GRE Math Rules Flashcards
Equations- Both Sides Rule
You can do anything you want to one side of the equation as long as you also do it to the other side of the equation.
Roots and Exponents
They balance each other out.
Order of Operations
PEDMAS - Parentheses, Exponents, Division, Multiplication, Addition, Subtraction
*when isolating a variable- reverse the order of PEDMAS
Equation Clean Up Moves
1) Get variables out of the denominators (multiple)
2) Simplify grouped terms (distribute)
3) Combine like terms
Fraction Bar Rule
Pretend there are parenthesis around the numerator and denominator of the fraction.
FOIL
First (A+b)(X+y)
Outer (A+b)(x+Y)
Inner (a+B)(X+y)
Last (a+B)(x+Y)
Factoring
pulling out a common term and rewriting the expression as a product
- variables with exponents
- numbers
- expressions with more than one term
Constant vs Coefficient
In a quadratic equation the constant is the third number without an unknown variable while the coefficient is the second number multiplied by variable X. (Ex: x² + 7x + 12 = 12 is the constant and 7 is the coefficient). The possible integers for X will be factors of 12, while the sum of the two possible integers will be 7. Thus the possible integers for X are -3 and -4 because (x+3)(x+4) is the factored version of the quadratic equation.
Quadratic Equation Rules
1) Before you factor a quadratic expression, you must make sure that the other side of the equation is 0.
2) If the sign of the middle term attached to X is positive, then the greater of the two numbers will be positive.
3) If the sign of the middle term attached to X is negative, the greater number will be negative and the smaller number will be positive.
Perfect Square Quadratic
One solution quadratic
(x+4)² = x is -4
(x-3)² = X is 3
0 in the denominator
If 0 is in the denominator, the expression becomes undefined. So 0 cannot be on the bottom of a fraction.
Multiplying and Dividing Inequalities
If you multiple or divide by a negative number, you must switch the direction of the inequality sign.
Absolute value
Describe how far away that number is from zero 0 = |number|. Treat the absolute value symbol like parentheses. |-3| = |3|
Solving:
1) Isolate the absolute value expression on one side of the equation.
2) Set upon the two equations with what’s inside the absolute value sign. Positive and negative versions.
3) Solve. Note there are two possible values for y.
Direct Formulas
The value of each item in a sequence is defined in terms of its item number in the sequence.
Recursive Formulas
Each item of a sequence is defined in terms of the value of previous items in the sequence. An = An-1 + 9
*If you do not know the value of any one term, then you cannot calculate the value of any other. (You need one domino to fall) A1 = 12
Sequence Problems
1) Determine which answer choice corresponds to the correct definition rule for a sequence.
2) Determine the value of a particular item in a sequence.
3) Determine the sum or difference of a set of items in a sequence.
*For the unit digit, use the pattern and just count up until you’ve repeated the pattern to the target number.
Inequality Techniques (Dos)
- DO think about inequalities as ranges on a number line.
- DO treat inequalities like equations when adding or subtracting terms, or when multiplying/dividing by a positive number on both sides of the ineqaulity.
- DO use extreme values t solve inequality range problems, problems containing both inequalities and equations, and many optimization problems.
- DO set terms with even exponents equal to O when trying to solve minimization problems.
Inequality Techniques DON’Ts
- DON’T forget to flip the inequality sign if you multiply or divide both sides of an inequality by a negative number.
- DON’T multiple or divide an inequality by a variable unless you know the sign of the variable.
- DON’T forget to perform operations on every expression when manipulating a compound inequality.
Fraction Denominator Rule
As the denominator of a number gets bigger, the value of the fraction gets smaller.
Common denominator
- Fraction adding and subtracting only works if you can add slices that are all the same size.
- Only the numerator changes once the common denominator is established.
- Then you have to simplify the final form.
Fraction Multiplication
Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
Reciprocals
Numbers that, when multiplied together, equal 1.
Importance because dividing a number is the exact same thing as multiplying by its reciprocal.
5/6 / 4/7 = 5/6 x 7/4
Fractions between 0 and 1
Multiplying = creates a product smaller than the original #
Division = creates a quotient or result that is larger than the original #
Splitting Up Fractions
You can never split the terms of the denominator.
