9) Overlapping sets Flashcards
Primary tool for handling TWO overlapping set Qs
Set-Matrix
> column headings
> row headings
>Total column and Total row
Note:
> the elements represented by the column headings must be MUTUALLY EXCLUSIVE
> the elements represented by the row headings must also be MUTUALLY EXCLUSIVE
e.g., Col heading 1 = Math; Col heading 2 = NOT Math
Row heading 1 = English; Row heading 2 = NOT English
Alternative tool: Simple two circle venn diagram with a box
Also Note…
> the row totals and column totals set the UPPER MAX BOUND (which can be useful for min/max questions)
—> inner quadrants must be less than or equal to their row and column totals
What is the formula for calculating the TOTAL number of UNIQUE elements involving TWO overlapping sets, A and B?
Visualize a set-matrix:
Formula #1:
Total # of unique elements = Only A + Only B + Both A and B + Neither A or B
> sum of the inner 4 quadrants = total
Formula #2:
Total # of unique elements = Total A + Total B - Both A and B + Neither A or B**
> don’t forget about the NEITHER A or B
Types of overlapping set Qs
(1) Numbers
(2) Percents
> only difference is the TOTAL figure is 100 (for assuming there are 100 units in total) OR assign a variable like x to represent the number # of items in the set
(3) Fractions
> Can assume Total figure is “x” where x represents the LCM of the denominators of all the fractions, or leave it as “x”
Note:
> pay attention to which totals you are computing
DS involving set-matrix - when is it sufficient vs not sufficient?
Remember that the inner 4 quadrants must add to the total
> know when the quadrants are all fixed
You need to fix at least 2/3 cells in each row to be able to solve the row
You need to fix at least 2/3 cells in each column to be able to solve the column
Note:
> If you are given all the row totals AND column totals, you just need to know ONE of the 4 inner quadrants to know them all
Complex set-matrix PS involving variables
Option 1: Solve system of linear equations (two variables, two unique equations)
Option 2) To handle multiple different equations, just try to find TWO EQUATIONS sharing an overlapping cell—> solve for one relationship (like a ratio), then plug into the other equation
Solving for a COMBINATION
of variables in a set (not just solving for a cell or percent, but a combination of cells)
e.g., if we are given two sets, how many elements belong exactly to ONE GROUP (only A + only B)
e.g., how many elements belong to not group A or not in group B?
Exactly ONE group = x + y
= Only A + Only B
Not A or Not B = SUM of the Not A + Not B columns, less duplicate for (Not A and Not B)
————> “Or” (treat as SUM)
= x + y + z
= (B and Not A + Not A and Not B) + (A and not B + Not A and Not B) - (Not A and Not B)
= (Not A only) + (Not B only) (Neither A and B)
> not equal to “neither A or B”
so an element that is only in A = part of Not B, along with Not B and Not A
Other tips:
> Not = 1 or total - Yes
Two overlapping sets: Number of members in either set
e.g., How many students learn either English or Math?
(A or B) = #(A) + #(B) - #(A and B)
Two overlapping sets: Maximization / minimization questions within sets
e.g., at a certain car dealership with 3000 cars, there are 1200 red cars and 1400 cars with a sunroof. At least 500 of the cars at the dealership are not red and do not have a sunroof. What is the greatest number of cars at the dealership that could be red AND have a sunroof?
Tip –> use the set-matrix to figure out realistic bounds (instead of relying on the equation where it can be harder to visualize)
> will notice that the max # of “other” cars is capped at 1600, making the max # of red and sunroof cars 1200
> need to make sure the numbers square up along each row and column
Three overlapping sets (A, B, C), best tool?
Number of elements that belong to exactly one category?
Number of elements that belong to exactly two categories?
Total number of unique items?
Three-circle venn diagram inside a box (for neither region)
> there are 7 sub “only” regions, + one neither region
Number of elements that belong to exactly one category = Only A + Only B + Only C
Number of elements that belong to exactly two categories= (A and B only) + (B and C only) + (A and C only)
Total number of unique items has a few different variations of the formula…
FORMULA 1 (“only” subparts) = #[A only + B only + C only] + #[A and B only + B and C only + A and D only] + #[A and B and C only] + #[Neither A nor B nor C]
> use if given individual sub components of “only 2”
FORMULA 2 (sum individual circles then subtract sub components of overlap) = #[A] + #[B] + #[C] - #[A and B only + B and C only + A and C only] - 2*#(A and B and C only) + #(Neither)
> Use when you are given lump of “only 2”
FORMULA 3 (sum individual circles than subtract overlap) = #[A] + #[B] + #[C] - #[A and B + B and C + A and C ] + #(A and B and C only) + #(Neither)
> Use when you are given lump of “2”
Note:
> Number of elements that belong to A and B only = Overlap A and B - Overall all 3 sets
Integrating algebra into overlapping sets Qs – finding the minimum number of ADDITIONAL quantities from this group VS ADDITIONAL quantities added to the group
e.g., There are only mid-cap mutual funds and small-cap mutual funds in a certain group of mutual funds. Of the total number of mutual funds, 20 are mid-cap funds, and 12 beat the SP 500. In addition, 5 of the mid-cap funds beat the SP 500, and 11 of the small-cap funds did not beat the SP 500. What is the minimum number of additional small-cap funds FROM this group that would have to beat the SP 500 so that over 50 percent of the small-cap funds beat the SP 500?
e.g., a certain bakery baked a batch of 500 cookies one day. Of those, 320 contained nuts, 230 contained chocolate chips, and 85 contained neither nuts nor chocolate chips. What is the fewest possible number of cookies with both chocolate chips and nuts that would need to be ADDED TO that batch so that cookies with both nuts and chocolate chips represented more than 3/5 of all the cookies in the batch?
Pay attention to the wording of questions involving ADDING
Type 1) “Additional quantities FROM the group”
> Wording means that you are looking to ADD to a cell, but the TOTAL numbers do not change (just SHIFTING between the groups)
Note:
> “minimum” DOES NOT mean “EQUAL” –> might have to ADJUST final number up to find true minimum
Set up the word problem:
(current no. of small-cap funds that beat SP 500 + x additional)/total number of small cap funds > 50%
x > 2
THEREFORE minimum is 3!!! (not 2)
Type 2: “Add TO the group”
> Wording means we add to the cell AND increase the total number in the sets
What is the most common mistake you tend to make on set-matrix or 3 overlapping set questions?
Math mistake
> double check your calculations (esp. using the three overlapping sets formula)
Hard DS involving two overlapping sets
Out of 200 dog owners, is the number of owners who run their dogs greater than the number of owners who walk their dogs?
(1) All of the dog owners who run their dogs also walk their dogs
(2) 75 dog owners walk their dogs
Set-matrix
> remember these set-matrices naturally create UPPER MAX BOUNDS that you can infer the max! (recall similar difficult DS involving ratios and multipliers)
Two sets tricky wording - “all of A also belongs to B”
Means there are not members of A only
A = A only + A and B, and A only = 0
So A = A and B
