Question 240232: Veronica High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.
Found 3 solutions by Theo, richwmiller, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! This is what I came up with.
ST = total number of seniors
FT = total number of seniors playing football
BT = total number of seniors playing baseball
LT = total number of seniors playing lacrosse
F = number of seniors playing football only.
B = number of seniors playing baseball only.
L = number of seniors playing lacrosse only.
FL = number of seniors playing football and lacrosse only.
FB = number of seniors playing football and baseball only.
BL = number of seniors playing baseball and lacrosse only.
FBL = number of seniors playing football and baseball and lacrosse.
We know that all seniors play on at least one sport.
The total number of seniors can be accounted for by the equation:
ST = F + B + L + FB + FL + BL + FBL
What we know so far:
ST = 85
FT = 74
BT = 26
LT = ???
F = ???
B = ???
L = ???
FB = 18
FL = 17
BL = 13
FBL = 0
We know that FT = 74
Since FT = F + FB + FL + FBL, we get 73 = F + 18 + 17 + 0 = F + 35.
This makes F = 39 which means that 39 seniors that play football only.
We assign 39 people to F.
Our tally is now:
ST = 85
FT = 74
BT = 26
LT = ???
F = 39 (removed ???)
B = ???
L = ???
FB = 18
FL = 17
BL = 13
FBL = 0
We know that BT = 26
We also know that BT = B + FB + BL + FBL
Since FB = 18 and BL = 13, and BT = 26 and FBL = 0, then the equation becomes:
26 = B + 18 + 13 + 0 which becomes:
26 = B + 31
We have 5 more people than total people on the team.
Since the team is composed of FB and BL only, and we are assuming that the numbers shown are correct, then we must have people who play Baseball AND Football.
These would be classified as FBL because they play on all 3 sports.
This means we have 5 people who being counted twice.
We subtract 5 people from FB and 5 people from BL and we add 5 people to FBL to get:
BT = B + FB + BL + FLB becomes:
26 = B + 13 + 8 + 5 because our category of FBL just became 5 from 0.
Baseball equation now becomes:
26 = B + 13 + 8 + 5 = B + 26
This becomes 26 = B + 26 which makes B = 0 which now accounts for all of the members of the baseball team.
The tally for Baseball is now:
B = 0
FB = 13 (down from 18)
BL = 8 (down from 13)
FBL = 5
BT = 0 + 13 + 8 + 5 = 26 (checks out ok because BT was given as 26).
Our overall tally now becomes:
ST = 85
FT = 74
BT = 26
LT = ???
F = 39 (removed ???)
B = 0 (removed ???)
L = ???
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5 (up from 0)
The equation for FT has to change because we dropped 5 FB and added 5 FBL.
The tally for Football is now:
F = 39
FB = 13 (down from 18)
FL = 17
FBL = 5
FT = 39 + 13 + 17 + 5 = 74 (checks out ok)
This does change the overall tally any further. It just corrects the account for Football.
We check the tally for total seniors again to see if anything is missing.
F = 39
B = 0
L = ???
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5
ST = 39 + 0 + L 13 17 + 8 + 5 = L + 82
We are missing 3 seniors.
The only category that is not completed is the number of seniors who only play lacrosse, and since all seniors must be playing at least one sport, these 3 seniors must be playing lacrosse.
We assign them to lacrosse by making L = 3.
Out tally for seniors now becomes:
F = 39
B = 0
L = 3
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5
ST = 39 + 0 + 3 + 13 + 17 + 8 + 5 = 85 (checks out ok)
The tally for the lacrosse team now becomes:
L = 3
FL = 17
BL = 8 (down from 13)
FLB = 5
LT = 3 + 17 + 8 + 5 = 33
LT is now equal to 33.
Our overall tally now becomes:
ST = 85
FT = 74
BT = 26
LT = 33 (removed ???)
F = 39 (removed ???)
B = 0 (removed ???)
L = 3 (removed ???)
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5 (up from 0)
Our tally for seniors now becomes:
F = 39
B = 0
L = 3 (up from 0)
FB = 13 (down from 18)
FL = 17
BL = 8 (down from 13)
FBL = 5 (up from 0)
ST = 39 + 3 + 13 + 17 = 8 = 5 = 85 (checks out ok).
All the seniors are accounted for.
The number of seniors who play on all three sports is 5.
The only part that is questionable is the statement "Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team."
What that means exactly is a mystery to me. I have computed the number of seniors who play all three sports. The number has to be 5 in order to make all other numbers check out. At least twice this number are members of the lacrosse team so I would have to say that the statement is satisfied.
The total accounting for all the seniors is now:
ST = 85 (given)
FT = 74 (given)
BT = 26 (given)
LT = 33 (calculated)
F = 39 (calculated
FB = 13 (calculated down 5 from 18 given)
FL = 17 (given)
FBL = 5 (calculated)
BL = 8 (calculated down 5 from 13 given)
B = 0 (calculated)
L = 33 (calculated)
Tally for Football is now:
F = 39
FB = 13
FL = 17
FBL = 5
FT = 39 + 13 + 17 + 5 = 74 (checks out ok)
Tally for Baseball is now:
B = 0
FB = 13
BL = 8
FBL = 5
BT = 0 + 13 + 8 + 5 = 26 (checks out ok)
Tally for Lacross is now:
LT = L + FL + BL + FBL
L = 3
FL = 17
BL = 8
FBL = 5
LT = 3 + 17 + 8 + 5 = 33 (checks out ok)
Tally for Seniors is now.
F = 39
B = 0
L = 3
FB = 13
FL = 17
BL = 8
FBL = 5
ST = 39 + 0 + 3 + 13 + 17 + 8 + 5 = 85 (checks out ok)
Answer by richwmiller(17219)  (Show Source):
You can put this solution on YOUR website! 
unfortunately the other tutor didn't meet all the criteria.
The number of students playing lacrosse is twice the number of students playing all three sports.
this is more or less trial and error.
But we know that the total playing all three can't be more than 13
because only 13 play baseball and lacrosse .
so working down from 13 .
13 doesn't work
12 doesn't work
11 works with 22 playing lacrosse.
here what we need to calculate
since 11 play all three 2 play baseball and lacrosse to get 13 paling base and lacrosse
since 11 play all three 6 play foot and lacrosse to get 17 playing foot and lacrosse.
since 11 play all three 7 play foot and base to get 18 playing both base and foot
since 22 play lacrosse we have 22-6- 11 -2 with 3 playing only lacrosse
since 26 play base we have 26-7-11-2 and we have 6 playing only base
since 74 play foot we have 74-7-11-6=50 playing only foot.
add them all up and we have 85=50+7+11+6+2+6+3=85
Answer by MathTherapy(10553)  (Show Source):
You can put this solution on YOUR website!
Veronica High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.
Number of seniors playing all sports: 
You can do the check!!
If you need a complete and detailed solution, let me know!!
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