15+10/5 (can split) = (15/5 + 10/5)
5/15+10 (DON’T SPLIT) = 5/25
15+10/5+2 (simplify but DON’T SPLIT) = 25/7
Smart Numbers
Pick smart numbers when no amounts are given in the problem. DO NOT pick smart numbers when any amount or total is given.
Powers of 10
1) When you multiple any number by a positive power of 10, move the decimal forward (RIGHT) the specified number of places.
2) When you divide any number by a positive power of 10, move the decimal backward (LEFT) the specified number of places.
**Negative powers of ten reverse the rules.
Heavy Division Shortcut
Move the decimals in the same direction and round to whole numbers.
1) Set up the division in fraction form.
2) Rewrite the problem, eliminating powers of 10.
3) Goal=need single digit to the left of the denominator. Move decimal to get just the single digits.
4) Focus on the whole numbers and solve.
Percents
Part/Whole = Percent/100
Benchmarks
To find 10% of any number, just move the decimal point to the left one place.
Percent Change vs. Percent of Original
Percent Change: Original + Change = New
Ex: New Price = 84 cents and Original Price = 80 cents
84/80 = 21/20 = x/100 so 2x=2,100 and x=105
Formula: Change/Original = Percent/100
Percent of Original: Original x (1 + Percent Increase/100) = New
Ex: 80x(1+5/100)= 80(1.05) = 84 is the new price
If the price was lower, than the equation is: Original x (1-Percent Decrease/100) = New
Successive Percents
Successive Percents cannot be added together. The change always creates a new price/number which takes the new percent change, not the original price/number. You can substitute smart numbers (100 in the case of percentages).
Simple Interest
Principle x Rate x Time = simple interest
Compound Interest
P(1+r/n) to the power of nt
P= principal R= rate (decimal) N= # of time per year T= # of years
Use of Fractions
1) good for canceling factor in multiplication and division
2) best way to exactly express proportions
- Fractions with denominators containing factors other than 2 and 5 = non-terminating and cannot be represented by decimals or %’s.
Use of Decimals
1) good for estimating results or comparing sizes when the basis of comparison is equivalent.
2) good for addition or subtraction
Use of Percents
1) good for addition and subtraction
2) good for estimating numbers or for comparing numbers because the denominator can be 100
Circumference Rule
Is the same anywhere if it gets through the center
Triangle Side Rule
The sum of any two side lengths of a triangle will always be greater than the third side length.
Triangle Angle Rule
The internal angles of a triangle must add up to 180 degrees
Triangle Side-Angle Relationship
Sides correspond to their opposite angles. The longest side is opposite the largest angle and the smallest side is opposite the smallest angle.
*You cannot determine how much shorter or longer the sides are.
Isosceles Triangle
A triangle that has two equal angles and two equal sides.
Equilateral triangle
A triangle that has three equal angles (all 60 degrees)
Area of a Triangle
1/2(base x height)
Right Triangle
Any triangle in which one of the angles is a right angle. Works well with the Pythagorean Theorem
Pythagorean Triples
3-4-5
(6-8-10)(9-12-15)(12-16-20)
5-12-13
(10-24-26)
8-15-17
30-60-90 Triangle
Two 30-60-90 triangles put together equal an equilateral triangle. The legs have the a specific ratio.
Short = 1 Long = √3 Hypotenuse = 2
Isosceles right triangle
Has one 90 degree angle and two 45 degree angles. The sides will be some multiple of:
Equal sides = 1
Hypotenuse = √2
Use of Right Triangles
Can help find the diagonals of other polygons such as squares, cubes, rectangles, and rectangular solids.
Rectangle Diagonal Rule
You must know either:
1) length and width
2) one dimension and the proportion of one to the other
4-sided Polygons
Squares, rectangles, parallelograms, and trapezoids
Interest: interior angles, perimeter, area
Polygon Properties
(n-2) x 180 = sum of interior angles of a polygon
Trangle = 3 sides and the sum of angles is 180 degrees
Quadrilateral = 4 sides and the sum of angles is 360 degrees
Pentagon = 5 sides and the sum of angles is 540 degrees
Hexagon = 6 sides and the sum of angles is 720 degrees
Area Formulas
Triangle: 1/2(base x height)
Rectangle: length x width
Trapezoid: [(Base1 x Base2)/2]x Height
Parallelogram: Base x Height
*If you forget one, you can simple cut the shape into rectangles/right triangles and solve.
