Math Techniques Flashcards
GCF and LCM
- Prime Factor
- Create box with all unique factors across top, two rows (one for each number)
- Input unique factors with correct exponents
4a. GCF - Multiply lowest values in each column
4b. LCM - Multiply highest values in each column
Converting Decimal to a Fraction
Place the number over 1*10^number of digits in the number.
.123 = 123/1000
Arc Length Calculation
Inner angle / 360 = arc length / circumference
Disguised 3-4-5 Right Triangle
30-40-50
60-80-100
Etc.
Sides-to-Degrees Calculation
(n-2)(180)
Where n is the number of sides of the shape
Combining Ratios
Combine ratios by multiplying or dividing common elements of the two ratios to get a common factor or multiple.
A:B = 5:6
B:C = 8:9
A:C = 2:3
Is x prime?
Test divisibility rules on primes less than square root
Area of an Equilateral Triangle
(√3 / 4)(side^2)
x + y Divisibility
If x and y are both divisible by d, then (x + y) is divisible by d
Negative Exponents
Turns the base into a reciprocal, does NOT change the sign of the base.
1 / 2^(-3) = 2^3
2^(-5) = 1 / 2^(5)
Mixed Roots
Can compare size / manipulate by multiplying to square of coefficient, then moving coefficient under the root.
2√13 = (√4)(√13) = √4*13 = √52
Parallel Symbol
A ∥ B
Standard Deviation
Never need to calculate
- Larger spaces = wider standard deviation
- All constituents of a set shifted by the same interval? No change to standard deviation
Normal Distribution
1st St. Dev = ~34% on either side
2st St. Dev = ~14% on top of 1st St. Dev
3st St. Dev = ~2% on top of 2st St. Dev
Terminating Decimals
Factor out 2s and 5s in denominator - any other factors and it will not terminate
Functions f(x)
Just plug and play
f(x) = x^2 and g(x) = x+3
f(g(1)) = 16
Rectangle
A square is a rectangle, but a rectangle is not necessarily a square. Length is defined as the longer side.
x^2 = 9
x = 3 OR -3
Circumference of a Circle
2πr
Colinear
Two points that lie on the same straight line. Any two points are colinear, three points may or may not be colinear.
Rate / Time Problems
Going opposite directions? Add rates. Then plug into R*T = D
Going the same direction? Subtract rates
Multiplying Decimals
Multiply the decimals by 10^x and then pull back out when finished. Can do with fractions as well.
Simplifying Factorials
9! / 6! = (9)(8)(7)(6!) / 6! = (9)(8)(7)
Can’t do the math? Shouldn’t do the math?
- Estimate
- Backsolve
- Use your own numbers
- Get creative
(w / x) / (y / z)
(w / x)(z / y)
Distance Formula
Rate * Time = Distance
Adding or Subtracting Exponents
23^13 - 23^12
Can ONLY factor, cannot subtract because these are not like terms
(23^12)(23-1) = (23^12)(22)
Percentages
is / of = % / 100
Set equal to each other and cross multiply
Right Triangle
a^2 + b^2 = c^2 applies
“Legs” are the shorter sides, not the hypotenuse
Percent Change
(New - Old) / Old
Semi-circles
If you take any point on a semicircle’s curved edge and draw lines to the corner points, it creates a 90 degree right triangle, with the diameter as the hypotenuse
If a is divisible by b, and b is divisible by c…
a is also divisible by c
If d has e and f as prime factors, then…
d is divisible by e, f, and (e)(f)
Every factor of a number (except 1)…
Is a product of a different combination of the number’s prime factors
Quadrilateral
360 degrees
Area of a Trapezoid = ((Base 1 + Base 2) / 2 )(Height)
Data Sufficiency Answers
A: Only I is sufficient
B: Only II is sufficient
C: I and II are only sufficient TOGETHER
D: I and II are both sufficient ALONE
E: I and II are NOT sufficient TOGETHER
Divisibility Rules: 4
If the final two digits of a number are themselves divisibly by 4, then the number is divisible by 4. 104, 288, 312
Is Zero negative or positive?
Neither - it is a neutral integer
Area of a circle
πr^2
3D Boxes
Vertices = Corners
Faces = Sides
Split into planes to find area, volume, etc.
