Question

In a party, boys shake hands with girls only but each girl shakes hands with everyone else. If there are total 40 handshakes, find the number (more than one) of boys and girls in the party.
can anyone ans this with description?

Answer 1

First we need to know the shake-hands algorithm:
X people shake hands with everyone else, then there are X(X-1)/2 handshakes.
Why, each person shakes hands with X-1 persons, that makes X(X-1) handshakes. But then we have counted twice, because we counted A shakes hands with B and B shakes hands with A, so twice. Therefore, for X people, there are X(X-1)/2 handshakes.

Next, all girls shake hands with boys, so that makes g*b handshakes.
All girls shake hands with girls, that makes g(g-1)/2 handshakes.
Question want 40 in total, so solve
g(g-1)/2+gb=40
How do we solve the problem, with two variables and one equation?
Not as hard as it seems, because we also know that g and b have to be positive integers.
One way to go about solving it is the brute force approach, by calculating x=g(g-1)/2+gb for g=1 to 10 and b=1 to 10.
The values of g and b that makes x=40 would be the solution.
Another way is to calculate, for each value of g, we TRY to solve for B as an integer.
For example,
g=1, then x=1(1-1)/2+1(b)=40, or b=40, that works, i.e. 40 boys, one girl!
But there MAY be other solutions, so keep trying!
g=2, then x=2(2-1)/2+2(b)=40, or b=19.5 nope.
....
Keep trying until you get at least one more solution!

Answer 2

yeah I have tried to ponder over this one but there is only one equation I can generate unfortunately.

Answer 3

thanks for your long answer mathmate

Answer 4

interestingly you should find 5 girls, 6 boys will suit the algorithm proposed.

Answer 5

you basically just need integer solutions G, B

Answer 6

It may be more helpful to rearrange your equation

\[b = \frac{ 40 }{ g } - \frac{ (g-1) }{ 2 }\]

This clearly shows your g should be a factor of 40 to start with.

Answer 7

... and g must be odd, so out of factors of 40={1,2,4,5,8,10,20,40}, there are only two that fit the bill!