Rectangular Solids and Cubes
3-D shapes formed from polygons
Surface Area = the sum of the areas of ALL the faces (six faces total)
Volume = Length x Width x Height
Parallelograms
Only 4-sided figures in which the opposite sides are parallel and equal and opposite angles are equal.
Rectangle Angles
All are right angles
Square Sides
All are equal
Max Area of a Quatrilateral
Square = has the largest area
Square = the smallest perimeter
Rule: a regular polygon with all sides equal and all angles equal will maximize area for a given perimeter and minimize perimeter for a given area.
Max Area of a Parallelogram
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.
Radius
Any line segment connecting to the center of the circle to any point on the circle is a radius. All have the same length.
Sector Area/Sector Length/Central Angle
1) Central Angle = part/whole - whole in this case is 100
2) Sector Area = (part/whole)(area of circle)
3) Arc Length = (part/whole)(circumference of circle)
Central Angle/360 degrees = Sector Area/Circle Area = Are Length/Circumference
Inscribed Angle
Has its vertex on the circle itself (rather than on the center). It is equal to half of the arc it intercepts.
Inscribed Angle = 1/2(sector angle)
- A triangle is inscribed in a circle if all the vertices of the triangle are points on the circle.
- If one side of the inscribed triangle is a diameter of the circle, then the triangle must be a right angle.
Cylinders - Surface Area
Made up of two circles and one rolled up rectangle.
Surface Area = 2 circles + rectangle =
2(πr²) + 2(πrh)
Surface Area Needs:
1) radius of the cylinder
2) height of the cylinder
Cylinders- Volume
Made up of two circles and one rolled up rectangle.
Volume = πr²h
Volume Needs:
1) radius of the cylinder
2) height of the cylinder
*Two cylinders can have the same volume but different shapes.
Properties of Intersecting Lines
1) a+b+c+d = 360 degrees because the angles form a circle
2) supplementary angles: interior angles that combine to form a line sum of 180 degrees.
3) vertical angles: angles found opposite of each other on intersecting lines, these are always equal
*applies to multiple intersecting lines
Exterior Angle
Of a triangle and is equal in measure to the sum of the two non-adjacent (opposite) interior angles of the triangle.
Triangle = a, b, and c
Exterior Angle of b = a+c
Point vs Number
Same thing on number lines!
Undefined line or curve
You cannot infer the location of a point based solely on visual cues.
Linear Equations
Y=mx+b
M= slope, how steep the line is B = y-intercept, where the line crosses the y-axis
Slope
Think of it as a fraction:
Numerator/denominator
Rise/run
Chang in y/change in x
Numerator: tells you how many unites you want to move in the y direction = how far up/down
Denominator: tells you how many unites you want to move in the x direction = how far left/right
No Intersection (Linear Equations)
If two lines do not intersect, than they are parallel = no (x,y) that will fit both equations
Finding the Distance Between 2 Points
Linear Equations
1) Draw a right triangle connecting the points
2) Find the length of the two legs of the triangle by calculating the rise and the run
3) use the Pythagorean theorem to calculate the length of the diagonal.
Diameter vs Other lines
The diameter is the largest line to pass through a circle. The farther a line passes from the center, the smaller the line will be
Square of the length
D² = 11² x 7² x 8²
= 121 x 49 x 64
= 234 (square of the length)
= 3√26 (final form)
Changes in x on the coordinate place
If the negative change of an equation is large it means the line is “falling” quickly
If comparing to another line, if the negative change in y is larger, than is is falling faster than the other line and thus is more negative
Circles within Circles
If circle 1 is inside circle 2 which is inside circle 3 and the points of origin all lie on the same line segment, than the two interior circle must add up to the same diameter as the larger circle.
C = π(Diameter 1 + Diameter 2) = π x Diameter 3
Pos/Neg Angles
NO Angles are always positive!
