Another Set Counting Question

• Jan 4th 2009, 02:00 AM
barc0de
Another Set Counting Question
This question is taken from a book. I have posted it because I would
like to understand how to complete it to do a similar problem for my homework.

In a survey of 1000 households, washing machines, vacuum
cleaners and refrigerators were counted. Each house had at least
one of these appliances.

542 no washing machine
294 had both a vacuum cleaner and washing machine
277 both a refrigerator and a vacuum cleaner
190 both a refrigerator and a washing machine.

How many households had all three appliances?

How many had only a vacuum cleaner?

Thanks for any guidance.
• Jan 4th 2009, 02:22 AM
mr fantastic
Quote:

Originally Posted by barc0de
This question is taken from a book. I have posted it because I would
like to understand how to complete it to do a similar problem for my homework.

In a survey of 1000 households, washing machines, vacuum
cleaners and refrigerators were counted. Each house had at least
one of these appliances.

542 no washing machine
294 had both a vacuum cleaner and washing machine
277 both a refrigerator and a vacuum cleaner
190 both a refrigerator and a washing machine.

How many households had all three appliances?

How many had only a vacuum cleaner?

Thanks for any guidance.

Draw a Venn diagram.

Put a pronumeral a, b, c, d, e, f, g in each part.

You can figure out where I did this to get:

a + b + c = 400 .... (1)

a + e + g = 380 .... (2)

c + f + g = 542 .... (3)

b + d = 294 .... (4)

d + f = 277 .... (5)

d + e = 190 .... (6)

a + b + c + d + e + f + g = 1000 .... (7)

Solve equations (1), (2), (3) ...... (7) simultaneously for a, b, c, .... g (not too tough).

d = 83.

There are probably easier ways but this is how I generally do it.
• Jan 4th 2009, 04:04 AM
barc0de
Is there an easier way?
• Jan 4th 2009, 04:19 AM
Plato
Quote:

Originally Posted by barc0de
Is there an easier way?

No. That way is very easy.
• Jan 4th 2009, 04:40 AM
Dream is not Utopia
An easier way
There is a quite easy way to do it, this question belongs to the set category.
<br><br>Note the following identity, if A, B, C are sets, then
n(A∪B∪C)=n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)
<br><br>in this question, let people having washing machine be set A, vacuum B, refrigerator C. so<br>n(A)=1000-542=458
<br>n(B)=1000-380=620
<br>n(C)=1000-400=600
<br>n(A∩B)=294<br>n(B∩C)=277<br>n(A∩C)=190
<br>n(A∪B∪C)=1000&nbsp; <br><br>n(A∩B∩C)=1000-(458+620+600)+(294+277+190)=83

<br><br>So next time you can just sub in the values into the equation for same kind of problem and it wont take too much time.
• Jan 4th 2009, 07:07 AM
barc0de
How would I then determine how many have a vacuum only?

I would expect n(B) - n(A n B) - n(B n C) - n(A n B n C) to give the correct result.

I get 620 - 294 - 277 - 83 = -34

Where am I going wrong?
• Jan 4th 2009, 09:51 AM
barc0de
I believe 83 to be correct but I cannot understand why the rest doesn't tally up.

Can anyone offer any guidance please?
• Jan 4th 2009, 11:21 AM
vg284
the best way to solve this problem is to use Ven diagram
• Jan 4th 2009, 11:28 AM
barc0de
I have done. But since the figures I have posted are the ones on the diagram it is no help
• Jan 4th 2009, 01:43 PM
mr fantastic
Quote:

Originally Posted by barc0de
I have done. But since the figures I have posted are the ones on the diagram it is no help

a = 57, b = 211, c = 132, d = 83, e = 107, f = 194, g = 216.
• Jan 4th 2009, 08:42 PM
Dream is not Utopia
Quote:

Originally Posted by barc0de
How would I then determine how many have a vacuum only?

I would expect n(B) - n(A n B) - n(B n C) - n(A n B n C) to give the correct result.

I get 620 - 294 - 277 - 83 = -34

Where am I going wrong?

you got wrong in the last step, coz following the above identity, it should be
n(B) - n(A n B) - n(B n C)+ n(A n B n C)=620-294-277+83=132

you can interpret it this way: as you have eliminated people who have both a vacuum and a washing machine+people who have both a vacuum and a refrigerator, you have eliminated twice people who have all the three appliances, so you should add back instead of eliminating it.