not sure how to summarize it...

Question

A man runs a theater where he charges 5 cents for men, 2 cents for women, and .1 cents for a child (10 for a penny). At the end of the evening, 100 people people attended and the till came to $1.00--you can tell how old my friend's father is.

What's the breakdown of men, women, and children?

Answer 1

Actually, my first thought was Donald's.  But when I went to give some examples of multiple solutions, I realized there really weren't any.  I kept getting negative numbers for one of the variables.  Then I realized this could be solved with reasoning.  The children MUST come in groups of 10, right?  So assuming 10 children, you can solve the other two equations...but you end up with 3M=negative, which can't be.  This continued to be the case up through 50 children.  But look what happens with 60 children:

 

5M+2W=94

M+W=40

2M+2W=80 (multiply second equation by 2)

3M=14

M=4.67 (can't have non integer numbers)

 

However, we are getting closer.  Continue these same steps for increasing groups of 10 children.  When you get to 70 children, you are left with:

 

5M+2W=93

2M+2W=60

3M=33

M=11, a nice integer!

 

With M=11, 2W=60-22=38, so W=19, and of course C=70.  This combination works.  

 

Men = 11

Women = 19

Children = 70

 

If we continue increasing children to 80 and 90, we get non integer numbers again, and of course if we use 100 children we are over our limit, so this is the only solution.

 

 

 

Answer 2

So, 100 cents = 5m + 2w + .1c

m+w+c = 100

You need one more constraint to solve this problem. You have 3 variables and 2 equations. To solve a set of simultaneous equations, you need 1 equation per unknown.