Simple Permutations
How many ways can n items be ordered? n!
Repeat entities? Divide n by # of repeats (x repeats, divide by x!) 3 repeats? Divide by 3!, or 6
Sum of Consecutive Numbers
(n/2)(First number + Last Number)
Exponent Rules (Addition or Subtraction)
Bases (even if the same) cannot be added or subtracted if the exponent values are different, because they are not like terms. Can factor out the bases though.
Adding or Subtracting Fractions
Simply multiple the denominators to get a common denominator and the numerators accordingly to get add or subtract.
Percentage Change Calculation
(New - Old) / Old
Three Overlapping Sets
Draw Venn Diagram, always subtract out center groups
Arc Length Formula
(Center Angle / 360) = (Arc Length / Circumference)
Divisibility Rules: 6
Must be divisible by 2 and 3
Sides of a 45-degree right triangle
x, x, x√2
Range
Largest value in a set minus the smallest value
Exterior Angles
Sum of opposite interior angles in a triangle is equal to the exterior angle opposite the third angle - look at flash card for diagram
Parallelogram
A shape consisting of two sets of parallel lines, angles opposite each other are equal. Area = base*height
Divisor
All the factors of a number and their negative counterparts
(√x)(√y) =
√xy
(√x) / (√y) =
√(x/y)
√x + √x =
2√x
√x + √y =
Nothing can be done to simplify here
(a / b)(c / d) =
ac/bd
The b root of x^a =
x^(a/b)
Inequalities
When dividing or multiplying by a negative number, the sign flips - may have to test multiple scenarios! All other operations by negatives, or positives do not flip sign
Area of a Triangle
1/2 * base * height
Compounded Percents
Calculate as several percent changes at a time
5-12-13 Right Triangle
May be disguised by proportional multiples
Divisibility Rules: 2
Any even number is divisible by 2
Trapezoids
Area of a trapezoid = ((Base 1 + Base 2)) / 2) * Height
Addition Method
Multiply an equation within a system of equations to get rid of one of the variables, then can add together to solve for the other variable
Can also subtract! Same effect
Substitution Method
For solving systems of equations. Solve for one variable in terms of the other, then plug the result into the other equation
Even Exponent?
The solution to x^2 = 9 could be 3, OR -3
Quant Problem Solving Strategies
- Estimate
- Backsolve
- Use Your Own Numbers
Sum of Equally Spaced Numbers
(Avg. of #s)(# of #s)
(Avg. of #s) = (Biggest + Smallest) / 2
(# of #s) = (Biggest - Smallest) / Spacing
Negative to positive? Cancel out corresponding additions on either side - -23 + 23 = 0, for instance
Favorite Probability Trick
Find the probability of what you’re NOT supposed to be solving for, and subtract it from 1
Data Sufficiency Danger Areas
- Negative Numbers
- Even Exponents
- Non-Integers
- Inequalities with Variables - don’t know sign, may not know direction of inequality symbol
Sub-Divided Group
Draw a visual tree diagram, use 100 as a base if working with percentages
Multiplying Fractions
(a/b)*(c/d) = ac/bd
Dividing Fractions
Multiply by reciprocal
(a/b) / (c/d) = ad/bc
GCF Facts
- The GCF of two numbers may be 1, as in the case of primes
- GCF can also be one of the two numbers, as in the case of 6 and 12
Similar Triangles
Triangles that have the same angles but different sizes have proportional perimeters and areas
Is 1 prime?
No
Divisibility Rules: 3
If the sum of the digits of a number sum to 3, then the number is divisible by 3
(Even)(Even) =
Even
(Even)(Odd) =
Even
(Odd)(Odd) =
Odd
Even +/- Even =
Even
Even +/- Odd =
Odd
Odd +/- Odd
Even
What to do if forget Even/Odd rules
Use 2 and 3 - all numbers, even negatives, behave the same way
(y^a)(y^a) =
(x*y)^a
Usually working backwards to find prime bases
(x^a)(x^b) =
x^(a+b)
(x^a)^b =
x^(ab)
x^a^b =
x^(a^b)
(x^a)/(x^b) =
x^(a-b)
Difference of Squares
x^2 - 16 = (x+4)(x-4)
May need to manipulate expression to get a difference of squares
Equivalent Equations
Set equations equal to the same value equal to each other, then can solve through substitution or addition. Can manipulate to make equations equivalent.
