GMAT Quant
Maximum Area of Polygons
Quadrilateral: - with a fixed perimeter, the SQUARE has the largest area. - of all quadrilaterals with a given area, a square has the minimum perimeter. - ***a regular polygon with all sides equal will maximize area for a given perimeter and minimize perimeter for a given area.
Rates reminders
RT=D or RT=W Use an RTD chart to solve All units must match! Always express distance over time! (Not the other way around)
Even exponents
hide the sign of the base since they always give a positive result; ***look for more than one possible solution! (4^2 = 16 AND (-4)^2 = 16)
Factor Foundation Rule
if a is a factor of b, and b is a factor of c, then a is a factor of c
Median
the middle score in a distribution; half the scores are above it and half are below it If odd number of terms - middle number If even number of terms - average of the 2 middle
Arc length
the portion of the circumference of the circle (use a fraction)
Inverse proportionality
the relationship when two quantities produce a constant value when multiplied Y=k/x or yx=k So y1x1 = y2x2
If (N) is a divisor of x and of y,
then (N) is a divisor of x+y
average speed
total distance divided by total time Will ALWAYS be closer to the slower of the two rates (spent more time traveling at that speed)
If xy>0
x and y are both positive OR both negative
If xy<0
x and y have different signs (one positive, one negative)
Vertical lines
x=a
(x^a)(x^b)
x^(a + b)
(x^a)/(x^b)
x^(a-b)
(x^a)^b
x^(ab) = (x^b)^a
(x + y)²
x² + 2xy + y²
(x - y)²
x² - 2xy + y²
(x + y) (x - y)
x² - y²
If x² - x < 0
x² < x, so 0 < x < 1 (it is a positive proper fraction)
Horizontal lines
y=b
slope-intercept form
y=mx+b
Linear Growth/Decay Formula
y=mx+b M = constant rate of growth/decay X = time B = value/quantity when time = 0
√x²
|x| (POSITIVE value of x)
Square root of 2 (√2)
~1.4 (Valentine's Day)
Square root of 3 (√3)
~1.7 (St. Patrick's Day)
√5
~2.25
x^(a/b)
(1/x^b)^a = (x^a)^1/b
10^(-3)
0.001
1/100
0.01
10^(-2)
0.01
1/50
0.02
1/25
0.04
1/20
0.05
1/12
0.08333...
10^(-1)
0.1
1/9
0.1111...
1/8
0.125
1/6
0.161616...
1/5
0.2
1/4
0.25
1/3
0.33333...
3/8
0.375
2/5
0.4
1/2
0.5
3/5
0.6
5/8
0.625
4/5
0.8
5/6
0.8333...
7/8
0.875
(-1)^even
1
x^0
1
a^0
1 (anything to the power 0 = 1)
Discriminant = 0
1 real solution
15^15
225
4!
24
5^2
25
√625
25
2^8
256
3^3
27
Diameter
2r
circumference
2πr
Surface Area of a Cylinder
2πr² + 2πrh
Cube root of 27
3
√9
3
Common Right Triangles
3-4-5 6-8-10 5-12-13 8-15-17
√900
30
2^5
32
6^2
36
2^2
4
Cube root of 64
4
√16
4
Ratios
Think about finding the unknown multiplier - use a table or algebra to solve
If GMAT asks "how many total factors"
Count ALL factors, not just primes Don't forget to count 1
If GMAT asks "how many prime factors"
Count each repeated factor only ONCE
Relative rates - bodies move toward each other
Create third RT=D equation for rate at which distance between them DECREASES (ADD rates together)
Relative rates - bodies move in same direction on same path
Create third RT=D equation for rate at which distance between them DECREASES (subtract rates - one is moving faster than the other)
Relative rates - bodies move away from each other
Create third RT=D equation for rate at which distance between them INCREASES (ADD rates together)
Unknown digits problems
Create variables (x y z) to represent unknown digits Box them in to differentiate Digits restricted - 0 to 9 Remember you can write them into formulas (if A is 2 digit number x y then A = 10x + y)
Comparing fractions
Cross Multiply Cross multiply the fractions and put each answer by the corresponding numerator For example: 7/9 vs. 4/5 (7 x 5) = 35 (4 x 9) = 36 Put 35 next to corresponding 7/9 and 36 next to corresponding 4/5. Since 36 is larger than 35, 4/5 > 7/9
Probability Tree
Draw diagram where possibilities emerge from events. (My words, not the text) Use to keep track of branching possibilities
Increase denominator of fraction, holding numerator constant
Fraction value decreases
Increase numerator of fraction, holding denominator constant
Fraction value increases
Adding same number to both numerator and denominator
Fraction value will be closer to 1 (If fraction was <1, it will increase in value closer to 1; if fraction was >1, it will decrease in value closer to 1)
Multiple ratios
Make a common term - find common element (multiple all pieces of ratio by same number like a common denominator)
Even +/- Odd
Odd
Odd x Odd
Odd
Perfect squares have this _____ number of total factors
Odd They only have even powers of primes
direct proportion
