequalities true are those that are true for BOTH inequalitites
4th Rule of Probability: Probability of A OR B
The probability of event A OR event B occurring is: the probability of event A occurring *plus* the probability of event B occurring *minus* the probability of both events occurring. P(A or B) = P(A) + P(B) - P(A and B)
Parallelogram
When divided gives you two right triangles. 1. Area = Base * Height 2. Perimeter = 2 (s1 + s2)
Bill runs a hot dog stand, and at the end of the day he has collected an assortment of $1, $5, and $10 bills. He discovers that the number of $1, $5, and $10 bills that he has is in the ratio of 10 : 5 : 1, respectively. How many $10 bills does he have? (1) The dollar value of his $1 bills equals the dollar value of his $10 bills. (2) Bill has a total of $225.
The question stem tells us that Bill has a stack of $1, $5, and $10 bills in the ratio of 10 : 5 : 1 respectively. We're trying to find the number of $10 bills. (1) INSUFFICIENT: Since the ratio of the number of $1 bills to $10 bills is 10 : 1, the dollar value of the $1 and $10 bills must be equal. Therefore statement (1) gives us no new information, and we cannot find the number of $10 bills. (2) SUFFICIENT: The problem states that the number of $1, $5, and $10 bills is in the ratio of 10 : 5 : 1, so let's use an unknown multiplier x to solve the problem. Using x, we can see that there are 10x $1 bills with a value of $10x. Furthermore, there are 5x $5 bills with a value of $25x. Finally, there are 1x $10 bills with a value of $10x. Statement (2) says that the total amount he has is $225, so we can set up an equation as follows: $10x + $25x + $10x = $225 $45x = $225 x = 5 Since there are 1x $10 bills this means that there are 5 $10 bills. The correct answer is B.
Similar Triangle Areas
The ratio of the areas of two similar triangles is the *square& of the ratio of corresponding lengths. Triangle ABC has sides AB = 2 and AC = 4. Each side of triangle DEF is 2 times the length of corresponding triangle ABC (DE = 4, DF = 8) Triangle DEF must have 22, or 4, times the area of triangle ABC.
All numbers greater than 1 or less than 0 -
The square becomes larger -With less than 0 numbers, its b/c 2 negatives combined together yield a positive - However is not the same case with absolute value
Sum of two primes
The sum of any two primes will be even ("add two odds"), unless one of those primes is the number 2. So if you see the sum of two primes that is odd, one of those primes must be the number 2
Subjunctive
The verb which follows the subjunctive indicator, infinitive form is used without the to.
Y-intercept is found by setting m=1 & x=0, b/c
The y-intercept is when x coordinate=0, (0,3)
Division
There are no guaranteed outcomes in division because the division of two integers may not yield an integer result ** rember that 3/4 is deemed to have a remainder of 3, 0 is an integer** Divided = Quotient * Divisor + Remainder
What is an integer?
They are numbers that: *can be negative or positive *they do NOT include fractions *zero is also an integer; it is not positive or negative
Balancing
Think of averages as what? The average of 3, 4, 5, and x is 5. What is x? 3 is 2 less than 5 4 is 1 less than 5 5 is the average x = 5 + 3 = 8
Explain this formula: P(A) + P(B) - P(A and B)
This formula works for all OR problems. However, it is a waste of time for problems in which events A and B cannot occurred together, b/c then P(A and B)=0
Complex Absolute Value Equations
Two primary types of complex absolute value equations: 1) Equation contains two or more variables in more than one absolute value expression. Not easy to slove with algebra and usually lack constants. (Topic coverd in "positive and Negatives Ch in Number Prob book) 2) The equation contains one variable and atleast one constant in more that one absolute value expression. Can be solved with an algebraic approch i.e I x-2 I = I 2x-3I, what are the possible values for x Since there is one variable (x) and three constants (-2, 2 and 3) should use algebraic approach. Only need to consider TWO cases year: - one in which neither expression changes sign (x-2) = (2x-3) - one in wich one expression changes signs (x-2) = -(2x-3) Must check validity of the solutions once you have solved equations
Combos Strategy
When a DS question asks for a combination of variables, dont try to solve for the value of each variable. Manipulate the statements to solve directly for the combination.
3-4-5, 5-12-13, 9-12-15
Three triangle length patterns
Compound Interest Formula - Compounding Annually
To compound annually: P = principal r = rate of interest (in decimal form) y = number of years New value = P (1 + r)^y
Compound Interest Formula - Compounding More Than Annually
To compound multiple times per year: P = principal r = rate of interest (in decimal form) y = number of years n = number of times per year (i.e., compounded every 3 months would be n = 4) FV = P (1 + r/n)^ny
reciprocals numbers
To determine if two number are reciprocals multiple the numbers in question; if this multiplication yields 1 then these numbers are reciprocals. For example, (Sqrt 7) / 2 and [2*(Sqrt 7)] / 7 (Sqrt 17) - 4 and (Sqrt 17) + 4
Figure out the probability for each individual event. Multiply the individual probabilities together.