Absolute Values
Keep track of negatives - try to backsolve
Quadratic Equation Terms
x^2 + 7x + 12 = 0
- Factors: (x+3)(x+4) = 0
- Roots (Solutions): x = -3 or x = -4
Difference of Squares - General Formula
x^2 - y^2 = (x-y)(x+y)
Is 0 an integer?
Yes
Positive Square Root Rule
Any time you see a square root in the GMAT, assume only the positive square root - can NOT be the negative square root
Conjugate
The conjugate of (x-√2) is (x+√2) - multiply denominators by these to get denominator off the bottom of fractions, GMAT does not like irrationals in the denominator
Work / Rate Reciprocal Rule
If we add machine rates together to find how much of the job they can complete in one hour, we can flip result to see how many units of time to complete job.
Formula for number of integers between x and y, inclusive
y-x+1 - Can use to find number of multiples between two numbers as well
Multiples of 5 between 358 and 81?
85 = 517
385 = 571
71-17+1 = 55
Rational Substitution
y - 13√y + 36 = 0
Substitute u for √y and solve
u^2 - 13u + 36 = 0
(a-b)^2 =
a^2 + b^2 - 2ab
(a+b)^2 =
a^2 + b^2 + 2ab
(a+b)(a-b)
a^2-b^2
Difference of Squares
Extraneous Roots
If roots are the answer to a data sufficiency equation, CHECK BOTH ROOTS by plugging back in - if one does not work, it is an extraneous root and is NOT a solution
Calculate the number of divisors
Prime Factor, then add 1 to exponents and multiply exponent values.
Looking for odd factors, or even factors? Sam process, but with only odd or even prime factors
Is a number a prime number?
- Cannot be >2 and even
- Not divisible by 3
- Not divisible by 5
- Not divisible by 7
Decimal Addition or Subtraction
Line up decimal points and add
Decimal multiplication
Product will have same total number of decimals as the sum of the decimal points of both factors
Decimal Division
Multiply top and bottom by the same 10^x to get whole numbers, then divide
Equilateral Triangle
Regular Triangle, all sides same length, all angles = 60 degrees.
Often need to bisect into two 30, 60, 90 right triangles.
Sides of a 30-60-90 Right Triangle
x, x√3, 2x
To Remember: Angles
- When lines intersect, angles on same side add to 180 degrees
- All angles in a parallelogram are the same if they face each other
Complex Combinations
Think total - options that don’t work, or one way that works * number of options
Circumference of a Circle
2πr
Integer
Any whole number, including 0
Units Digit Multiplication
Just multiply the units digit to find what the units digit will be. Typically works in patterns for large exponents
Average Rate or Speed Calculation
TOTAL Distance / TOTAL time
May need to find totals first
Cubed Roots
As with square roots, find prime factors and reduce by triple occurrences under the root symbol
Divisibility Rules: 8
If the last three digits are themselves divisible by 8, then the number is divisible by 8. 6,216 = 216/8 = Yes
Divisibility Rules: 9
If the digits sum to something divisible by 9, the number is divisible by 9 and 3
Divisibility Rules: 7
No clean rule - long division
Proportion
Ratios or fractions that are equal to each other. Cross multiply and divide.
