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)
