OG 11- diagnostic test - problem 6

[Pugalenthi]

A marketing firm determined that, of 200 households surveyed, 80 used neither brand A nor brand B soap, 60 used only brand A soap, and for every household that used both brands of soap, 3 used only brand B soap. How many of the 200 households surveyed used both brands of soap?
(a) 15
(b) 20
(c) 30
(d) 40
(e) 45

Here is my logic (applying the venn diagram logic)...
P(A u B) = p(A) + p(B) - p(A n B) + neither p(A n B)
200 = 60 +3x -x +80
2x = 60
X = 30
But the answer given in the book is A. What am I missing? Any help would be much appreciated.

[aneesh.kg]

You missed the word 'only'.

Notice: '60 used only brand A soap' and '3 used only brand B soap'

Calculate
n(A) = n(only A) + n(A and B) = 60 + x
n(B) = n(only A) + n(A and B) = 3x + x
and then plug these in the formula that you used.

n(A OR B) = 200 = (60 + x) + (3x + x) - (x) + 80
x = 15

Alternate Method:
When the data is given in terms of 'only', you can use this formula:
n(A OR B) = n(Only A) + n(only B) + n(A AND B) + n(neither A OR B)
200 = 60 + 3x + x + 80
x = 15

P.S.: If you're not sure where both the formulae come from, you can make a Venn-Diagram with two groups and figure out.

[transfer9858]

I don't understand the formulas- I think they are very confusing. I have been doing the "3 line by 3 line" method- and I don't think I have gotten a ven diagram question wrong since then. With this method- this question took me 20 seconds- and I am not quick at all. Try it out- I attached a pic of it-

So all you need to do is solve 4x=60 which gives you 15. Hope it helps.