To determine multiple-event probability where each individual event must occur in a certain way.
Multiple Event Probability
To determine multiple-event probability where each individual event must occur in a certain way: • Figure out the probability for each individual event. • Multiply the individual probabilities together.
Quadratic Formula
To find roots of quadratic equation: ax^2+ bx + c = 0 x = [−b ± √(b^2 − 4ac)]/2a
Indistinguishable Events (i.e., anagrams with repeating letters)
To find the number of distinct permutations of a set of items with indistinguishable ("repeat") items, divide the factorial of the items in the set by the product of the factorials of the number of indistinguishable elements. Example: How many ways can the letters in TRUST be arranged? (5!)/(2!) = 60 5! is the factorial of items in the set, 2! is the factorial of the number of repeat items ("T"s)
What is the units digit of 7^2*9^2*3^3?
To find the units digit of a product or a sum of integers, only pay attention to the units digit of the numbers you are working with. Drop any other digits.
Powers and Roots
To multiply one radical by another, multiply or divide the numbers outside the radical signs, then the numbers inside the radical signs. Example: 12√15/(2√5) = (12)/2 √15/√5 = 6√3 Example: (6√3 )2√5 = (6 × 2)(√3√5) = 12√15
DS: Solving a System of Equations Rule
To solve a system of n variables, you need n distinct linear equations. Example: What is the value of y? Given: x + y = 1 => insufficient without another distinct equation
Common Function Types - Inverse Proportionality
Two quanities change by reciprocal factors. Cutting the input in half will actually double the output. Tripling the input will cut the output to 1/3 of its orginal value y = k/x (where x= input and y is output) Can also be expressed as xy = k
Multiple Event Probability
Two things to do: • Find the total number of possible outcomes. • Find the number of desired outcomes. Write them out if necessary.
FOIL Method with Quadratics with Roots
Use FOIL Method with Quadratics with Roots n − 4√n + 4 => (√n − 2) (√n − 2) => x2 − 4x + 4
Number Added or Deleted
Use the mean to find number that was added or deleted. • Total = mean x (number of terms) • Number deleted = (original total) - (new total) • Number added = (new total) - (original total)
Word Problems
Use these steps on every problem: 1. Identify what they want 2. Identifiy what they give you 3. Represent relationships as equations 4. Slove the algebra
Foodmart customers regularly buy at least one of the following products: milk, chicken, or apples. 60% of shoppers buy milk, 50% buy chicken, and 35% buy apples. If 10% of the customers buy all 3 products, what percentage of Foodmart customers purchase exactly 2 of the above products? (A) 5% (B) 10% (C) 15% (D) 25% (E) 30%
Use this equations: 1. b1 + 2b2 + 3b3 = 145% 2. b1 + b2 + b3 = 100% 2. b3= 10%
Volume of a Sphere
V = (4/3)(pi)(r^3)
Idioms
Verbs + infinitives: allow A to do X choose to do X decide to do X forbid A to do X persuade A to do X try to do X to prohibit A from doing X
4/3 TT r ^3
Volume of a sphere
Bag A contains red, white and blue marbles such that the red to white marble ratio is 1:3 and the white to blue marble ratio is 2:3. Bag B contains red and white marbles in the ratio of 1:4. Together, the two bags contain 30 white marbles. How many red marbles could be in bag A? 1 3 4 6 8
We are told that bag B contains red and white marbles in the ration 1:4. This implies that WB, the number of white marbles in bag B, must be a multiple of 4. What can we say about WA, the number of white marbles in bag A? We are given two ratios involving the white marbles in bag A. The fact that the ratio of red to white marbles in bag A is 1:3 implies that WA must be a multiple of 3. The fact that the ratio of white to blue marbles in bag A is 2:3 implies that WA must be a multiple of 2. Since WA is both a multiple of 2 and a multiple of 3, it must be a multiple of 6. We are told that WA + WB = 30. We have already figured out that WA must be a multiple of 6 and that WB must be a multiple of 4. So all we need to do now is to test each candidate value of WA (i.e. 6, 12, 18, and 24) to see whether, when plugged into WA + WB = 30, it yields a value for WB that is a multiple of 4. It turns out that WA = 6 and WA = 18 are the only values that meet this criterion. Recall that the ratio of red to white marbles in bag A is 1:3. If there are 6 white marbles in bag A, there are 2 red marbles. If there are 18 white marbles in bag A, there are 6 red marbles. Thus, the number of red marbles in bag A is either 2 or 6. Only one answer choice matches either of these numbers. The correct answer is D.
If 72 cupcakes must be divided equally among the students in a certain class, how many students are in the class? (1) If the number of students is reduced by one-third, each student will receive 3 more cupcakes. (2) If the number of cupcakes is doubled, each student will receive 12 cupcakes.
We can begin by rephrasing the question; because we know that the 72 cupcakes must be shared equally among the students, we must have the following relationship: We can pick variables to represent both the number of students and the number of cupcakes each student will get: let s represent the number of students, and let p represent the number of cupcakes per student. Thus, we have the following equation: The original question asked for the value of s. Solving for s, we can rephrase this question as "what is 72/p?" Thus if we can determine the value of s directly or p we have sufficient data to answer the question. (1) SUFFICIENT: If the number of students is reduced by one-third, then two-thirds of the students will remain; the new number of students would therefore be (2/3)s. In this scenario, each student would then receive 3 more cupcakes, or p + 3 cupcakes each. Translating the general equation from the stem using this new information yields the following equation: Substituting the equation, p = 72/s, from the stem into the equation from statement (1) yields: (2) SUFFICIENT: If there are twice as many cupcakes, then there are 144 cupcakes and each student will receive twelve of them, so we know p = 12. Translating the statement yields the following equation: # of cupcakes = (# of students) × (# of cupcakes per student) 144 = s(12) s = 12 The correct answer is D.
What is the sum of the digits of the positive integer n where n < 99? 1) n is divisible by the square of the prime number y. 2) y4 is a two-digit odd integer.
We cannot rephrase the given question so we will proceed directly to the statements. (1) INSUFFICIENT: n could be divisible by any square of a prime number, e.g. 4 (22), 9 (32), 25 (52), etc. (2) INSUFFICIENT: This gives us no information about n. It is not established that y is an integer, so n could be many different values. (1) AND (2) SUFFICIENT: We know that y is a prime number. We also know that y4 is a two-digit odd number. The only prime number that yields a two-digit odd integer when raised to the fourth power is 3: 34 = 81. Thus y = 3. We also know that n is divisible by the square of y or 9. So n is divisible by 9 and is less than 99, so n could be 18, 27, 36, 45, 54, 63, 72, 81, or 90. We do not know which number n is but we do know that all of these two-digit numbers have digits that sum to 9. The correct answer is C.
Wendy walks 1 mile per hour more slowly than Maurice
Wendy --> r -1 miles per hour Maurice --> r miles per hour
1. Value 2. y/n 3. a range of numbers
What are the three DS question types?
180(n-2)
What does the Sum of the angles in a Regular Polygon formula look like?
Representing Even's and Odd's Algebraically
What is the remainder when "a" is divided by 4? 1) a is the square of an odd integer 2) a is a multiple of 3 Stmt 1: An arbirtary odd inter can be writter as 2n+1, where n is a integer. Therefore stmt 1 is (2n+1)^2 = 4n^2 + 4n + 1. The 1st two terms are multiples of 4 and thus have a remainder of 0. The "+1" would give the term a remainder of 1. Sufficent Stmt 2: Can be proven by picking examples. When 3/4 rmainder is 3, when 6/4 the remainder is 2
Odds & Evens
When you multiply integers, if ANY of the integers is even, the result is EVEN - If ther eare TWO even integers in a set of integers being multiplied together, the result will be divisible by 4 ( 2*5*6 = 60 (divisible by 4) - If there are THREE even integers in a set of integers being multiplied together, the result will be divisible by 8 (2*5*6*10 = 600 (divisible by 8)
Immediately UNFACTOR or vice versa
When you see an equation in factored form in a question?
Testing Inequality Cases
When you see inequalities with zero on one side of the inequality, should consider using positive / negative analysis Common inequality statements and their translations: xy > 0 : "x and y are both positive OR both negative" xy < 0 : x and y have DIFFERENT signs ( one positive, one negative) x^2 - x < 0 : 0 < x < 1
Data sufficiency statements are always true & are new pieces of information:
Will never contradict each other - Key is to take info and add to question stem, to ANSWER the question
Consider work done in one hour. Inverse of the time it takes everyone working together = Sum of the inverse of the times it would take each person working individually.
Work problem rule
Is X<Y Then 1/X > 1/Y?
Yes, when X and Y are positive OR negative. Otherwise, when X is negative and Y is positive: 1/x< 1/y
If x is an even integer, is x(x +1)(x+2) divisible by 4?
Yes. Setup prime boxes.
Is 456 divisible by 8?
