Math Techniques Flashcards
1
Q
GCF and LCM
A
- 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
2
Q
Converting Decimal to a Fraction
A
Place the number over 1*10^number of digits in the number.
.123 = 123/1000
3
Q
Arc Length Calculation
A
Inner angle / 360 = arc length / circumference
4
Q
Disguised 3-4-5 Right Triangle
A
30-40-50
60-80-100
Etc.
5
Q
Sides-to-Degrees Calculation
A
(n-2)(180)
Where n is the number of sides of the shape
6
Q
Combining Ratios
A
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
7
Q
Is x prime?
A
Test divisibility rules on primes less than square root
8
Q
Area of an Equilateral Triangle
A
(√3 / 4)(side^2)
9
Q
x + y Divisibility
A
If x and y are both divisible by d, then (x + y) is divisible by d
10
Q
Negative Exponents
A
Turns the base into a reciprocal, does NOT change the sign of the base.
1 / 2^(-3) = 2^3
2^(-5) = 1 / 2^(5)
11
Q
Mixed Roots
A
Can compare size / manipulate by multiplying to square of coefficient, then moving coefficient under the root.
2√13 = (√4)(√13) = √4*13 = √52
12
Q
Parallel Symbol
A
A ∥ B
13
Q
Standard Deviation
A
Never need to calculate
- Larger spaces = wider standard deviation
- All constituents of a set shifted by the same interval? No change to standard deviation
14
Q
Normal Distribution
A
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
15
Q
Terminating Decimals
A
Factor out 2s and 5s in denominator - any other factors and it will not terminate
16
Q
Functions f(x)
A
Just plug and play
f(x) = x^2 and g(x) = x+3
f(g(1)) = 16
17
Q
Rectangle
A
A square is a rectangle, but a rectangle is not necessarily a square. Length is defined as the longer side.
18
Q
x^2 = 9
A
x = 3 OR -3
19
Q
Circumference of a Circle
A
2πr
20
Q
Colinear
A
Two points that lie on the same straight line. Any two points are colinear, three points may or may not be colinear.
21
Q
Rate / Time Problems
A
Going opposite directions? Add rates. Then plug into R*T = D
Going the same direction? Subtract rates
22
Q
Multiplying Decimals
A
Multiply the decimals by 10^x and then pull back out when finished. Can do with fractions as well.
23
Q
Simplifying Factorials
A
9! / 6! = (9)(8)(7)(6!) / 6! = (9)(8)(7)
24
Q
Can’t do the math? Shouldn’t do the math?
A
- Estimate
- Backsolve
- Use your own numbers
- Get creative
25
(w / x) / (y / z)
(w / x)(z / y)
26
Distance Formula
Rate * Time = Distance
27
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)
28
Percentages
is / of = % / 100
Set equal to each other and cross multiply
29
Right Triangle
a^2 + b^2 = c^2 applies
"Legs" are the shorter sides, not the hypotenuse
30
Percent Change
(New - Old) / Old
31
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
32
If a is divisible by b, and b is divisible by c...
a is also divisible by c
33
If d has e and f as prime factors, then...
d is divisible by e, f, and (e)(f)
34
Every factor of a number (except 1)...
Is a product of a different combination of the number's prime factors
35
Quadrilateral
360 degrees
Area of a Trapezoid = ((Base 1 + Base 2) / 2 )(Height)
36
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
37
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
38
Is Zero negative or positive?
Neither - it is a neutral integer
39
Area of a circle
πr^2
40
3D Boxes
Vertices = Corners
Faces = Sides
Split into planes to find area, volume, etc.
41
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
42
Sum of Consecutive Numbers
(n/2)(First number + Last Number)
43
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.
44
Adding or Subtracting Fractions
Simply multiple the denominators to get a common denominator and the numerators accordingly to get add or subtract.
45
Percentage Change Calculation
(New - Old) / Old
46
Three Overlapping Sets
Draw Venn Diagram, always subtract out center groups
47
Arc Length Formula
(Center Angle / 360) = (Arc Length / Circumference)
48
Divisibility Rules: 6
Must be divisible by 2 and 3
49
Sides of a 45-degree right triangle
x, x, x√2
50
Range
Largest value in a set minus the smallest value
51
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
52
Parallelogram
A shape consisting of two sets of parallel lines, angles opposite each other are equal. Area = base*height
53
Divisor
All the factors of a number and their negative counterparts
54
(√x)(√y) =
√xy
55
(√x) / (√y) =
√(x/y)
56
√x + √x =
2√x
57
√x + √y =
Nothing can be done to simplify here
58
(a / b)(c / d) =
ac/bd
59
The b root of x^a =
x^(a/b)
60
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
61
Area of a Triangle
1/2 * base * height
62
Compounded Percents
Calculate as several percent changes at a time
63
5-12-13 Right Triangle
May be disguised by proportional multiples
64
Divisibility Rules: 2
Any even number is divisible by 2
65
Trapezoids
Area of a trapezoid = ((Base 1 + Base 2)) / 2) * Height
66
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
67
Substitution Method
For solving systems of equations. Solve for one variable in terms of the other, then plug the result into the other equation
68
Even Exponent?
