each man in a gang of 100 investors has investments ineither equities or securities or both. Exactly 2525 of the investors in equities haveinvestments in securities, exactly 4040 of the investorsin securities have investments inequities. How many have investmentsin equities?  Choose the correct choice from the following answers:	A) 70	B) 80	C) 90	D) 100	E) 120

Rationale: the investors can be categorized into three groups : ( 1 ) those who have investments in equities only . ( 2 ) those who have investments in securities only . ( 3 ) those who have investments in both equities and securities . let xx , yy , and zz denote the number of people in the respective categories . since the total number of investors is 110 , we have x + y + z = 110 x + y + z = 110 - - - - - - - - - - - - - ( 1 ) also , the number of people with investments in equities is x + zx + z and the number of people with investments in securities is y + zy + z . since exactly 25 % 25 % of the investors in equities have investments in securities , we have the equation 25100 × ( x + z ) = z 25100 × ( x + z ) = z 25100 × x = 25100 × z 25100 × x = 25100 × z x = 3 zx = 3 z - - - - - - - - - - - - - - - - - - - ( 2 ) since exactly 40 % 40 % of the investors in securities have investments in equities , we have the equation 40100 × ( y + z ) = z 40100 × ( y + z ) = z ( y + z ) = 5 z 2 ( y + z ) = 5 z 2 y = 3 z 2 y = 3 z 2 substituting equations ( 2 ) and ( 3 ) into equation ( 1 ) yields 3 z + 3 z 2 + z = 1103 z + 3 z 2 + z = 110 11 z 2 = 11011 z 2 = 110 z = 110 × 211 = 20 z = 110 × 211 = 20 hence , the number of people with investments in equities is : x + z = 3 z + z = 3 × 20 + 20 = 60 + 20 x + z = 3 z + z = 3 × 20 + 20 = 60 + 20 = 80 b
Answer: B