Yes; because 456 is divisible by 2 three times. (456/2 = 228, 228/2 = 114; 114/2 = 57). Always check the last three digits of a larger number.
Is 48 divisible by 6?
Yes; because it is divisible by 2 (it ends with an 8, which is even) AND by 3 (4+8, which is divisble by 3).
Is 4,185 divisible by 9?
Yes; because the sum of the digits is divisible by 9 ((4+1+8+5 = 18)/9 = 2)
Is 28 divisible by 4?
Yes; because you can divide it by 2 twice. Or, when the last two digits are divisible by 4.
Is 72 divisible by 3?
Yes; the sum of its digits are 9, which is divisible by 3.
A number is divisible by 11 if -
You can add alternating digits, and then subtract those two numbers; & if divisible by 11 (0, counts) 4983 = (4+8)-(9+3)= 0
Multiplying or dividing two equations
You can multiply or divide two complete equations together, because when you do so, you are doing the same thing to both sides of the equation. ie if xy^2 = -96 and i / xy = 1/24, what is y? Can simply multiply equations together: xy^2 (1 / xy) = -96 ( 1 / 24)...y = -4 If (a / b) = 16 and (a / b^2) = 8, what is ab? Can divide equations to solve: (a / b) / (a/b^2) = 16 / 8 B^2a / ba = 2 , b=2
Manipulating Compound Inequalities
You can perform operations on a compound inequality as long as you remeber to perform those operations on every term in the inequality, not just the outside terms ie ( x + 3 < y < x + 5 ----> x <y-3 <x+2)
Is there a condition on when to multiple or divide an inequality by a variable?
You cannot multiple or divide an inequality by a variable, unless you the sign of the number of that the variable stands for.
Any given ratio can not be added or subtracted too unless ____?
You have the original amounts
If you have a radical in the denominator, then ______ (b/c it is not standard format)
You have to multiply both Numerator (dividend) & Denominator (divisor) by radical to remove from the denominator.
Properties of Zero
Zero is an even integer. Zero is neither positive nor negative. Zero is a multiple of every number. Zero is a factor of no number.
What's the area of a Trapezoid?
[(Base1 + Base2)*Height]/2
IF TWO similar triangles have corresponding side lenghts in ratio a:b, then their areas will be in ratio of?
a^2:b^2
For similar SOLIDS with corresponding sides in ratio a:b their volumes will be in ratio of?
a^3:b^3
When we multiply terms with the same base, we ___ the exponents.
add
Yes or no plugging in checklist
first try plugging in a normal number for your variable. The number you pick must satisfy the statement itself. If it doesn't plug in another number. The number will yield an answer to the question either yes or no. But you're not done yet Now try plugging in a different number for your variable. This time he might try one of the weird numbers, such as 01 or negative number for a fraction. If the statement still answers the question the same way that he could begin to suspect that the statement yield a consistent answer and that you're down to choices A or D If you plug in a different number and get a different answer this time a yes after getting a no or a know after getting a yes then the statement does not definitively answer the question and you're down to selections BC or T Now repeat this checklist with statement 2
Guessing with data sufficiency rules
focus on what you do know. It is a mistake to guess that random. Even if you have no idea what a statement saying use what you do know to narrow down your choices and gas more intelligently.
Volume of Cylinder
height * π *r^2
about 1/3
how many DS questions are Y/N?
Combining Inequalities : Line Em Up
ie. If x > 8, x < 17, and x+5 < 19 1. solve any inequalities that need to be solved. In this example only the last must be solved (x < 14) 2. Simplify the inequlities so that all the inequality symbols point in the same direction, preferably to the left (less than) 8 < x x < 17 x < 14 therefore, 8 < X < 14. ** Notice that x<14 is more limiting than x < 17 (whenever x < 14 it will be less than 17)**
never assume
if you are answering intermediate or difficult data sufficiency questions, you can never assume anything.
In Exponential Growth, a quantity is...?
is multiplied by the constant each period of time:Y(t)= Yo * k^t
A number is divisible by 8 if -
it is still even after dividing by 2, twice
Always cover-up statement 2
just because one statement seems to agree with the other doesn't mean they are necessarily saying the same thing. Just remember when you look at statement 2 always cover-up statement 1 and forget you ever saw it.