Square: Relationship between Area & Perimeter
Area = nP²
Area = s²
Perimeter = 4s
S = sides
Divisibility Properties
Divisible by:
2 if the integer is even
3 if the SUM of the integer’d units is divisible by 3
4 if the integer is divisible by 2 twice/the two-digit number at the end is divisible by 4
5 if the integer ends in 0 or 5
6 if the integer is divisible by both 2 and 3
8 if the integer is divisible by 2 three times in succession/the three digit number as the end is divisible by 8
9 if the SUM of the integer’s digits is divisible by 9
10 if the integer ends in 0
*If none fit, check for 7 before assuming it is prime
Factor Language
2 is a factor of 6 2 is a divisor of 6 2 divides 6 6 is a multiple of 2 6 is divisible by 2 2 goes into 6
1,2,3,6 are factors of 6
Prime Factors
The unique DNA of a number, every number has a unique prime factorization
All the factors of an integer are just different domination so far the prime numbers that mae up the prime factorization of that integer
Prime Factorization Rule
1) If a is divisible by b, and b is divisible by c, then a is divisible by c as well (12/6 and 6/3 so 12/3 works too)
2) if d is divisible by two different primes, e and f, than d is also divisible by e x f (20 is divisible by 5 and 2 as well as 5x2=10)
Fewer Factors. More Multiples
1) factors divide into an integer
(old integer is greater than or equal to the new integer)
2) positive multiples multiply out = infinite number of possibilities
(old integer is less than or equal to the new integer)
Factoring Terminology
12 is divisible by 3 12 is a multiple of 3 12/3 is an integer 12=3n, where n is an integer 3 is a divisor of 12 3 is a factor of 12 3 divides 12 12/3 yields a remainder of 10 3 "goes into" 12 evenly 12 items can be shared among 3 people so that each person has the same # of items
Unknown Base
Roots and Exponents
x^3 = 8 3^√x^3 = 3^√8 x = 2
**Square root each side with the power of 3 to cancel out the power of 3 for x
Unknown Square Root
Roots and Exponents
√x = 6 (√x)² = 6² x = 36
**Square both sides to cancel out he square root.
Unknown Exponent
Roots and Exponents
2^x= 8
*Rewrite the equation so that the same base is on each side of the equal sign.
2^x 2^3 3^x+2 = 27
x = 3 OR 3^x+2 = 3^3
x = 1
Rates and Work
Rate x Time = Distance or Rate x Time = Work
Rate = Work/Time or Rate = Distance/Time
Time = Work/Rate or Time = Distance/Rate
Average
Sum/number of terms = Average
A = s/n
A x n = S
Factorials to 6!
1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24 5! = 5 x 4 x 3 x 2 x 1 = 120 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
Probability
of desired or successful outcomes / total # of possible outcomes
Overlapping Sets
c = a-e Total = a+b-e+f d = b-e Total = c+d+e+f
a = Salaried b = In Operations c = Salaried BUT not in Operations d = In Operations BUT not Salaried e = Salaried and in Operations f = neither (not salaried and not in operations)
RTD Chart Units
They all must match!
Multiple Rate Problems or Relative Rate Problems
Two bodies are traveling at the same time:
1) the bodies are moving toward each other
2) the bodies are moving away from each other
3) the bodies are moving in the same direction on the same path
Relative Rates: Example 1
Two people increase the distance between themselves (they are moving toward each other).
Person 1: r = 5
Person 2: r = 6
The rate they decrease the distance = Person 1 + Person 2 = Combined Rate
5+6 = 11mph
Relative Rates: Example 2
Two cars increase the distance between themselves (they are going away from each other).
Car 1: r = 30
Car 2: r = 45
The rate they increase the distance between each other = Car 1 + Car 2 = Combined Rate
30 + 45 = 75mph
Relative Rates: Example 3
Persons X and Y decrease the distance between themselves (they are on the same path going the same direction).
Person 1: r = 8
Person 2: r = 5
The rate they decrease the distance between each other = Person 1 - Person 2 = Combined Rate
8-5 = 3mph
Average Rate Rule
If an object moves the same distance twice, but at different rates, then the average rate will never be the average of the two rates given for the two legs of the journey.
The average rate will be closer to the slower of the two rates than the faster rate because the object spent more time traveling at the slower rate.
Average Rate Process
1) Pick a smart # or the distance (whatever numbers you have for the rates should dictate the smart #)
2) Find the time for each part of the trip = “going” and “returning”
3) Solve for the total time and total distance by adding the time variables together and adding the distance variables together.
4) RT= D, plug in the total sum variables for T and D, then divide D by T to get R.
Ratio Relationships
1) Part-Part Relationship (Ex: men to women)
2) Part-Whole Relationship (Ex: 2 men to every 7 employees)
Limits of Ratios
The relationship that ratios express is division. They only express a relationship between 2 or more items; they do not provide enough information on their own to determine the exact quantity for each item.
