GMAT
decimals multiplication
# of digits to the right of the decimal point = # of decimal places in product so multiple without and then move decimal ex) .8*.5 = .40 (40 and move 2 decimal places)
if each team plays the other team exactly one, how many games?
# of teams * 2
for uniformly spaced set (multiples): # of terms
# of terms = (range/spacing) + 1
for consecutive integers: # of terms
# of terms = range + 1
Total Profit as a Percent of Total Cost
((total revenue / total initial cost)) - 1
approximate: measurements obtained for the interior dimensions of a rectangular box are 200 cm by 200 cm by 300cm. If each of the three measurements has an error of at most 1 centimeter, which of the following is the closest maximum possible difference, in cubic centimeters, between the actual capacity of the box and the capacity computed using these measurements?
(1*200*300) + (1*200*300) + (1*200*200) --> works same way with error of 2, etc.
1-(10^-8)=
(1-(10^-4)) (1+(10^-4))
third side of triangle if know the other 2
(difference of other two sides) < third side < (sum of other two sides)
median formula
(n+1)/2 th where "n" is the number of items in the set and "th" just means the (n)th number.
percent change
(new-old)/old *100
rephrase question: "set has a mean of 75, how many #s are equal to 75?"
-how many #s have a residual of zero (meaning distance from the mean of zero); must balance #s below and above the mean -if given all numbers are < or equal to the mean; you know all are actually equal to the mean because the residual wouldn't balance to zero
Remember when picking potential values of n and arranging sets around the mean that: -when n is an even number of consecutive integers: -when n is an off number of consecutive integers:
-the median/average will = an integer + .5 -the median/average will be a whole number
0! = ?
1
approximate square root 2 and square root 3
1.4 and 1.7
If you roll three standard, six-sided dice, what is the probability that the sum of the rolls will be 6?
1/36 explanation: # of ways for rolls to sum to 6= 6, # of total outcomes = # of ways to role 3 six-sided die= 6*6*6=6^3--> 6/(6^3)=1/36
0-9 = how many options
10 options!
5% or 20,000
100
How to solve: How many positive integers less than 100 are neither multiples of 2 nor multiples of 3 ?
100- (multiples of 2+Multiples of 3 - multiples of 6) *remember to subtract the multiples of 3 and 2 (=multiples of 6) because those are overlap
2^10
1024
two digit integer ab can be written as?
10a + b
factors of 143
11*13=143
multiples of 12
12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144,156
2^7
128
2^4
16
14^2
196
First 10 prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 with 2 being the only even prime
66.7%=
2/3
6^3
216
3^5
243
16^2
256
2^8
256
how many letters in the alphabet
26
17^2
289
smart numbers example - ratio of 2/1 or 2:1
2x+1x = 3x (use 2, 1, and total is 3); one part is 2/3 and other part is 1/3
30-60-90 triangle
30-60-90 s-s sqr root 3 - 2s
if right triangle with hypotenuse 2X a leg, think of?
30-60-90 triangle !
2^5
32
7^3
343
80% as a simplified fraction
4/5
45-45-90 triangle
45-45-90 s-s-s * sqr root 2
how many different ways can the word LEVEL be arranged?
5!/ (2!*2!)
smart numbers - percents
50 or a 100
2^9
512
8^3
512
2^6
64
2/3 = _%
66 2/3%
9^3
729
2*2*2=?
8
2^3
8
sum of terms
= average * # of terms
if question asks for one solution of a quadratic equation?
= one root (x-a) (x-b) --> either give a solution
How many ways can you roll a pair of dice and get an even product?
=(ways to roll an odd and an even) OR (ways to roll an even and an even) =(possibilities even or odd*possibility the other)+(possibility even*possibility even) =(6*3) + (3*3)= 27 OR P(not odd) = 36 - prob (odd)
inscribed angle
=1/2 of equivalent central angle
how to transform 83 1/3 percent so easier to solve
=250/3
# of ways to roll 3 six-sided die
=6^3
PS Question Shortcut: 9*9*8*7
=8*7=56 --> should have a units digit of 6. if only one digit with this, that's your answer!
for consecutive integers/ uniformly spaced set (multiples): average of outer most terms = ?
=average of the entire set
how to compute possible values for "sum of n consecutive positive integers is _. what is n?"