GCF- Greatest Common Factor -
largest number that divides evenly into each number of a given set(Prime factorization, then pick out factor's in common)
if a first object may be chosen in m ways and a second object may be chosen in n ways, then there are mn ways of choosing both objects
multiplication principle
Pay $10 per CD for the first 2 CDs, then $7 per additional CD
n = # of CDs bought T = total amt. paid ($) T = $10 x 2 + 7 x (n - 2) (assuming n > 2)
Mixture Problem: How many liters of a solution that is 15% salt must be added to 5 liters of a solution that is 8% salt so that the resulting mixture is 10% salt?
n = total liters of solution 0.15n + 0.08(5) = 0.1(n + 5) 15n + 40 = 10n + 50 5n = 10 => n = 2 liters
Combinations - Oder Matters
n! / (n-k)! n = Total number of items k = number chosen
Permutations -- Order Matters
n!/(n-k!)
combinatorics -- order does not matter
n!/(n-k!)k!
Dividing Rules: Given M/n = X r. Y Then: if you subtract Y from the divisor n, (n-Y); the result A can be added to M and divide evenly by n
n-Y = Result + M / n = integer
Simplifying a question
sometimes it helps to restate a data sufficiency problem in order to simplify what. For example quotes did candidate X received more than 40% of the 30,000 votes?" Could be simplified to read quotes did candidate X received more than 12,000 votes?".
1.4
sq. rt(2)
1.7
sq. rt(3)
sqrt (3^2)=
sqrt (16) Sqrt 2
When drawing the height down an equilateral triangle, the height of both new triangles =
sqrt(3) / 2*side
When we divide the terms with the same base, we ___ the exponents.
subtract
If you are given: 5 - 12 + 7, you must
subtract the 5 & 12 1st, because they are two different operations, order matters
Greatest Common Factor (GCF)
the largest divisor of two more integers
A number is divisible by 4 if -
the last 2 digits are divisible by 4
Least Common Multiple
the smallest multiple of two or more integers
A number is divisible by 3 if -
the sum of its digits are divisible by 3
A number is divisible by 9 if -
the sum of its digits are divisible by 9
Divide 4999 by 15 => 333 integers
to determine the number of integers less than 5000 that are evenly divisible by 15...?
A satellite is composed of 30 modular units, each of which is equipped with a set of sensors, some of which have been upgraded. Each unit contains the same number of non-upgraded sensors. If the number of non-upgraded sensors on one unit is 1/5 the total number of upgraded senors on the entire satellite, what fraction of the sensors on the satellite have been upgraded?
u = upgraded across satellite n = non-upgraded per module u/(30n + u) n/u = 1/5 --> 5n = u
ex. 56!, how many 14's - where 14^p is a factor?
well 14= 2*7, so # of 7's will determine - use 7 because its a prime 7-14-21-28-35-42-49-56 = 8, 7's; but 49 is actually 7*7, so 2 7's;answer is 9 7's This works because after 56! is broken down all the way into its prime fatorization - there will be a total of 9 7's used
Inequalities and Absolute Value
when I x + b I = c, the center point of the graph is -b. This equation tells us that x must be exactly c units away from -b. Similarly, for I x + b I < c, the center point of the graph is -b and the "less than" symbol tells us that x must be less than c units away from -b
yes or no data sufficiency problems
when yes or no questions involve variables, there's a good way to keep everything straight. Here is an example: Is x and integer? Statement one: 5X is a positive integer statement 2: 5X equals one for the first statement you can plug into different numbers that make statement one both true and false in answering the question.
x^3 < x^2. Describe the possible values of x.
x < 1 (except 0)
dogs to cats is 2 to 3
x dogs to y cats = 2 dogs to 3 cats or x dogs/y cats = 2/3
Is .4x > .3x? And you know x>0 then you know...
x's cancel out and the question is .4 > .3?
Fractional Exponents
x^(r/s) = s root of (x^r) Ex: 4^(3/2) = sqrt(4^3)
Squaring inequalities: x < y and x & y > 0 then you know....
