Question 1207374: A survey of 1,174 tourists visiting Orlando was taken. Of those surveyed:
250 tourists had visited LEGOLAND
268 tourists had visited Universal Studios
53 tourists had visited both the Magic Kingdom and LEGOLAND
100 tourists had visited both the Magic Kingdom and Universal Studios
77 tourists had visited both LEGOLAND and Universal Studios
32 tourists had visited all three theme parks
79 tourists did not visit any of these theme parks
How many tourists only visited the Magic Kingdom (of these three)?
Found 3 solutions by mccravyedwin, ikleyn, Edwin McCravy:
Answer by mccravyedwin(406)   (Show Source):
You can put this solution on YOUR website!
Put the clues in reverse order:
Of those surveyed:
79 tourists did not visit any of these theme parks
32 tourists had visited all three theme parks
77 tourists had visited both LEGOLAND and Universal Studios
100 tourists had visited both the Magic Kingdom and Universal Studios
53 tourists had visited both the Magic Kingdom and LEGOLAND
268 tourists had visited Universal Studios
250 tourists had visited LEGOLAND
A survey of 1,174 tourists visiting Orlando was taken.
The answer to "How many tourists only visited the Magic Kingdom (of these
three)?" will be the answer to region g.
79 tourists did not visit any of these theme parks
So put h=79 outside all three circles.
32 tourists had visited all three theme parks
So put e=32
77 tourists had visited both LEGOLAND and Universal Studios
The other 77-32=45 go in region b, so b=45
100 tourists had visited both the Magic Kingdom and Universal Studios
The other 100-32=68 go in region f, so f=68
53 tourists had visited both the Magic Kingdom and LEGOLAND
The other 53-32=21 go in region d, so d=21
268 tourists had visited Universal Studios
The other 268-45-32-68=123 go in region c, so c=123
250 tourists had visited LEGOLAND
The other 250-45-32-21=152 go in region a, so a=152
A survey of 1,174 tourists visiting Orlando was taken.
The other 1174-152-45-123-21-32-68-79 = 654, so g=654
That's the answer, 654.
Edwin
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
A survey of 1,174 tourists visiting Orlando was taken. Of those surveyed:
250 tourists had visited LEGOLAND
268 tourists had visited Universal Studios
53 tourists had visited both the Magic Kingdom and LEGOLAND
100 tourists had visited both the Magic Kingdom and Universal Studios
77 tourists had visited both LEGOLAND and Universal Studios
32 tourists had visited all three theme parks
79 tourists did not visit any of these theme parks
How many tourists only visited the Magic Kingdom (of these three)?
~~~~~~~~~~~~~~~~~~
In this problem, you are given a universal set V of all 1,174 tourists.
There are also 3 its basic subsets
L of 250 tourists visited LEGOLAND,
S of 268 tourists visited Universal Studios,
M of x tourists visited Magic Kingdom.
We also are given info about their in-pair intersections sets
ML of 53 tourists visited both the Magic Kingdom and LEGOLAND,
MS of 100 tourists visited both the Magic Kingdom and Universal Studios,
LS of 77 tourists visited both LEGOLAND and Universal Studios.
Finally, we are told that
- the triple intersection LUM has 32 tourists
and
- the union (L U S U M) has 1174 - 79 = 1095 tourists.
Notice that the number x in the set M is unknown, and our goal is to find it.
As soon as we find it, we easy will find the answer to the problem's question (in one line).
To find x, use the Exclusion-Inclusion principle. It is this formula, which
connects the number of elements in subsets
n(L U S U M) = L + S + M - ML - MS - LS + LSM. (1)
+----------------------------------------------------+
| Notice the interior symmetry of this formula |
| which is its beauty |
+----------------------------------------------------+
Now substitute the given values into equation (1)
1095 = 250 + 268 + x - 53 - 100 - 77 + 32. (2)
It is an equation for a single unknown x.
Simplify the equation
1095 = x + (250 + 268 - 53 - 100 - 77 + 32)
1095 = x + 320
and find x
x = 1095 - 320 = 775.
Thus the number of tourists in subset M is 775.
These are those who visited Magic Kingdom.
But the problem asks about the number of those who visited Magic Kingdom, ONLY.
To get it, we should subtract from x=775 the numbers ML and MS and add the number LSM.
So, the ANSWER to the problem's question is
n(Magic Kingdom ONLY) = M - ML - MS + LSM = x - ML - MS + LSM = 775 - 53 - 100 + 32 = 654.
Solved.
From this post, you learn on HOW TO apply the Inclusion-Exclusion principle
to solve this problem (and thousand other similar problems).
----------------------
On inclusion-exclusion principle, see this Wikipedia article
https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
To see many other similar (and different) solved problems, look into the lessons
- Counting elements in sub-sets of a given finite set <<<---=== this lesson is a pre-requisite to start
other lessons contain similar solved problems
- Advanced problems on counting elements in sub-sets of a given finite set
- Challenging problems on counting elements in subsets of a given finite set
- Selected problems on counting elements in subsets of a given finite set
- Inclusion-Exclusion principle problems
in this site.
Happy learning (!)
/\/\/\/\/\/\/\/\/
The most natural method of solving this problem (and many other similar problems)
is Inclusion-Exclusion principle.
As I showed/explained to you in my post above, it is an equation, which
connects the numbers of elements in the set and its subsets.
The formulas and the principle itself are very beauty part of Math,
of elementary set theory and logic.
The major goal of this and similar problems is to teach students to these basics.
There are other ways to solve.
One of the other ways is to draw Venn diagram and analyze it step by step.
Probably, it is a good way to start learning this subject for middle school students.
But in my view, it is the same, or almost the same, as to transfer matches from
one matchbox to another. Probably, you will solve the problem to the end
in this way, but will learn nothing - or almost nothing.
Another way is to construct a system of linear equations in 8 equations and 8 unknowns,
but, again, this can hardly be considered as a reasonable way.
Only the way via Inclusion-Exclusion principle is the most natural mathematical method for such problems.
Doing this way, you learn everything what is needed to know to a Precalculus student
- basics of the elementary set theory and relevant logic - for this subject.
As a conclusion, think how will you solve similar problem if 4 basic subsets are given
with all auxiliary necessary information about their intersections?
Will you transfer matches from 16 matchboxes, from one to the other ?
Or will you construct and solve system of equations of the size 16 ?
Notice that Inclusion-Exclusion will give you the solution easy using the same scheme.
What if you are given a problem with 5 basic subsets ?
I have no more questions. Hope everything is as clear as a sunny day.
Answer by Edwin McCravy(20054)  (Show Source):
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