a relationship between two variables in which their ratio is constant Y=kx or k=y/x So y1/x1 = y2/x2
non-terminating decimal
After fully reducing the fraction, if the denominator has ANY prime factor besides 2 or 5, then it will NOT terminate
Cube root of 1000
10
√100
10
30^2
900
Sum of Consecutive Integers
(1) Find the average by using the first and last term to find the middle of the set (2) Count the number of terms (remember to add 1 if inclusive!) (3) Multiply the average by the number of terms to find the set
Absolute value equations
(1) Isolate the expression within the absolute value brackets (2) Remove the absolute value brackets and solve for the equation in 2 cases. Case 1: x=a (x is positive). Case 2: x=-a (x is negative) (3) Check to see whether each solution is valid by putting each one back into the original equation and verifying that the two sides of the equation are in fact equal.
Area of a sector of a circle
(n/360)(πr²), where n is the central angle.
x² - y²
(x + y) (x - y)
x² + 2xy + y²
(x + y)²
x² - 2xy + y²
(x - y)²
(x/y)^a
(x^a)/(y^a)
(x^a)(y^a)
(xy)^a
midpoint formula
(x₁+x₂)/2, (y₁+y₂)/2
Reciprocals of Inequalities
*If you know the signs of the variables, you should flip the inequality UNLESS x and y have DIFFERENT signs. If x < y, then: 1/x > 1/y when x,y positive 1/x > 1/y when x,y negative 1/x < 1/y when x neg and y pos
When to work backwards
- Answer choices are numerical and "nice" numbers - question asks for a discrete number - Don't if numbers are large or ugly or if they ask for combo of variables
(-1)^odd
-1
Inequalities with absolute values
-Visualize the problem with a number line -Generally has more than 2 possible solutions -If the equation is within the absolute value brackets, just shift the line! -Remember to consider two scenarios - positive when you remove brackets and solve as is OR negative when you reverse WHOLE sign within brackets, remove them and solve -NEVER just drop brackets and change signs - must change whole expression!
Optimization Problems
1) Linear functions: extremes occur at the boundaries (the smallest and largest possible x) 2) Quadratic functions: whatever value of x makes the squared expression equal to 0 is the value that maximizes or minimizes the function; the resulting value is the max/min Focus on the largest and smallest possible values Test extremes to determine the right combo
Finding GCF and LCM Using Prime Columns
1. Calculate prime factors of each integer 2. Create column for each prime factor found within any of the integers 3. Create row for each integer 4. For each cell in table, place prime number raised to a power - the # times that column's prime factor appears in the prime box of the row's integer 5. GCF = product of LOWEST count of each prime factor (remember that a^0=1) LCM = product of HIGHEST count of each factor
Finding GCF and LCM using Venn diagrams
1. Factor numbers into primes 2. Create Venn Diagram 3. Place each shared factor into middle 4. Place nonshared factors into other areas 5. GCF = product of primes in the middle LCM = product of all primes in the VD
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 3. Subtract possibilities of #2 from #1 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
Anagram Grid - combinatorics
1. Label number of columns for how many number of members of group 2. Categorize each number of group on bottom / put the choices that there could be on bottom row *use only letters * for questions where it's saying only a certain number could be part of the group - use Y/N as your letters 3. Make a fraction: Numerator = factorial of largest number on top Denominator = product of factorials of each different kind of letter on bottom row 4. Simplify and cancel out numbers
Counting Total Factors
1. Make the prime factorization of the # 2. See how many possible occurrences there are of each prime factor (N + 1 where N is the power to which the prime appears (because it could occur 0 times!)) 3. Multiply # of occurrences for each prime Ex: 2000 = 2^4 x 5^3 5 possible 2's and 4 possible 5's --> 5 x 4 = 20 factors
Creating numbers with a certain remainder
1. Set up remainder relationship (Dividend = (Quotient x Divisor) + Remainder 2. Perform plugging in numbers to get possible numbers 3. Notice patterns 4. Pick answer
Combining inequalities
1. Solve any inequalities that need to be solved 2. Line up variables 3. Combine Discard any less limiting inequalities Watch relationships - sometimes not possible to combine into one Signs must face the same direction!