x^2
If you divide or multiply both sides of an inequality by a negative #,
the direction of the inequality symbol changes
a number is divisible by 2 if
the last digit is divisible by 2
Root Rules
1. (√10)^2=10 2. √10^2=10 3. √-10^2=10 4. √2= ~1.4 5. √3= ~1.7 6. the √ of a bigger number is alwasy bigger than the √ of a smaller number 7. The √ of a number >1 is smaller than the org number (√2<2) 8. the √ of a number betwee 0 and 1 is bigger than the orginal (√2/3>2/3) 9. √5^12=5^6 (take the √ of a postive number rasied to a power and rewrite the √ as an exponet of 1/2, the mult exponets 10. 125^2/3= 3√125^2
Decimals
1. Can quickly figure out the number of decimal places to be in a solution. (ie. the result of a cubed decimal is 3 times the number of decimal places in the original decimal.) - Ex: (.04)^3 = 2 places times 3 (exponet) means there should be 6 places after the decimal 2. Likewise the number of decimal places in a cube root is 1/3 the number of decimal places in the orginal decimal - Ex: 3√.000000008 = .002 (org has 9 places (9/3=3 places in solution)
FDP's
1. Convert decimal to a fraction: put the digits to the right of the decimal point over the appropriate power of 10 then simplify (ie. .036 = 36/1000) 2. If deonominator of fraction only contains 2's and 5's as factors convert to a power of 10 (ie. 7/8 = (7*125)/(8*125)=875/1000 = .875 3. Mult. by power of 10 move decimal to the right 4. Divide by a power of 10 move decimal the number of the exponet places to the left (.6 / 10^3 = .0006) 5. Multiplying by a negative power of 10 flip the power and change the sign (ie. .004 * 10^(-3) = .004 / (10^3) = .000004
Slot Method
1. First, draw empty slots corresponding to each of the choices you have to make 2. then fill in each slot with the number of options for that slot. Fill in whatever orders makes the most sense, picking the most restriced choices first 3. Finally multiply the numbers in the slots to find the total number of combinations
Fractions - Add/Sub
1. Have addition or sub in the numberation can split into two fractions 2. Add/Sub in denominator can pull out a factor that will cancel in the numeration but NEVER split into two 3. Don't cancel on add and sub fractions
Probability - Independent Events
1. If X and Y are independent events, to determine the probability that event X AND Y will both occur, MULTIPLY the two probabilities together
Probability - Mutually exclusive events
1. If X and Y are mutually exclusive events (meaning that the two events cannot both occur) to determine the probality that event X OR Y will occur, ADD the two probalities together 2. If both events can potenitally occur together P( X or Y) = P(X) + P(Y) - P(X and Y)
Arrangements with constraints
1. If the problem has unusual constraints, try counting arrangements without constraints first. Then subtract the forbidden arrangements 2. For problems in which items or people must be next to each other, pretend that the items "stuck together" are actually one larger item ie. Greg, Marcia, Peter, Jan, Bobby, and Cindy go to a movie and sit next to each other in 6 adjacent seats in the front row of the theater. If Marcia and Jan will not sit next to each other, in how many diff arrangements can 6 ppl sit? - Ignore constraints for now. There are 6! ways to seat everyone. = 720 - Since JM are "stuck" together the arragement can be viewed as seating 5! =240 - Each of 120 ways rep two diff posibilities because they are "stuck together" (120*2)=240 - Finally, do not forget that those 240 possibilities are the ones to be excluded from consideration. The number of allowed seating arrangements is therefore 720-240= 480
Divisibility & Add/ Sub
1. If you add or subtract multiples of an integer, you get another multiple of that integer 2.If you add a multiple of N to a non-multiple of N, the result is a non-multiple of N (same holds true for subtraction) 3. If you add two non-multiples of N, the result could be either a multiple of N or a non multiple pf N
Factorials & divisibilty
1. N! is a multiple of all the integers from 1 to N 2. In a quotient of two factorials, the smaller factorial cancels complete (ie 8! / 5! = 8*7*6 = 336
Percent Change
1. Orginal % + Change% = New% (all % of orginal value) 2. %Change = New Value / Org. Value 3. To find a new value from the %change and the org value need to find the new "% of" using Org + Change = New, then multi by org value (ie. $60 + 25% = 100% + 25% = 125% as new %; 125% = 5/4 = (5/4) * 60 = $75 4. To find %more than or less than treat like %increase or decrease prob (ie. $230 is what % more than $200? = 30/200 = 15%)
Strategies For Problem Solving Q's
