A math riddle, but also a question on semantics

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In a family, each son has as many brothers as sisters. However, each daughter has twice as many brothers as sisters. How many children are there altogether in the family?
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A friend had this as a question, apparently they expected the answer 7. Thing is, as I understand it who is to say there are any children? Also, what if there is one son, who has no siblings. Does the "each" imply in a purely mathematical sense that there isn't an empty set of brothers and sisters for a first born?
If you believe that there must be an answer which isn't 0 or 1 to be a valid one, could you give your formula for how to calculate the number of children for any positive number of siblings for the first born?

thanks alot Smile

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yes,

That is the result of an unique solution.

From the problem statement one would derive the following relationships.

Let b be the # of boys and g be the # of girls

part 1-'each son has as many brothers as sisters' then
b-1=g (you are not your own brother)

and part 2- 'each daughter has twice as many brothers as sisters' then
g-1=2b (nor can you be your own sister)

so now you have two independent linear equation in two unknowns

so by substitution, you can determine that b=4 and g=3 and the total number of siblings (b+g) has to be 7.

Rap

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well 4-1=3 is true, but 3-1=2*4 isn't, so it only works for one of the equations. How would you find a simple way to get all possible number of children in a linear equation? Besides, any idea if from the way the question is structured whether or not you could answer 0 (for there being no children) or 1 (if the child has no brothers or sisters)?

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sorry I mis-typed the second eqn

I meant

g-1=b/2

Rap

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0 or 1 are not answers to this question. If you look at it this way:

b - g = 1
.5b - g = -1

Forming the matrix

1 -1 1
.5 -1 -1

Reduces to

1 -1 1
0 1 3

Therefore there is ONLY 1 answer NOT zero. I assume you may not understand this, but to see why there really is just one answer linear algebra is the best way.

To see it visually, simply replace "g" and "b" with "x" and "y" and graph the two equations:

y - x = 1 or x = y - 1
and
.5y - x = -1 or x = (1/2)y + 1

Do this on graph paper and you'll notice that he intersection of the two lines is at the point (3,4) or 3 girls and 4 boys. These lines will never intersect anywhere else.

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A math riddle, but also a question on semantics

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