Functions with unknown constants
1. Solve for unknown constant with givens 2. Rewrite function with solved constant 3. Solve function for new input variable
Work Backwards Strategy
1. Start with B or D 2. Narrow - if B is right, choose B If need to go smaller, choose A 3. If need to go bigger, try D Repeat steps with D, C, and E
Symmetry problems
1. Substitute the function in each function OR 2. Pick number for x and see which produces equal output
When to estimate
1. When problem explicitly asks for approximate number 2. When answers are far apart 3. When answers cover certain "divided characteristics" Use to at least eliminate some choices. Glance at answers first!!!
5/4
1.25
4/3
1.333...
7/4
1.75
x^(-a)
1/(x^a)
10^2
100
10^3
1000
2^10
1024
√121
11
√144
12
5!
120
11^2
121
5^3
125
2^7
128
√169
13
√196
14
12^2
144
√225
15
2^4
16
4^2
16
√256
16
13^2
169
14^2
196
2!
2
Cube root of 8
2
√4
2
Discriminant > 0
2 real solutions
Any product of 3 consecutive numbers is divisible by
2, 3 AND 6
First ten prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
√400
20
20^2
400
isosceles right triangle
45-45-90 degrees 1:1:square root of 2 sides Remember they make up a square - use to find a square diagonal
7^2
49
2.25^2
5
Cube root of 125
5
√25
5
2^9
512
3!
6
√36
6
25^2
625
2^6
64
4^3
64
8^2
64
√49
7
6!
720
2^3
8
√64
8
9^2
81
3^2
9
√81
9
perpendicular bisector
A line that is perpendicular to a segment at its midpoint (divides it in half) Have negative reciprocal slope -1/m1 = m2 or m1*m2=1
equilateral triangle
A triangle with three congruent sides and angles (60)
To multiply two exponential expressions that have the same base, keep the base and ___ the exponents
Add *make sure the bases are the same!!!
If GMAT asks "how many total prime factors (length)?" (Length = number of products whose product is x)
Add exponents of the prime factors If no exponent, count it as one
Machines working togerher
Add rates together! (Over same unit of time!)
Angles in a triangle
Add up to 180
Combinatorics - "OR" (how many choices)
Add!
Area of equilateral triangle
A=(s²√3)/4
Area of a rhombus
A=1/2 x d1 x d2
Area of a trapezoid
A=1/2(b1+b2)h
Area of a circle
A=πr²
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.t
Square root of x squared
Absolutely value of x! |x| (so x could be positive OR negative)
Standard deviation = 0
All numbers are identical
Parallel lines cut by a Traversal
All opposite angles equal All acute angles equal All obtuse angles equal Watch out for disguised figures - Z shape can be extended
Count consecutive multiples
All values are multiples of the increment (4, 8, 12, 16) [(Last - First) / Increment] + 1
Recursive Sequence
a sequence in which each term is determined by one or more of the previous terms Question will give value of at least one term - use that to find desired
Count consecutive numbers (inclusive)
B - A +1 *add one before you are done!!! (how many integers from 73-419 = 419-73 + 1)
Discriminant
b²-4ac
Hidden Integer Constraint
Be careful certain situations where you must have integer (ex. You can't split a person)
Strange Symbol Formulas
Break operations down one by one Watch for symbols that invert the order of operation Always perform procedures in parenthesis first
Percent Change
Change/Original
negative exponent
a number with a negative exponent should be moved to the denominator of a fraction and the exponent switched to positive. Or flip to numerator if denominator is the one with the negative exponent
Percent
Divide by 100
Divisibility rule for 6
Divisible by both 2 and 3
Smart Numbers Guidelines
Do not choose 0 or 1 Do not pick numbers that appear elsewhere If multiple, choose different numbers, with different properties Follow all constraints
Divisibility rule for 10
Ends in 0
Divisibility rule for 5
Ends in 5 or 0
Is
Equals
Even +/- Even
Even
Even x Even
Even
Even x Odd
Even
Odd +/- Odd
Even
Divisibility rule for 2
Even number
linear sequence problems
Find the number of "jumps" between the term you have and term you want, multiple by the number added each time
Multiply/divide inequality by negative number
Flip the sign
If there is a square root in denominator
If just square root, multiply both numerator and denominator by that square root If it's square root and another term (ex. Sqrt(2) + 7) multiply both numerator and denominator by the CONJUGATE (Sqrt(2) - 7)
Inscribed Triangle in a Circle
If one of the sides of a triangle is the diameter of a circle, it MUST be a RIGHT TRIANGLE
Maximum area of parallelogram or triangle
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. - if given 2 sides of a triangle, to maximize the area, make those sides the base and height and make the angle between 90 degrees. - you can maximize the area of a rhombus with a given side length by making it a square.
