How many people attended the party?

Question

A scientist wants to research the potential spread of germs by contact. She knows that the number of possible handshakes within a group of n people is given by the equation H=(1/2)(n^2-n). Everyone at a party shook hands. There were 66 handshakes in all. How many people attended the party?

Answer 1

H=(1/2)(n^2-n)

66 = (1/2)(n^2-n)

Multiply both sides by 2 

(2)66 = (2)(1/2)(n^2-n)

132 = n^2 - n

 

This is a quadratic equation.  We solve this be setting it equal to zero and factoring

Subtract 132 from both sides

0 = n^2 - n - 132

Factor

0 = (n - 12)(n + 11)

Set each factor equal to zero

n - 12 = 0

n = 12

or

n + 11 = 0

n = -11

Since you cannot have -11 people, we throw out this answer. 

 

12 people attended the party!