Question 1201973: In survey of 900 workers 600 owned houses,500 owned cars and 345 owned boat,300 owned cars and houses,250 owned houses and boats and 270 owned cars and boats 200 owned the three.find how many workers don't owned any of items, find how many workers owned only two of the items
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Set H = houses
Set C = cars
Set B = boats
Let's draw out a venn diagram
The uppercase letters represent the circle names/labels
The lowercase letters are the 8 regions I'll talk about down below.
There are 8 distinct regions- a = houses only
- b = houses and cars, but no boats
- c = cars only
- d = houses and boats, but no cars
- e = all three
- f = cars and boats, but no houses
- g = boats only
- h = none of the three items mentioned
Example: If someone owns a house and a boat, then they are in region d.
We have these given facts
| Number | Statement | | Fact 1 | Survey of 900 workers | | Fact 2 | 600 owned houses | | Fact 3 | 500 owned cars | | Fact 4 | 345 owned boat | | Fact 5 | 300 owned cars and houses | | Fact 6 | 250 owned houses and boats | | Fact 7 | 270 owned cars and boats | | Fact 8 | 200 owned the three. |
Fact 8 is where we'll start.
This tells us that 200 goes in region e at the very center.
It's where all three circles overlap.
We'll then use Fact 7 and Fact 8 to determine there must be 270-200 = 70 people who own a car and a boat, but not a house.
The value 70 goes inside region f.
Next use Fact 6 and Fact 8 together to determine there are 250-200 = 50 people who own a house and a boat, but no car.
The value 50 goes in region d.
Next up will be Fact 5 and Fact 8 used together. We can say there are 300-200 = 100 people who own a house and a car, but no boat.
The value 100 goes in region b.
Here's what we have so far
b = 100
d = 50
e = 200
f = 70
Focus your attention on circle H.
The values in this circle must add to 600 (due to Fact 2)
That means...
a+b+d+e = 600
a+100+50+200 = 600
a+350 = 600
a = 600-350
a = 250
There are 250 people who own a house, but not a car and also not a boat.
Now move your attention to circle C.
We use Fact 3 to say the following:
b+c+e+f = 500
100+c+200+70 = 500
c+370 = 500
c = 500-370
c = 130
Use a similar idea for circle B. Refer to Fact 4
d+e+f+g = 345
50+200+70+g = 345
320+g = 345
g = 345-320
g = 25
We have these values
a = 250, b = 100, c = 130
d = 50, e = 200, f = 70
g = 25
which leads to this nearly completed venn diagram
The last thing to figure out is what we put in region h.
Let's add up all of the values in the circles ('a' through 'g')
a+b+c+d+e+f+g = 250+100+130+50+200+70+25 = 825
This represents the total number of people who have at least one (i.e. one or more) item mentioned.
Then we use Fact 1 to determine how many people have none of those three items
900-825 = 75
There are 75 people who own neither a house, nor a car, nor a boat.
This value goes in region h.
Here's the completed venn diagram
where,
a = 250, b = 100, c = 130
d = 50, e = 200, f = 70
g = 25, h = 75
Now onto the questions:
Question 1) how many workers don't own any of those items?
Answer: 75
Reason: See region h of the venn diagram
Question 2) how many workers own only two of the items?
Answer: 220
Reason: b+d+f = 100+50+70 = 220
Ignore region e because that subset of people own all 3 items.
Answer by ikleyn(52776)  (Show Source):
You can put this solution on YOUR website! .
In survey of 900 workers 600 owned houses, 500 owned cars and 345 owned  boats,
300 owned cars and houses,250 owned houses and boats and 270 owned cars and boats.
200 owned the three.
(a) Find how many workers don't owned any of items.
(b) Find how many workers owned only two of the items.
~~~~~~~~~~~~~~~~~
(a) You are given a universal set of 900 workers and three its basic subsets
- H (owned houses) of 600 workers;
- C (owned cars) of 500 workers;
- B (owned boats) of 345 workers.
You also are given the info about in-pair intersections of these three subsets
- CH (owned cars and houses) of 300 workers;
- HB (owned houses and boats) of 250 workers;
- CB (owned cars and boats) of 270 workers.
You also are given the info about the triple intersections of these three subsets
- CHB (owned all three items) of 200 workers.
Use the Inclusion-Exlusion principle. The Inclusion-Exclusion formula is
n(H U C U B) = n(H) + n(C) + n(B) - n(HC) - n(HB) - n(CB) + n (CHB) =
= 600 + 500 + 345 - 300 - 250 - 270 + 200 = 825.
Thus 825 workers own at least one of three items.
Hence, the rest 1000-825 = 175 workers don't have any of items. ANSWER
(b) To find the number of workers who own only two of the items, subtract triple intersection from
the corresponding in-pair intersection.
You will get:
n(CH_only) = n(CH) - n(CHB) = 300 - 200 = 100 (owned cars and houses only);
n(HB_only) = n(HB) - n(CHB) = 250 - 200 = 50 (owned houses and boats only);
n(CB_only) = n(CB) - n(CHB) = 270 - 200 = 70 (owned cars and boats only).
These sets CH_only, HB_only and CB_only are disjoint (i.e. have empty intersection; THEREFORE
the number of workers owned only two of the items is the sum 100 + 50 + 70 = 220. ANSWER
Solved.
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To see many other similar (and different) solved problems on Inclusion-Exclusion principle, see the lessons
- Counting elements in sub-sets of a given finite set
- 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.
On Inclusion-Exclusion principle, see this Wikipedia article
https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
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