1. Picking Numbers - when you come across a variable or other unknown, consider Picking Numbers 2. When you see a percent question with uspecified values, pick 100 for the unknown 3. Pick numbers when you have variables in the answer choices 4. On "which of the following" questions start with E and work up 5. When numbers are given in the answer choices try to backslove (start with B or D first) - if you first answer is incorrect, its helpful to know whether the correct answer is larger or smaller
Combinations - Repated Items
1. The number of arrangements of a set of items is the factorial of the total number of items divided by the factorial(s) corresponding to sets of indistinguishable items
Imporant Quadratic Rules
1. To factor -2x^2 + 16x - 24 factor out -2 from all terms. The facotr normally. = -2(x-6)(x-2). Same applies is the term is negative. -x^2+9x-18 = -(x^2-9x+18) 2. If you cannot pull out a common factor from x^2 then keep coefficient on one or even both x's. (i.e 2x^2 - x - 15 = (2x+..)(x+...) experiment with factor pairs of the constant 3. To solve (x - 7)^2 = 625 take the positive and negative sq root of both sides. (ie x - 7 = 25 or x-7 = -25) (x=32 or x=-18) 4. To solve x^3=x must factor. (ie x^2 - x = 0...x(x^2 - 1)=0...x=0 or -1 or 1. **x^3 should alert that there could be 3 solutions** 5. Expressions that only differ by a sign change are only different by a factor of -1 (ie y-x / x-y = -(x-y) / (x-y)...cancel and get -1 6. (y+7)^2 + (y+7) / (y+8) = (y+7)[(y+7)+1)] / (y+8) ...can cancel and solve
Remainders
1. You can add & subtract remainders directly, as long as you correct excess or negative remainders (excess remainders are lager than or equal to the divisor) (remainder must be non negative and less than the divisor) 2. You can multiply remainders, as long as you correct excess remainders 3. If x divided by y yields a remainder of 0 (commonly referred to as "no remainder" then x is divisible by y (x/y) 4. if told x/y has a remainder of 5, it is possible that x=5...quotient would be 0
Exponent Rules
1. a^2*a^3=a^5 (Mult with same base add exponents) 2. a^5/a^3=a^2 (Div with same base subtract exponents) 3. a^0=1 (anything to zero=1) 4. a^-2= 1/a^2 (negative exponents are the recipricoal) 5. 2a^-2/3=2/3a^2 (good to manipulate equations - when move a term from top to bottom of fraction switch the sign of the exponent) 6. (a^2)^4=a^8
Probability - "1 - x" trick
1. sometimes the probality of the desired event not happening may be much easier to calc. 1 - Prob event doesnt happen = Prob that it does
sqrt (1/4) =
1/2 square root of a # smaller than 1, but positives is larger than the original fraction
how to determine which fraction is larger?
1/2 vs 5/9, cross multiply, and see which has a larger top number, that one is larger
3^-2
1/3^2 = 1/9
(4y)^2
4^2 x y^2 = 16 y^2
Powers of 5
5^1=5 5^2=25 5^3=125
FDP's contd...
6. Multiply decimal and a big number then trade decimal places from the big number to the decimal (ie. 50,000 * .007 = 50*7) 7. Divide two decimals move the decimal points in the same direction to eliminate decimals as far as you can (.002 / .0004 = 20 / 4) ( always go with larger number of moves, can always add zeros to the other number)
Exponet Rules Contd..
7. If you have diff bases that are numbers try breaking the bases down to prime factors. Can express everything in terms of one base (Ex: 2^2*4^3*16=2^3*(2^2)^3*2^4=2^12 8.(ab)^3=a^3b^3 9. (a/b)^4=a^4/b^4 10. a^3b^3=(ab)3 11. 2^3+2^5=2^3(1+2^2) (add or subtract terms with same base pull out common factor) 12. 2^3+6^3=2^3*(3*2)^3=2^3*3^3*2^3=2^3*(1+3^3) (add or subtract terms with diff bases break down bases and pull out common factor)
Divisibility Rules
Divisible by...: 2 - Last digit even 3 - Digits add up to a multiple of 3 4 - All numbers that end w/ "00" or Last two digits multiples of 4 or divisible by 2 twice 6 - multiple of 2 & 3 (divisible by 2 & 3) 7 - take the LAST digit and double. Subtract answer from remaining digits- If answer is divisible by 7 or 0 (EX: 161= 1*2 = 16-2 = 14) 8 - last 3 digits=multiple of 8 9 - if digits add to multiple of 9 12 - multiple of 3&4
Find LCM
Consider the multiple of the larger integer until you find on thats of the smaller EX: 12&40 40*1=40 40*2=80 40*3=120 (120 is a multiple of 12 and is the LCM) or find the prime factorization 40= 2,2,5,2 12=2,2,3 LCM = 2*2*2*3*5=120
Odd and Even Rules
Even +/- Even = even even +/- odd = odd odd +/- odd = even Even * even = even even * odd = even odd * odd = odd ** Only way to get odd is "even +/- odd" or "odd*odd"
PEMDAS - Exponents
Exponets come before everything else EXCEPT (). EX: -3^2=-(3^2)=-9 but (-3)^2=9
Fractions - Division
Flip and then multiply
Overlapping Sets
For problems involving only two categorizations or decisions set up a "doublbe-set Matrix" : a table whos rows correspond to the options for one decision, and whose columns correspond to the options for the other decision. The last row and colum contain totals. The bottom right corner contains the total number of everything or everyone in the problem ** Make sure col and row labels represent OPPOSITE situations (ie not possible to be A and Not A at the same time)**
