SOLUTION: A bag contains 100 cards with letters on them. There are 50 cards with a letter A on them, 25 cards with a letter B, and 40 cards with a letter C. Referring to the scenario above

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Question 1042269: A bag contains 100 cards with letters on them. There are 50 cards with a letter A on them, 25 cards with a letter B, and 40 cards with a letter C.
Referring to the scenario above, if only 10 cards have both letters A and B, how many cards definitely have only the letter C and NOT the letter A or B?
Found 2 solutions by josmiceli, Edwin McCravy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
I used a Venn diagram with 3 intersecting circles.
If I start with just the A and B circles alone and
allow them to intersect, the intersection has 10
cards in it.
---------------
The part of circle A that is outside the intersection
has 50 - 10 = 40 cards in it.
---------------
The part of circle B that is outside the intersection
has 25 - 10 = 15 cards in it
---------------
So far 40 + 10 + 15 = 65 cards are accounted for.
Now I slide the C circle into the intersection of
A and B. This doesn't change the results I already have.
---------------------------------------------
There are 100 cards total, so 100 - 65 = 35
MUST be outside circles A and B
-------------------------------
35 cards have only the letter C and not A or B
--------------
Hope I got this right -you can get another opinion, too

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

A bag contains 100 cards with letters on them. 
So

d+e+f+g+h+i+j = 100

There are 50 cards with a letter A on them, 
d+e+g+h = 50

25 cards with a letter B, 
e+f+h+i = 25

and 40 cards with a letter C.
g+h+i+j = 40

Referring to the scenario above, if only 10 cards
have both letters A and B, 
e + h = 10

How many cards definitely have only the letter C
and NOT the letter A or B?
We want only the value of j.

So we have this system of equations:



Multiply eq.1 by -1



Add eq. 1 to eq. 2 



Multiply eq. 3 by -1



Add equation 5 to eq. 3

  

Multiply equation 2 by -1

 

Add eq. 3 to eq. 2:

 

The second equation tells us that j = 35.

The other tutor is correct.

Edwin