Set Theory

[prasantthakur01]

There is a shortage of tubelights, bulbs and fans in a village. It is found that
"¢ All houses do not have either tubelight or bulb or fan.
"¢ exactly 19% of houses do not have just one of these.
"¢ atleast 67% of houses do not have tubelights.
"¢ atleast 83% of houses do not have bulbs.
"¢ atleast 73% of houses do not have fans.
What percentage of houses have
all three items tubelight, bulb and fan missing?
(ii)exactly 2 items missing?

[winniethepooh]

IMO - The answer is (i) = 42 and (ii)=39.

Let there be 100 houses.
So there must be in all at least 300 items to fulfill 100% needs of t-lite(t), bulb(b) and fan(f).

Shortage = 67+83+73 = 223 items.

Also, exactly 19 % houses do not have 1 item.
So 223- 19 = 204 account for houses having none of these things and houses having only 1 of these things.

Lets deduct people not having at least 2 of these things = 100 - 19 = 81.
204 - [81(2)]= 42.(even after deducting people missing at least 2 items 42 remains.
Therefore, 42 houses don't have any of the items {Answer for (i)}


Out of 100 houses :

19 don't have exactly 1 item.
42 don't have any of the items
So, 100-61 = 39 houses miss 2 items.{Answer for (ii)}

[hagan]

IMO - The answer is (i) = 42 and (ii)=39.

Let there be 100 houses.
So there must be in all at least 300 items to fulfill 100% needs of t-lite(t), bulb(b) and fan(f).

Shortage = 67+83+73 = 223 items.

Also, exactly 19 % houses do not have 1 item.
So 223- 19 = 204 account for houses having none of these things and houses having only 1 of these things.

Lets deduct people not having at least 2 of these things = 100 - 19 = 81.
204 - [81(2)]= 42.(even after deducting people missing at least 2 items 42 remains.
Therefore, 42 houses don't have any of the items {Answer for (i)}


Out of 100 houses :

19 don't have exactly 1 item.
42 don't have any of the items
So, 100-61 = 39 houses miss 2 items.{Answer for (ii)}

a more step by step explanation will be very helpful, am not getting it

[winniethepooh]

If you indicate which part you ain't getting, it'll be easier for me to explain!

But I don't think you need to bother, this isn't going to be asked in Gmat for sure. Nothing of this sort is added in our portion, this is more of logical reasoning.

[knight247]

Refer to this post for all direct formulas that can be used for solving venn diagram problems

https://www.beatthegmat.com/formulas-for ... 16479.html


Let the total number of households be 100
All houses do not have either tubelight or bulb or fan.
So we have,
19 houses do not have just one of these.
67houses do not have tubelights.
83houses do not have bulbs.
73houses do not have fans.


(i) Number of ppl lacing all three utilities

Lets use our regular formula

P(TuBuF) = P(T) + P(B) + P(F) - {P(TnB) + P(BnF) + P(FnT)} + P(TnBnF)

Let P(TnB) + P(BnF) + P(FnT) = X

We Get,

P(TuBuF) = P(T) + P(B) + P(F) - {X} + P(TnBnF) ....(1)

keep that aside for now.

Now, it is given that 19 houses do no have just one of these. Which means we are looking at the higlighted area of the venn diagram i.e. the number of ppl in exactly one set. Understand this part and the rest is just formula substitution.

No of persons in exactly one set = P(T) + P(B) + P(F) - 2P(TnB) - 2P(BnF) - 2P(FnT) + 3P(TnBnF)

(since P(TnB) + P(BnF) + P(FnT) = X)

No of persons in exactly one set = P(T) + P(B) + P(F) - 2{X} + 3P(TnBnF)

19=223-2{X}+3P(TnBnF)

2{X}=204+3P(TnBnF)

{X}={204+3P(TnBnF)}/2

Substitute this value of X in (1)

100=223-{204+3P(TnBnF)}/2+P(TnBnF)

-123={-204-3P(TnBnF)}/2+P(TnBnF)

-246=-204-3P(TnBnF)+2P(TnBnF)

-246=-204-P(TnBnF)

P(TnBnF)=42=Number of ppl lacking all three utilities

Substituting this value back in in statement (1)

100=223-{X}+42
{X}=123+42=165



(ii)Number of ppl lacking exactly two utilities

No of persons in exactly two of the sets = P(TnB) + P(BnF) + P(FnT) - 3P(TnBnF)={X}-3P(TnBnF)
=165-126=39

Refer to the link above for all set related formulas. Its just simple substitution.