=sum / n = mean = median (evenly spaced set); set is arranged around sum/n must be an integer or an integer (n is odd) + 1/2 (if n is even) draw chart with n, sum/n, set and eliminate any values not an integer or integer + 1/2
DS problem asks whether a quadrilateral is a parallelogram. rephrase the question.
A parallelogram is a quadrilateral with opposite sides parallel and equal in length. Rephrase the question as "Are both sets of opposite sides equal?"
Area of a trapezoid
A=1/2(b1+b2)h or 2 triangles area plus the rectangle bases of trapezoid are parallel
diagonal of a square
Diagonal=side(√2) D=s√2
If we're looking for arrangements...?
Don't divide because order matters
When given a how many question, what should you consider?
If the things we're counting have to be integers.
Dividend, divisor and quotient
In a division sentence like 12÷3 = 4, the number you are dividing, 12, is called the dividend. The number you are dividing by, three, is called the divisor. The result is called the quotient. if doesn't divide evenly then remainder left
If order doesn't matter (ie- choosing things out of a bag, etc.)?
Less possible choices. Selecting as a group. Divide the number of choices ! for which the order doesn't matter= getting rid of the different ways to arrange choices that are the same as a group. get rid of duplicates. Code= "choosing at the same time"
what can you infer from: "The intersection of sets M and N contains exactly 0.4m elements."?
M must be a multiple of 5 for because a set contains a whole exact number of integers (.4 = 2/5)
Is 1 a prime number?
NOOOO!
product of 2 consecutive integers will always be:
Odd*Even = Even
20 blue marbles, 20 red marbles, 20 yellow marbles. Joni randomly selects a marble and puts it back 5 times. Probability she picks red 4 times?
Order matters because increases probability. Here she puts them back so total each time. 1 way = (1/3)*(1/3)*(1/3)*(1/3)*(2/3)=2/(3^5) this can happen 5 ways so 5*((2)/(3^5))=10/(3^5)
Dividing with inequalities
Remember can't divide other side by a negative. Pay attention to DS givens if know a<b and want to divide by a-b can't because it's negative
"If...," Data Sufficiency Questions
Remember that the given gives hints/constraint ex) if a<b, is a+b<0 -> remember a must be less than b
What is the sum of all n digit positive integers that can be formed? - formula with repetition allowed and repetition not allowed
Repetition allowed: (sum of possible digits)*(111...n times)*n^n-1 Repetition not allowed: (sum of possible digits)*(111...n times)*(n-1)!
parallel lines have
Same slope different y intercept
Question: Jeff drove two laps around a track, Was his average speed less than 60 mph for the two laps? (1) Average speed for first lap was 20 mph (2) Average speed for second lap was 120 mph
Say the distance around the track is 60 miles. So the TOTAL distance travelled = 120 miles If Jeff's average speed were 60 mph, then he would travel the two laps in 2 hours. If Jeff's average speed were LESS THAN 60 mph, then he would travel the two laps in MORE THAN 2 hours. So, we can REPHRASE the target question as "Was Jeff's TOTAL travel time greater than 2 hours?" Statement 1: 60 miles/20 mph = 3 hours. So, after the first lap, Jeff's travel time is already greater than 2 hours. This means that, regardless of what happens during second lap, we already know that Jeff's travel time will be greater than 2 hours. Statement 1 is sufficient Statement 2: 60 miles/120 mph = 0.5 hours. So, it took Jeff 0.5 hours to complete the second lap. If it took Jeff 0.5 hours to complete the first lap, his travel time would be LESS THAN 2 hours. If it took Jeff 10 hours to complete the first lap, his travel time would be GREATER THAN 2 hours. Statement 2 is not sufficient
Time to complete one job is
The reciprocal of the rate
On the number line, 0 is closer to x - 1 than to x: what does this look like/ make you infer?
Visualize midpoint between x-1 and x. All of the points to the left of the midpoint are closer to x - 1 than to x; all of the points to the right of the midpoint are closer to x than to x - 1. Statement (1) indicates that 0 is to the left of the midpoint. The variable x is to the right of the midpoint and of 0, so x must be positive.
