If each shook hands w/ all else, 300 total shakes, how many

Okay, I'm in Algebra 1 and we are working factoring and princple of zero. We have a chapter of word problems and I surprised myself by figuring all of them out, that is, except one. I have worked on this problem for over 3 hours and can't figure out how to begin. Maybe one of you can point me in the right direction. Here's the question:

Everyone in a meeting shook hands with every other person. There were 300 handshakes in all. How many people were in the room?

I have tried everything I could think of and to no avail. All I need is help setting it up. I can solve it from there. A similar problem was a breeze to do, so I don't know why I am having so much problem with this one. Please help before I'm ready for a straight jacket!

Thanks,

Re: This problem is driving me bonkers!

persons..... handshakes
....2............... 1
....3................2
....4...............6 [3+2+1]
....5...............10 [4+3+2+1]
etc
n persons handshakes = (n-1)+(n-2)+.....1
n persons handshakes= {[n+1]+1] [n-1]/2
n persons handshakes= n[n-1]/2
set it equal to 300
300=n[n-1]/2
n^2-n-600=0
n=1+/-[1+2400]^1/2 all over 2
n=1+/- 49 all over 2
n=25 or -24
n=25 answer

please check math
Arthur

Re: This problem is driving me bonkers!

wendys8861 said:

... I have worked on this problem for over 3 hours and can't figure out how to begin.

Click to expand...

Hello Wendy:

Your statement above must be a gross exaggeration. If you've spent three hours on this exercise, then obviously you've begun. (I would like to have seen some of your work and reasoning; then I would know what you've already tried.)

Did you try modeling with diagrams?

Draw a large circle on several sheets of paper. Imagine that this circle is a room.

Start with two people. Put two dots anywhere on the circle. These dots represent the two people. Connect them with a line. The line represents their handshake.

On the next sheet of paper, put three dots on the circle. Draw lines so that there is one handshake connecting each pair of people. These three lines represent the three handshakes.

Put four dots on the next circle. Connect each pair with a line. These six lines represent the six handshakes.

Put five dots on the next circle. Connect each pair with a line. There are ten lines total, so there are ten handshakes with five people.

These numbers form a sequence: 1, 3, 6, 10, ...

Continue until you recognize a pattern and are able to write a formula for generating the nth number in the sequence.

Set your formula equal to 300, and solve for n to answer your exercise.

Cheers,

~ Mark

Re: This problem is driving me bonkers!

Whoops, I forgot to check whether or not Arthur is online. If I had checked, then I would not have bothered to post anything, since Arthur generally completes the entire exercise.

~ Mark :|

Re: This problem is driving me bonkers!

Hello, wendys8861!

Another approach . . .

Everyone in a meeting shook hands with every other person.
There were 300 handshakes in all.
How many people were in the room?

Click to expand...

Let's say there are \(\displaystyle n\) people.
Call them: .\(\displaystyle A,B,C,\hdots, N\)

Select any person; there are \(\displaystyle n\) choices.

Then he can shake hands with any of the other people; there are \(\displaystyle n-1\) choices.

. . \(\displaystyle \text{It seems that there are }n(n-1)\text{ handshakes.}\)


\(\displaystyle \text{But this list contains }\;\begin{array}{ccc}AB\!:& A\text{ shakes hands with }B \\ BA\!: & B\text{ shakes hands with }A \end{array}\quad\hdots\quad\text{same handshake}\)

So the list is twice as long as it should be.

. . \(\displaystyle \text{The number of handshakes is: }\:\frac{n(n-1)}{2}\)


\(\displaystyle \text{We are told that there were 300 handshakes: }\;\frac{n(n-1)}{2} \:=\:300\)

\(\displaystyle \text{Multiply by 2: }\;n(n-1) \:=\:600 \quad\Rightarrow\quad n^2 - n - 600 \:=\:0\)

\(\displaystyle \text{Factor: }\;(n - 25)(n + 24) \:=\:0 \quad\Rightarrow\quad n \:=\:25,\;-24\)


Therefore, there were 25 people.

Re: This problem is driving me bonkers!

Another approach: google "number of handshakes" :idea: