Probelm Involving Sets

[bretania]

I'd like to hear what you think is the right answer for this one:

In a marketing survey for products A, B, and C, 100 people were asked which of the products, if any, they use. Of the people surveyed, a total of 55 use A, a total of 40 use B, and a total of 45 use C. If all of the people surveyed use at least one of those products and 30 percent of the people surveyed use exactly two of those products, how many of the people use only one of these products ?

A.50
B.55
C.60
D.65
E.70

Comments: My answer is C but another friend says it is D.

[clock60]

let it be
a -who use A only
b-who use B only
c-who use C only
x-who use A+C only
y-who use A+B only
z-who use C+B only
w-who use all three A+B+C
now we have equation a+b+c+x+y+z+w=100 according the condition of the task.then 30% of the surveyed use exactly two of the products
it means that x+y+z=30 (30%from100)
and it follows that a+b+c+w=70 (100-(x+y+z)) 100-30=70
accrding to the task total use of A is 55 then
a+x+y+w=55
total use of B is 40 then b+y+z+w=40
total use of c is 45 then c+x+z+w=45
let us write all equation together below
a+x+y+w=55 (1)
b+y+z+w=40 (2)
c+x+z+w=45 (3)
(1)+(2)+(3)= a+b+c+2x+2y+2z+3w=140 but x+y+z=30 and as a result
a+b+c+3w=80 (4) according our mentioned above
a+b+c+w=70 (5) substract (4)-(5)= 2w=10 w =5
put w=5 in(5) a+b+c=70-w a+b+c=65 so the answer is D

[bretania]

Wow! Clock60, Thanks so much. This solution is so comprehensive.
I'm convinced now that it's 65.

I'm just a bit concerned if I could do all these calculations in the actual test if a similar prob comes up. this will take me prob 4-5 mins (conservative estimate).