When is it worth it to test if more than one set of pairs work in a DS problem with 2 variables
When there is no upper limit on possible values, there is a good chance more than one pair work
every other multiple of 2 will be:
a multiple of 4
a line drawn through a triangle, parallel to one of the triangle's sides creates?
a second similar triangle (equal angles, sides in proportion)
geometric sequence
a sequence in which each term is found by multiplying the previous term by the same number
Probability 1 thing happens OR another happens
add probabilities (same with different combinations that fit certain criteria)
if solving when absolute values contain variables
always plug solutions back into equation to confirm they work
for consecutive integers: sum of set =
average * (# of terms) = (average outer terms) * (range +1)
x increases 200 percent more than y increased, y increased by 1. x increased by ?
be careful trap "200 percent more" so not 2*1 but ((2*1)+1)
area of a parallelogram
bh (base x height)
remainder
decimal part of quotient * divisor
decimals and exponents (increase or decrease when?)
decimals b/w 0 and 1 decrease as their exponent increases negative decimals increase as their exponent increases because become smaller (closer to zero) or positive
inscribed triangle with one side equal to diameter of circle
diameter= hypotenuse (greatest possible length) so angle across from diameter = 90 degrees
divisible by 6 if?
divisible by 2 and 3
Productof 6 consecutive integers will always be:
divisible by six --> set will always have, a number that is one more than a multiple of three, a multiple of three, and one less than a multiple of three
When order does matter in combinatorics?
don't divide by the number of choices because different orders are different choices ex) how many "codes", "combinations", etc. possible
integer is divisible by 10 if
ends in 0
an integer is divisible by 2 if
even
even/odd addition or subtraction
even +/- even = even odd +/- odd = even even +/- odd = odd
sum of four consecutive integers must be:
even because O + E + O + E = O+O+E+E = E+E = E
even/odd multiplication or division
even*even = even even *odd = even odd *odd = odd No Guarantee with Division!!
median of a set of #s is the average when
evenly spaced set
f(x)=x^2 + x f(2x)=?
f(2x)=(2x)^2 + 2x = 4x^2 + 2x remember to square the 2!
procedure for finding SD
find difference between each term in the set and the mean of the set
when a probability question uses "at least" or "at most"?
find probably it won't happen; then use 1 - probability it won't happen
To find the closest a list of numbers is to a number (short cut if fractions)
find the difference of all from that number - the smallest fraction difference will be the closest
in general, a number is _ its remainder?
greater than
Odd numbers divided by 4
have a R of 1 or 3
perpendicular lines have
have negative reciprocal slopes
altitude
height of a shape
if a ds statement proves something is not possible or gives a "no" answer, then?
if the question was a yes or no then that is sufficient
probability pathways
if there are 4 ways to reach desired outcome: this way or this way or this way or this way --> sum up 4 ways if its probability x and y happens - then multiply
how does the sum of two #s tens digits impact their sum's hundreds digit?
if they sum to > or equal to 10 there is carry over so the hundreds digit is greater than or equal to the sum of their tens digit ex) 19+21 = 40 ex) 19+23 = 42
how do you know if (x+1)/(y+1)> x/y
if y>x explanation:( not necessary) multiple both sides of inequality by y(y+1) ... end up with y>x?
is zero an integer? positive or negative?
integer but not positive or negative
"Prime" means
integer that has exactly two factors
the height of an object
is perpendicular to the base making a 90 degree angle so can form right triangles (look for this and if special triangles apply ie- pythagorean triples)
The sum of two positive integers is __? - what should this make you think about?
is the sum odd or even? if its odd, must be an odd + even (can apply this to other problems about the sum, difference, product of TWO integers)
positive integer divided by 10 has a remainder equal to?
it's units digit ex) 14/10 --> R is 4
Guessing probability answer choices hint (problem solving)
look for answer pairs - usually the GMAT puts the inverse since probabilities add up to 1 ex) 16/30 and 14/30
sum of the first m odd whole numbers =?
m^2
for consecutive integers/ uniformly spaced set (multiples): average =
median = sum of terms/ number of terms
how to tell if larger SD by looking at set
more spread out!