The solution to x^2 = 9 could be 3, OR -3
69
Quant Problem Solving Strategies
1. Estimate
2. Backsolve
3. Use Your Own Numbers
70
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
71
Favorite Probability Trick
Find the probability of what you're NOT supposed to be solving for, and subtract it from 1
72
Data Sufficiency Danger Areas
1. Negative Numbers
2. Even Exponents
3. Non-Integers
4. Inequalities with Variables - don't know sign, may not know direction of inequality symbol
73
Sub-Divided Group
Draw a visual tree diagram, use 100 as a base if working with percentages
74
Multiplying Fractions
(a/b)*(c/d) = ac/bd
75
Dividing Fractions
Multiply by reciprocal
(a/b) / (c/d) = ad/bc
76
GCF Facts
1. The GCF of two numbers may be 1, as in the case of primes
2. GCF can also be one of the two numbers, as in the case of 6 and 12
77
Similar Triangles
Triangles that have the same angles but different sizes have proportional perimeters and areas
78
Is 1 prime?
No
79
Divisibility Rules: 3
If the sum of the digits of a number sum to 3, then the number is divisible by 3
80
(Even)(Even) =
Even
81
(Even)(Odd) =
Even
82
(Odd)(Odd) =
Odd
83
Even +/- Even =
Even
84
Even +/- Odd =
Odd
85
Odd +/- Odd
Even
86
What to do if forget Even/Odd rules
Use 2 and 3 - all numbers, even negatives, behave the same way
87
(y^a)(y^a) =
(x*y)^a
Usually working backwards to find prime bases
88
(x^a)(x^b) =
x^(a+b)
89
(x^a)^b =
x^(ab)
90
x^a^b =
x^(a^b)
91
(x^a)/(x^b) =
x^(a-b)
92
Difference of Squares
x^2 - 16 = (x+4)(x-4)
May need to manipulate expression to get a difference of squares
93
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.
94
Absolute Values
Keep track of negatives - try to backsolve
95
Quadratic Equation Terms
x^2 + 7x + 12 = 0
1. Factors: (x+3)(x+4) = 0
2. Roots (Solutions): x = -3 or x = -4
96
Difference of Squares - General Formula
x^2 - y^2 = (x-y)(x+y)
97
Is 0 an integer?
Yes
98
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
99
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
100
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.
101
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
102
Multiples of 5 between 358 and 81?
85 = 5*17
385 = 5*71
71-17+1 = 55
103
Rational Substitution
y - 13√y + 36 = 0
Substitute u for √y and solve
u^2 - 13u + 36 = 0
104
(a-b)^2 =
a^2 + b^2 - 2ab
105
(a+b)^2 =
a^2 + b^2 + 2ab
106
(a+b)(a-b)
a^2-b^2
Difference of Squares
107
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
108
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
109
Is a number a prime number?
1. Cannot be >2 and even
2. Not divisible by 3
3. Not divisible by 5
4. Not divisible by 7
110
Decimal Addition or Subtraction
Line up decimal points and add
111
Decimal multiplication
Product will have same total number of decimals as the sum of the decimal points of both factors
112
Decimal Division
Multiply top and bottom by the same 10^x to get whole numbers, then divide
113
Equilateral Triangle
Regular Triangle, all sides same length, all angles = 60 degrees.
Often need to bisect into two 30, 60, 90 right triangles.
114
Sides of a 30-60-90 Right Triangle
x, x√3, 2x
115
To Remember: Angles
1. When lines intersect, angles on same side add to 180 degrees
2. All angles in a parallelogram are the same if they face each other
116
Complex Combinations
Think total - options that don't work, or one way that works * number of options
117
Circumference of a Circle
2πr
118
Integer
Any whole number, including 0
119
Units Digit Multiplication
Just multiply the units digit to find what the units digit will be. Typically works in patterns for large exponents
120
Average Rate or Speed Calculation
TOTAL Distance / TOTAL time
May need to find totals first
121
Cubed Roots
As with square roots, find prime factors and reduce by triple occurrences under the root symbol
122
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
123
Divisibility Rules: 9
If the digits sum to something divisible by 9, the number is divisible by 9 and 3
124
Divisibility Rules: 7
No clean rule - long division
125
Proportion
Ratios or fractions that are equal to each other. Cross multiply and divide.