Things to Remember: Scientific Notation
Just two things multiplied together, can reduce or split apart as need be
Calculation for slope of a line
Rise / Run
Negative Exponents
Just take reciprocal and make exponent positive - do not change the sign of the base! If part of a product, can move it to top or bottom of a fraction if needed
Composite Numbers
Any number that is composed of multiple primes factors
Probabilities: And vs. Or
And: Multiply
Or: Add
Circles: Interior Angles
Triangles from the center have radius as sides, so makes an isosceles triangle
Divisor
All the factors of a number and a their negative counterparts
Area of a Circle
πr^2
Rate / Time Questions
Find a common time period if working together - use fractions to get to same denominator of time, then add or subtract as needed
Compound Interest Calculation
Usually don’t need to calculate:
Principle(1+(Interest Rate/# of Periods per Year))^((# of Periods per Year)(# of Years))
Scientific Notation
Can move decimal back and forth by changing the 10^x exponent. Can split the factors up as well if needed
Line Slope Properties
Parallel lines have the same slope, perpendicular lines have negative reciprocal slopes
Area of a Trapezoid
((Base 1 + Base 2) / 2) * Height
Convert a Fraction to a Decimal
Long Division
Basic Concepts: Geometry
- Do NOT trust the diagram
- Geo problems are susceptible to estimation
- Always be looking to extend lines or subdivide shapes if needed
Bisection
To divide something exactly in half
Polygons: Interior Angles Total Calculation
(n-2)*180, where n is the number of sides
A “Regular” polygon is one that has sides of all the same length and all the same angels
Volume of a Cylinder
Find area of circle on top/bottom, then multiply by height
Perpendicular
⊥, negative reciprocal slopes
No order in smaller group
n! / (n-k)!(k!), where n is the total number in the group and k is the smaller number being chosen
Properties of a Square
A quadrilateral, four 90-degree angels, all sides same length.
Can bisect into two 45-degree right triangles
Arc Length Formula
(Center Angle / 360) = (Arc Length / Circumference)
Basic Probabilities Calculation
Acceptable Outcomes / Total Possible Outcomes
Always between 0 and 1
Isosceles Triangle
Triangles in circles are isosceles, if the center is at an angle - 2 sides are radius
Divisibility Rules: 5
Ends in a 0 or a 5
Hexagon
6 sides, (6-2)(180) = 720 total interior degrees
Order in Smaller Group
n! / (n-k)!
Perfect Square
The square of an integer
Perfect Cube
The cube of an integer
Shortcut for Addition or Subtraction
Do math for just units digit - is that enough to narrow it down to one answer?
Simplify…
…before you multiply
Prime #: # of Factors
2
Square of Primes: # of Factors
3
Composite Number: # of Factors
4 or more
First Several Primes
2, 3, 5, 7, 11, 13, 17, 19
When two numbers DONT share any prime factors…
Their LCM is always their product
When two numbers DO share prime factors…
Their LCM is less than their product - you have to strip out any common factors
if x is divisible by a and b, then…
…x is divisible by the LCM of a and b
This means if asked if a certain number MUST be divisible by another, it’s really an LCM question
Two factors of x with Primes in common?
Combine, eliminating any overlap in prime factors to find the LCM
Two factors of x with no primes in common?
The LCM is the product of those numbers
Negative Number raised to an even power
Always positive
Negative Number raised to an odd power
Always negative
Even Exponents…
…Hide the Sign of the base
Negative Exponents
Creates a reciprocal value with a positive exponent. You can move multiplied terms above or below division lines by changing sign of the exponent. Negative exponents do NOT change the sign of the base.
When a positive power to a negative power…
Multiply the exponents - you now have a negative exponent
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
(y - x) / (x - y)
GMAT Disguise - factor out -1 from top or bottom to get factors you can simplify
Quadratics in Fraction?
Factor and Cancel! Avoid fractional coefficients at all costs
When should you divide by a variable?
Only if you are SURE the variable is not 0
Quadratics with higher powers?
Factor out variable, then treat as third factor
x^3 - 3x^2 + 2x = 0
x(x^2 - 3x + 2) = 0
x(x - 2)(x - 1) = 0
Roots are 0, 1, and 2
Substitution (y + 1)^2 = 16
Substitute u for (y + 1)
u^2 = 16
u = 4 or -4
y + 1 = 4
y + 1 = -4
Solve
Quadratic Roots
Roots are possible solutions, but the variable is not equal to both simultaneously
Simplify Quadratics before factoring
Roots are answers to factors, not simplified coefficient
3(x + 3)(x+4)
x = -3, -4
Multiplication FOIL
(102)(301) = (100)(300) + (100)(1) + (2)(300) + (2)(1)
Allows you to work with easier numbers
If you have a variable in an exponent…
Make bases the same to solve equation. Usually must prime factor bases and then get equal to each other through exponent rules
2^y = 2^4
y = 4
Exceptions are bases of 0, 1, or -1