how to get rid of radical in the denominator
multiple by its complement ex) 2+(3)^(1/2) * 2- (3)^(1/2)
common denominator of fractions with factorials in the denominator
multiple by term to raise denominator to the largest factorial ex) largest denominator is factorial of 9! then multiple term with denominator of 8! by 9/9
to get rid of decimals
multiply by multiple of 10 (10,100, etc)
Probability 1 thing happens AND another happens
multiply probabilities
combining exponential terms w/ the same bases
multiplying = add exponents; divide = subtract exponents; nested = multiply
sum of two and three consecutive integers (algebraically)
n + (n+1+ = 2n + 1 and n + (n+1) + (n+2) = 3n +1
product of all integers from 1 to n
n!
nCr= ?
n!/r!(n−r)! = ways to choose n things from r
x^2/t^2 = 4/9, given x and t are positive integers, can you solve for x and t?
no because only know x and t are multiples of 2 and 3 --> multiple possibilities without more information 2 and 3, 4 and 6, etc.
can you make conclusions about the SD with statements about the mean?
no! because need to no how far each term is from the mean (the spread)
sum of 6 consecutive integers must be:
odd because O+E+O+E+O+E=O+O+O+E+E+E= O+E= O
sum of two consecutive integers must be:
odd because odd+even = odd
Rhombus
parallelogram with four congruent sides and opposite angles equal and opposite sides are parallel and diagonals bisect each other and meet in the middle at a right angle
smart numbers - variable under a square root sign or cube root
perfect squares or perfect cubes
If you roll two standard 6-sided die, what is the probability that the product of the two numbers facing up is even?
probability NOT ODD*ODD = 1 -(probability roll an odd and roll another odd) = 1 - ( (3/6)*(3/6))= 1-(1/4)=3/4
when to work backwards? and how?
problem solving friendly #s in choices question asks for a single value -start with b or d if in ascending order then can move from b to d and if one is too big and one too small, choose c. (the one in the middle)
"for every $3 in profits earned last year the company earned $3.96 this year" simplifies to?
ratio of profits last year/ ratio of profits this year = $3/$3.96
(280^2)-(200^2): simplified?
recognize special product even though no variables, is still the difference of two squares so equals: (280-200)(280+200)
conversion question using cubic units: what should you remember
remember to cube conversion factor given if not already cubed example: If the volume of a small container is 14,520 cubic millimeters, what is the volume of the container in cubic centimeters? (1 millimeter = 0.1 centimeter) * change to: 1 mm^3 = (.1)^3 cm^3= .001 cm^3
if a gmat question asks "what is the closest ..."
signal to find a way to approximate
method to organize days of the week problems
slots
smart numbers with lots of multiplication or exponents
smaller #s (2, 3, etc.)
r^4 = 9 --> how to solve
sort both sides: r^2 = 3 sqrt again r= +/-sqrt 3
The sum of an evenly spaced set or consecutive multiples set with an even increment is?
sum of a pair * # of pairs
for uniformly space set multiples: sum of terms=
sum of terms = ((range/spacing)+1)*(average of outer terms)
sum of the first m even whole numbers =
sum of the first m even whole numbers = m(m+1)
divisible by 9 if?
sum of the integer's digits are divisble by 9
DS questions about points on a circle - possible test cases?
test point on the circle, outside the circle, and inside the circle and see if they satisfy conditions or not to determine if sufficient
in an evenly spaced set each concentric pair of numbers has?
the same sum
in an evenly spaced set, the sum of the set is?
the sum of a pair * # of pairs
If X is 75% more than Y and the numbers must be integers- what can you infer
then X= 7/4Y so you can infer that Y is divisible by 4, so a multiple of 4. and X is a multiple of 7. and total is a multiple of 11 because 7/4Y +4/4Y= 11/4Y
not a right triangle but know two sides and all angles
then know the third side if use trig
smart numbers ratios or fractions
total = denominator and numerators = parts
divisible by 5 if?
units digit is 5 or 0
negative bases and exponents
unless the negative sign is inside ( ) it does not distribute ex) -2^4 = -1 * 2^4 = -16; (-2)^4 = -1^4 * -2^4 =16
smart numbers if you'll need to divide?
use a multiple as the divisor
smart #s rate and work problems
values that will give integers when calculating the rate r = d/t
if you know the range of a set of numbers is zero, do you know the SD?
yes SD is zero because all values are the same!
all numbers in a set are positive integers, what can a number in the set not equal?
zero (not positive or negative even though an integer) and any negative integers