Proportional Ratios
If two quantities have a constant ratio, they are directly proportional to each other.
Ratios and Cross Multiplication
NEVER cancel factors diagonally across an equals sign.
Ex: 4 girls : 7 boys and there are 35 boys
4/7 = x/35 (both 7 and 35 can be simplified to 1 and 5)
4/1 = x/5
Uses for the Unknown Multiplier Technique
When to use it:
1) the total items is given
2) neither quantity in the ratio is equal to a number or a variable expression.
Ratio = 3 men : 4 women, total people in the room = 56
3x + 4x = 56
7x = 56
x = 8
3(8) = 24 and 4(8) = 32, thus 24+32 = 56
unknown multiplier
the variable x, used to reduce the number of variables making the algebra easier.
Multiple Ratios
Ex: What is the least # of (x items) in the group?
You can change ratios to have common terms corresponding to the same quantity.
1) The two ratios will be interconnected in some way. The item that connects the other two items is the key. Whatever the key item numbers are, you must find the lowest common multiplier for the key item numbers. The corresponding item in each ration will have to be multiplied by the same number as the corresponding key item number.
(3 cars : 2 buses) x 5 = 15 for the key item 15: 10 : none
(5 cars : 4 bikes) x 3 = 15 for the key item 15: none: 12
2) The resulting ratio will have three values: (15:10:12), which to get the actual lowest # of items in the group, the new ratio needs a positive Unknown Multiplier which can produce the least # of action figures.
15(1) + 10(1) + 12(1) = 37
Multiple Ratios
Ex: ratio of buses to bikes when you have two ratios such as (3 cars : 2 buses) and (5 cars : 4 bikes)
You can change ratios to have common terms corresponding to the same quantity. If you combine the ratios, the you can find the second rat of buses to bikes.
15: 10:12 is the combined ratio
12: 10 is the ratio for bikes to buses = 6:5
Weighted Averages
a weighted average of only two values will always fall closer to whichever value is weighted more heavily.
Ex: 2 shots 15% alcohol vs. 3 shots 20% alcohol
*It will be closer to 20% because it’s weighted more.
Odd Number List (Median)
A list with an odd number of values means a median that is the unique middle value when the data is arranged in increasing or decreasing order.
- the median will be found in the list of numbers!
Even Number List (Median)
A list with an even number of values means a median that is the arithmetic mean of the two middle numbers when the data are arranged in increasing or decreasing order.
- the median is a new value that is not in the list, unless the two middle values are equal.
Set vs. List
Set: all the # are different
List: repeat values are allowed, but not required.
Dataset = list!
Small Standard Deviation
Indicated that a list is clustered closely around the average (mean) value.
Large Standard Deviation
indicated that the list is spread out widely, with some points appearing far from the mean.
Variance
the square of the STD
Range
The difference between the largest number in the list and the smallest number in the list.
Quartiles and Percentiles
There are always four quartiles.
Quaritle Marker Q1 = the average of the highest item in Q1 and the lowest value in Q2
(Q1 highest item + Q2 lowest item)/2
Quartile Marker Q2 = the same as the median of the list
(Q2 highest item + Q3 lowest item)/2
Normal Distribution/Gaussian Distribution
Looks like the classic bell curve, rounded in the middle with long tails, and symmetric around the mean. The mean is the median.
Properties of Parabolas
1) Quadratic equation is ax² + bx + c
2) a, b, and c are constants and a doens’t = zero.
3) The x-intercepts of the parabola are the solutions of the equation ax² + bx + c = 0
4) a = positive, then the parabola opens upward and the vertex is the lowest point.
5) a = negative, the the parabola opens downward and the vertex is the highest point.
6) The two x-intercepts are equidistant from the line of symmetry.
STD and Normal Distribution Properties
- 1 STD from mean
- 2 STD from mean
- 3 STD from mean
1) Roughly 2/3 of the sample will fall within 1 STD of the mean. (1/3 will be below the STD and 1/3 will be above the STD)
50% - (1/2)(2/3) = 17% or 17th Percentile
50% + (1/2)(2/3) = 83% or 83rd Percentile
2) 96% of the sample will fall within 2 STD of the mean. (48% below the STD and 48% above the STD)
50% - (1/2)(96%) = 2% or 2nd Percentile
50% + (1/2)(96%) = 98% or 98th Percentile
3) Only about 1/1000 (0.1%) of the sample is 3 or more SD below and above the mean.
