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