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 N=1/2(n^2-n). Everyone at a party shook hands. There were 253 handshakes in all. 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 N=1/2(n^2-n). Everyone at a party shook hands. There were 253 handshakes in all. How many people attended the party?

Answer 1

N=1/2(253^2-253)

N = 1/2(64009-253)
N = 1/2(63756)
N =31878

Answer 2

To find out how many people attended the party, we need to solve the equation N=1/2(n^2-n) given that there were 253 handshakes.

First, let's rewrite the equation as 1/2(n^2-n) = 253.

Next, we can simplify the equation by multiplying both sides by 2 to get rid of the 1/2 fraction: n^2-n = 506.

To solve this quadratic equation, we can rearrange it as n^2 - n - 506 = 0.

Now, we can solve this quadratic equation for n by factoring it or using the quadratic formula.

Let's use factoring method: We need to find two numbers that multiply to -506 and add up to -1.

After some trial and error, we find that the numbers are -23 and 22.

So, the equation can be rewritten as (n-23)(n+22) = 0.

Therefore, n-23 = 0 or n+22 = 0.

By solving these, we find that n = 23 or n = -22.

Since the number of people cannot be negative, the correct value for n is 23.

Therefore, 23 people attended the party.

Answer 3

To find the number of people who attended the party, we need to solve the equation N=1/2(n^2-n), where N represents the total number of handshakes (253 in this case).

Let's substitute N=253 into the equation:

253 = 1/2(n^2 - n)

To simplify the equation, we multiply both sides by 2:

2 * 253 = n^2 - n

506 = n^2 - n

Rearranging the equation to set it equal to zero, we have:

n^2 - n - 506 = 0

Now we can solve this quadratic equation by factoring or using the quadratic formula.

Factoring the equation gives:

(n - 23)(n + 22) = 0

Setting each factor equal to 0, we have two possible solutions:

n - 23 = 0, which leads to n = 23

n + 22 = 0, which leads to n = -22

Since we are looking for a positive value for the number of people, we discard the negative solution (-22).

Therefore, the number of people who attended the party is 23.