Fraction Conversions
Fraction Decimal Percent 1/100 0.01 1.0% 1/20 0.05 5.0% 1/10 0.1 10.0% 1/8 0.125 12.5% 1/5 0.2 20.0% 3/8 0.375 37.5% 2/5 0.4 40.0% 3/5 0.6 60.0% 5/8 0.625 62.5% 4/5 0.8 80.0% 7/8 0.875 87.5% 6/5 1.2 120.0% 5/4 1.25 125.0% 3/2 1.5 150.0%
Divisibility
If a number is NOT divisible by 3 it is NOT divisible by 6 or 9
Combinations - Multiple arrangements
If a problem requires you to choose two or more sets of items from separate pools, count the arrangements separately, then multiply the numbers of possibilities for each step
Standard Deviation
Indicates how far from the mean the data points typically fall - Small STD indicates that a set is clusters closely around the average - Large STD indicates that the set is spread out widley, with some points appearing far from the mean - Variance = sq of the STD
Ratio Problems
Manipulate ratio problems to get unit answer needed for question (set up to cancel out others)
Age Problems
On age problems must add number of years to BOTH sides of equations
3 Set Problems : Venn Diagrams
Problems that involve 3 overlapping sets can be solved by using a Venn Diagram. The three overlapping sets are usually 3 teams or clubs, and each person is either on or not on any given team or club. * Work from the inside out. It is easiet to begin by filling in a nuber in the innermost sections. Then fill in numbers in the middle sections. Then Fill in outermost sections* ** Should only be used for problems that involve 3 sets **
Exampble LCM Problem
Q: If the LCM of a and 12 is 36, what are the possible values of "a"? - First notice that "a" cannot be larger than 36. The LCM of two or more integers is always ATLEAST as large as any of the integers. Therefore max value of a is 36. - Next LCM of 12 and a contains two 2's. Since LCM containes each prime facotr to the power it appers the MOST, we know that a cannot contain more than two 2's. It does not necessarily contain any twos, so "a" can contain zero, one or two 2's. - Finally, observe thta the LCM of 12 and a contains two 3's. But 12 only contains ONE 3. The 3^2 factor in the LCM must have come from the prime factorization of a. Thus we know that a contains exactly two 3's - since "a" must contain exactly two 3's and contain no 2's, one 2, two 2;s, a could be 3*3=9, 3*3*2=18, or 3*3*2*2=36
Remainders example
Q: When positive integer A is divided by positive integer B, the result is 4.35. Which of the following coulb be te remainder when A is divided by B? (a) 13 (b) 14 (c) 15 (d) 16 (e) 17 .35 = Remainder / Divisor = R / B .35/100 = 7/20 = 7/20 * R/B Finally can cross multiple: 7B = 20R Now, since both B and R are integers, we can see that R must contain a 7 in its prime factorization, otherwise, there is no way for 7 to appear on the left side. Thus, R must be a multiple of 7
Overlapping Sets - Unknowns
Read the problem very carefully to determine whether you need to use algebra to represent unknowns
Special Quadratic Expressions
Square of a sum : (x+y)^2 = x^2 + 2xy + y^2 Square of a difference: (x-y)^2 = x^2 - 2xy = y^2 Difference of Squares: (x+y)(x-y) = x^2 - y^2
LCM of Large Numbers
To find LCM of large #'s use prime factorization and multiply greatest power of the common multiples EX: 150&225 150= 2*3*5^2 225= 3^2*5^2 LCM=2*3^2*5^2 = 2*9*25 = 450 = LCM
Work Formula
Work = Rate x Time For combined work problems simply add each individuals rate togeter to plug into W=R*T
A number is divisible by 3 if
sum of digits can be divided by three
A number is divisible by 4 if
the # formed by the last 2 digits is divisible by 4 e.g. 3028
Weighted Average
Wtd Avg. = (weight)(data point) + (weight)(data point) / sum of weights 1. a weighted avg of only two values will fall closer to whichever value is weighted more heavily 2. If you know the ratio of the weights, you know the weighted average - and vice versa - ie. " a mixture of "lean" ground beef (10%) fat and "super lean" (4%) fat contains twice as much lean beef as super lean. What is the % of fat in mixture?" Ratio of lean to super lean is 2:1, so we can use 2 and 1 as the weights. Eq= (10%)(2) + (4%)(1) / 2 + 1 = 24% / 3 = 8%
% increase problems
amount of increase/ original amount
slope = y/x
the difference in y / difference in x coordinates
a number is divisible by 6 if
it can be divided by 2 and 3, thus 318 is is, because it is even and 3 + 1+ 8 /3
consecutive even integers
n,n+2,n+4,n+6...
how do you multiply 14.3X .232?
remove decimal points and multiply like normal, than add the number of decimals together and add that amount of decimals to the answer.
permutations
single source but order matters
1/0=
undefined
odds that at least one thing will happen
will- 1-wont
odds something wont happen
wont = 1-will
