# A company with 500 employees wants to have its group health insurance premiums lowered. Here is one question from the questionnaire that every employee filled out and returned to the HR department: What have you done in the past year to improve your health? (circle all that apply) A. I have eaten healthier. B. I have exercised regularly. When the results were tabulated, the HR clerk reported that 140 people circled response A, 290 circled response B, and 50 people circled both responses A and B. What percent of company employees circled neither A nor B?

May 19, 2015

Reading the question, it says "140 people circled response A" it does not state that they had only circled response A, that tells me they of those 140 people that circled response A, 50 of them had circled response B too...

thus $A \mathmr{and} \text{not} B = 140 - 50 = 90$
and $B \mathmr{and} \text{not} A = 290 - 50 = 240$
and $A \mathmr{and} B = 50$

Which we can use to calculate $A \text{or} B$

$\left(A \cup B\right) = \left(A \cap B '\right) + \left(B \cap A '\right) + \left(A n B\right)$

$\left(A \cup B\right) = 90 + 240 + 50 = 380$

now we know how many did circle either, and we can use that result to calculate whom circled neither. and here you must keep in mind that there are 500 employees in total.

$\text{not} \left(A \cup B\right) = 500 - 380 = 120$

Thus we can say that 120 people chose to not circle either A or B.