Full revolution of a spinning wheel
Is equal to its circumference
The product of any k consecutive integers is always divisible by
K!
Multiply/divide inequality by positive number
Keep the sign * NEVER multiply or divide an inequality by variable unless you KNOW the sign of the number that variable stands for
Divisibility rule for 8
Last 3 digits are divisible by 8 (or can be divided by 2 three times)
Divisibility rule for 4
Last two digits are divisible by 4 (or can be divided by 2 twice)
Unknown digits problems
Look at answer choices first to limit search Use given constraints Focus on units digit Test remaining answer choices
Sequences
Look for patterns (ex. If asked to find units digit, do a few and find the pattern)
terminating decimal
Most reduced fraction for a terminating decimal/fraction - you can have will only have prime factors of 2s and/or 5s in them!
Multiplying very large decimal and very small decimal
Move decimals the same # places but in OPPOSITE directions, then multiply
Of
Multiply
Combinatorics - "AND" (how many choices, make two decisions)
Multiply!
Inequalities with even exponents
Must consider TWO scenarios!! (Flip the sign as needed)
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.
Parabolas for ax² + bx + c, if |a| is large
Narrow curve
Negative raised to odd power
Negative number
Odd roots
Only one solution, also keep sign of the base
Given square root on GMAT
Only use the positive root
Domino Effect
Outcome of first event affects probability of subsequent Check to see whether object is replaced and fix subsequent probabilities If you have multiple possibilities but they are ALL equivalent - calculate probability of ONE case and multiply it by number of cases
Negative raised to even power
Positive number
Raising a decimal to a higher power
Rewrite the decimal as product of integer and power of 10 and then distribute the exponent!
Roots reminder
Roots are numbers raised to a fractional power
Simplifying roots
Separate the number into its prime factors and take out matching pairs: square root of 20 = square root of 2 x 2 x 5= 2* square root 5 *only works when roots are connected by multiplication or division!!! Never by addition or subtraction
30-60-90 triangle
Sides: x, x√3, 2x Two of these make up one equilateral
Similar Triangles
Similar triangles have the same shape: corresponding ANGLES are EQUAL and corresponding SIDES are PROPORTIONAL
If given result and asked for remainder or quotient
Solve using principles of multiples (set up equation, cross multiply and solve) Answer must be a multiple of something - check choices
Integer constraints with inequalities
Solve/simplify equations Combine equations if needed Substitute as necessary
To divide two exponential expressions that have the same base, keep the base and ___ the exponents.
Subtract *make sure the bases are the same!!!
Two machines - one undoing the work of another
Subtract rates!
Intersecting Lines properties
Sum of angles = 360 Interior angles = 189 Opposite angles are equal
Divisibility rule for 9
Sum of digits divisible by 9
Divisibility rule for 3
Sum of digits is divisible by 3
sum of interior angles of a polygon
Sum=180(n-2), where n is the number of sides
Negative fractional exponents
Take the reciprocal first!
Complex Absolute Value Equations (2 or more variables in more than 1 absolute value expression OR variable and constant in more than 1 absolute value expression)
Test TWO cases 1. One in which NEITHER expression changes sign 2. One in which ONE expression changes sign You MUST check validity of solutions by plugging back in
large standard deviation
The data points are spread out over a wider range of values
Remainder
The leftover amount when a number cannot be divided evenly
Quotient
The number of times that the divisor goes into the dividend completely
Dividend
The number that is being divided
Divisor
The number that is dividing
Denominator or divisor is a power of (10-1) (9, 99, 999...)
The numerator tells you the repeating digit
Perfect cubes' prime factorization
They only have powers of 3 in their prime factorization
If a number has prime factorization (a^x)(b^y)(c^z)
Then that number has (x+1)(y+1)(z+1) different factors
The Last Digit Shortcut
To find the units digit of a product or sum of integers, only pay attention to the units digit. 1. Drop any digits but the ones unit from all numbers. 2. Multiply/add all the ones digits. 3. Take the ones digit of the final product.
Interest formula
Total Amount = P (1 + r/n)^nt
What
Unknown value
If equation has squared variable and YOU take the square root
Use both positive and negative solutions!
Two sets, three choices
Use double set matrix and if no other options and choices don't overlap
Multiple trips or travelers
Use multiple RT=D relationships Pay attention to relations between equations Use minimum necessary number of variables Put into an RTD table
Population Problems
Use population chart Make sure one of the rows says NOW Work forward, backward or both Pick smart number as necessary
(1-x)probability trick
Use shortcut when the thing not happening has smaller probability than it happening Solve for that and then use formula 1 - P(A) = P(Not A) or P(A) + P(Not A) = 1
Ratios
Use unknown multiplier to solve for part:part or part:whole as needed
"How many"
Usually signals combinatorics: 1. decision 1 OR decision 2 - ADD 2. decision 1 AND decision 2 - MULTIPLY
Three overlapping sets
Venn Diagrams ALWAYS start with INNER most circle
Arranging groups with no restrictions
Ways to arrange = n! Where n is number of distinct objects
Nested Exponents (a^2)^3
When raising a power to a power, combine exponents by multiplying (a^6)
Parabolas for ax² + bx + c, if |a| is small
Wide curve
Average of even number of consecutive integers
Will NOT be an integer
Average of odd number of consecutive integers
Will be an integer
Parabolas for ax² + bx + c, if a < 0
Will open down
Parabolas for ax² + bx + c, if a > 0
Will open up
Fractional Exponents
Within the exponent fraction, the numerator tells us what power to raise the base to, and the denominator tells us which root to take Numerator = power to raise to Denominator = root to take
Compound functions
Work from the INSIDE OUT! -Start by solving for the inner function -Use the result of the inner function as the new input variable for the outer function. *Changing the order of a compound function changes the answer -- f(g(x)) =/= g(f(x))
Squaring Inequalities
You CANNOT square both sides of an inequality unless you know sign of BOTH sides If both sides are NEGATIVE, FLIP the sign when you square If both sides are POSITIVE, DON'T flip the sign If one side positive and one side negative or unclear, DON'T SQUARE!!!
Never subtract or divide two inequalities
You can multiply as long as all possible values are positive
Add or subtract a multiple of N to a non multiple of N
You get a NON multiple of N
inscribed angle
an angle whose vertex is on a circle and whose sides contain chords of the circle An inscribed angel is 1/2 of the arc it intercepts!
Mean
average Average = sum / number of terms ***average x number of terms = sum
Pythagorean Theorem
a²+b²=c²
If two similar triangles have corresponding side lengths in ratio a:b, then areas will be in ratio...
a²:b² Holds true for any similar polygons
Area of a parallelogram
base x height
Diagonal of a square
d=s√2
Main Diagonal of a Cube
d=s√3
small standard deviation
data is close to the mean
N! divisibility
divisible by all integers from 1 to N
Rectangular prism diagonal
d² = x² + y² + z²
Exterior angle of a triangle
equals sum of opposite (non adjacent) interior angles
slope
m=(y2-y1)/(x2-x1)
Discriminant < 0
no real solutions
Evenly Spaced Sets
sequences of numbers whose values go up or down by the same amount/increment (4,7, 10, 13, 16) Mean and median are equal = (First + Last)/2
Work problems
simmilar to distance but ... think how much of the job can be done in one hour (as a fration of the whole). R=W/T or R*T=W
Volume of a cylinder
πr²h