SOLUTION: I am stuck on this problem can any one show me how to do this especially the VENNGRAM In a survey of 160 dancers, 98 knew the Salsa, 120 knew the Merengue, and 86 knew the Cha C

Question 293564: I am stuck on this problem can any one show me how to do this especially the VENNGRAM
In a survey of 160 dancers, 98 knew the Salsa, 120 knew the Merengue, and 86 knew the Cha Cha. Of these, 77 knew the Merengue and Salsa dances, 51 knew the Salsa and Cha Cha, and 70 knew the Merengue and Cha Cha. Forty-seven dancers knew all three dances.

(a) How many knew only the Salsa?
(b) How many knew only the Merengue?
(c) How many knew exactly one of the three dances?
(d) How many knew the Merengue and Cha Cha, but not the Salsa?
(e) How many knew none of these dances?

Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
the "VENNGRAM" is three intersecting circles, surrounded by a box

the box represents the 160 dancers surveyed
___ the circles represent the Salsa, Merengue, and Cha Cha

start on the INNERMOST parts of the circles, and work your way out

47 dancers knew all three, so this number goes in the INTERSECTION OF ALL THREE CIRCLES

70 knew the Merengue and Cha Cha, so this number goes in the intersection of the M and CC circles
___BUT ___ there are already 47 in the intersection from the people who knew all three
___ so 23 (70 - 47) are added to the M and CC intersection [question(d)]

similarly for S and CC ___ 4 people added
___ and M and S ___ 30 people added

86 knew the Cha Cha ___ BUT ___ 47 knew all three; 23 knew M and CC; and 4 knew S and CC
___ so 12 (86 - 47 - 23 - 4) are added to the CC circle [part of question(c)]

similarly for Salsa ___ 17 added [question(a)] and [part of question(c)]
___ and Merengue ___ 20 added [question(b)] and [part of question(c)]

(e) 160 dancers MINUS everyone in the three circles
___ 160 - 47 - 23 - 4 - 30 - 12 - 17 - 20 = ?
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