Fundamental Counting Principal
If you must make a number of separate decisions, then multiply the numbers of ways to make each individual decision to find the number of ways to make all the decisions.
2 bread types x 3 filling types = 6 possible sandwiches
Simple Factorials and (n!)
simple factorial: the number of ways of putting n distinct objects in order, if there are no restrictions, is n!
n! + the product of all the positive integers from 1 through n; inclusive.
Multiple Arrangements Rule
If the GRE requires you to choose two or more sets of items from separate pools, count the arrangements separately- perhaps using a different anagram grid for each. Then multiple the number of possibilities for each step.
Ex: Alpha Sigma Pi delegation = 3 seniors and 2 juniors. The sorority has 12 seniors and 11 juniors.
1) Pool of 12 seniors and 3 are chosen = 12! / (9! x 3!)
2) Pool of 11 juniors and 2 are chosen = 11! / (9! x 2!)
Possibilities of Seniors = (12x11x10)/6 = 220
Possibilities of Juniors = (11x10)/(2x1) = 55
SO…. 220 x 55 = 12,100 different delegation possibilities
Anagramming Words Rule
If two letters are repeated, they create indistinguishable elements that have to be accounted for. (Ex: Level)
5! = number of letters
2! x 2! = the two repeating letters than need to be addressed, you cannot “uncount” these!
5!/(2!x2!) = 30
Probability Rule for 1 and 0
A probability of 1 means that the event must occur 100% of the time. A probability of 0 means that the event is impossible and thus occurs 0% of the time.
More than 1 Event: “AND”
To determine the probability that event X ad event Y will both occur, MULTIPLY the two possibilities together. Assume X and Y are independent events.
Ex: Flip a coin twice and get heads both times =
(1/2)(1/2) = (1/4)
(1/2) = chance it will be heads and heads
“Both this AND that” Probability
Through this phrasing, you define success in a more constrained way, so the probability of success will be lower.
More than 1 Event: “OR”
To determine the probability that event X OR event Y will occur, ADD the two probabilities together. Assume that X and Y are mutually exclusive.
Ex: A fair die rolled once will land on either 4 or 5.
(1/6) + (1/6) = (1/6) = (1/3)
(1/6) = prob of 4 (1/6) = prob of 5
(1/3) = chance it will be 4 or 5
independent events (Probability)
the likelihood of one occuring does not depend on the likelihood of the other occuring.
mutually exclusive events (Probability)
the two events cannot both occur.
“Either the OR that” Probability
through this phrasing, you define success in a less constrained way, so the probability of success will be higher.
“OR” Probability for events that are not completely mutually exclusive.
Subtract out the probability that both events occur.
Ex: 20 balls, 10 white with integers 1-10, 10 red with integers 11-20. 1 ball selected, white OR even number.
(1/2) of the balls are white
(1/2) are even
(5/20) are white and even
(1/2) + (1/2) - (5/20) = 3/4
p(white or even) = p(white) + p(even) - p(white and even)
Probability of Failure
If on the GRE, “success” contains multiple possibilities- especially if the wording contains phrases such as “at least” and “at most” - then find the probability that success does not happen.
1 - x
Probability of Success + Probability of Failure = 1
Probability of Failure = x
Ex: 3 rolls and at least one will be 6
(5/6) = probability it won’t be 6
(5/6)(5/6)(5/6) = 125/216 = x
1-(125/216) = 91/216 chance one roll will be 6
Domino Effect Problems
The outcome of the first event will affect the probability of a subsequent event.
Ex: Drawing red blocks out, but not putting them back in. 10 blocks, 3 red
(3/10) = probability first round that it’s red
(2/9) = probability second round that it’s red
(3/10) x (2/9) = 6/90 = 1/15 chance red for both draws
Optimization
1) Be aware of explicit constraints in the text as well as hidden constraints (often real-world, like humans are always a positive whole #)
2) In most cases, you can max or min quantities by choosing the highest or lowest values of the variables that you can select.
Overlapping Sets
People/Items that fit in both categories lessen the number of people/items in just one category. Thus, all other things being equal, the more people/items in “both”, the fewer in “just one” and the more in “neither”.