126
Things to Remember: Scientific Notation
Just two things multiplied together, can reduce or split apart as need be
127
Calculation for slope of a line
Rise / Run
128
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
129
Composite Numbers
Any number that is composed of multiple primes factors
130
Probabilities: And vs. Or
And: Multiply
Or: Add
131
Circles: Interior Angles
Triangles from the center have radius as sides, so makes an isosceles triangle
132
Divisor
All the factors of a number and a their negative counterparts
133
Area of a Circle
πr^2
134
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
135
Compound Interest Calculation
Usually don't need to calculate:
Principle(1+(Interest Rate/# of Periods per Year))^((# of Periods per Year)(# of Years))
136
Scientific Notation
Can move decimal back and forth by changing the 10^x exponent. Can split the factors up as well if needed
137
Line Slope Properties
Parallel lines have the same slope, perpendicular lines have negative reciprocal slopes
138
Area of a Trapezoid
((Base 1 + Base 2) / 2) * Height
139
Convert a Fraction to a Decimal
Long Division
140
Basic Concepts: Geometry
1. Do NOT trust the diagram
2. Geo problems are susceptible to estimation
3. Always be looking to extend lines or subdivide shapes if needed
141
Bisection
To divide something exactly in half
142
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
143
Volume of a Cylinder
Find area of circle on top/bottom, then multiply by height
144
Perpendicular
⊥, negative reciprocal slopes
145
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
146
Properties of a Square
A quadrilateral, four 90-degree angels, all sides same length.
Can bisect into two 45-degree right triangles
147
Arc Length Formula
(Center Angle / 360) = (Arc Length / Circumference)
148
Basic Probabilities Calculation
Acceptable Outcomes / Total Possible Outcomes
Always between 0 and 1
149
Isosceles Triangle
Triangles in circles are isosceles, if the center is at an angle - 2 sides are radius
150
Divisibility Rules: 5
Ends in a 0 or a 5
151
Hexagon
6 sides, (6-2)(180) = 720 total interior degrees
152
Order in Smaller Group
n! / (n-k)!
153
Perfect Square
The square of an integer
154
Perfect Cube
The cube of an integer
155
Shortcut for Addition or Subtraction
Do math for just units digit - is that enough to narrow it down to one answer?
156
Simplify...
...before you multiply
157
Prime #: # of Factors
2
158
Square of Primes: # of Factors
3
159
Composite Number: # of Factors
4 or more
160
First Several Primes
2, 3, 5, 7, 11, 13, 17, 19
161
When two numbers DONT share any prime factors...
Their LCM is always their product
162
When two numbers DO share prime factors...
Their LCM is less than their product - you have to strip out any common factors
163
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
164
Two factors of x with Primes in common?
Combine, eliminating any overlap in prime factors to find the LCM
165
Two factors of x with no primes in common?
The LCM is the product of those numbers
166
Negative Number raised to an even power
Always positive
167
Negative Number raised to an odd power
Always negative
168
Even Exponents...
...Hide the Sign of the base
169
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.
170
When a positive power to a negative power...
Multiply the exponents - you now have a negative exponent
171
Square of a Sum
(x + y)^2 = x^2 + 2xy + y^2
172
Square of a Difference
(x - y)^2 = x^2 - 2xy + y^2
173
Difference of Squares
(x + y)(x - y) = x^2 - y^2
174
(y - x) / (x - y)
GMAT Disguise - factor out -1 from top or bottom to get factors you can simplify
175
Quadratics in Fraction?
Factor and Cancel! Avoid fractional coefficients at all costs
176
When should you divide by a variable?
Only if you are SURE the variable is not 0
177
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
178
Substitution (y + 1)^2 = 16
Substitute u for (y + 1)
u^2 = 16
u = 4 or -4
y + 1 = 4
y + 1 = -4
Solve
179
Quadratic Roots
Roots are possible solutions, but the variable is not equal to both simultaneously
180
Simplify Quadratics before factoring
Roots are answers to factors, not simplified coefficient
3(x + 3)(x+4)
x = -3, -4
181
Multiplication FOIL
(102)(301) = (100)(300) + (100)(1) + (2)(300) + (2)(1)
Allows you to work with easier numbers
182
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
