How many men, women, and children are there?

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Homework Statement
Solve the puzzle-problem below:
Alcuin of York, 775. One hundred bushels of grain are distributed among 100 persons in such a way that each man receives 3 bushels, each woman 2 bushels, and each child 1/2 bushel. How many men, women, and children are there?
Relevant Equations
None.
Proof: Let x be the number of men, y be the number of women
and z be the number of children.
Then we have 3x+2y+0.5z=100
such that x+y+z=100.
Note that x+y+z=100 gives us z=100-x-y.
Substituting this result into 3x+2y+0.5z=100 and multiplying it
by 2 produces: 5x+3y=100.
Consider the Diophantine equation 5x+3y=100.
Applying the Euclidean Algorithm produces:
5=1(3)+2
3=1(2)+1
2=2(1)+0.
Now we have gcd(5, 3)=1.
Note that 1##\mid##100.
Since 1##\mid##100, it follows that the Diophantine equation
5x+3y=100 can be solved.
Then we have 1=3-1(2)
=3-1(5-3)
=2(3)-1(5).
This means 100=100[2(3)-1(5)]
=200(3)-100(5).
Thus, xo=200 and yo=-100.
All solutions in the integers are determined by:
x=200+(3/1)t=200+3t for some integer t,
y=-100-(5/1)t=-100-5t for some integer t.
Therefore, x=200+3t and y=-100-5t.
To find all solutions in the positive integers of the
Diophantine equation 5x+3y=100, we solve for t:
x=200+3t##\geq##0 ##\land## y=-100-5t##\geq##0
t##\geq##-200/3 ##\land## t##\leq##-20.

Now I'm stuck. I know I need to combine these two inequalities and solve for integers t in order to find the solution. But it seems like after I solve the inequality, I found out that the integers t are negative values. Can anyone please find where my mistake(s) are in this problem? Thank you.
 
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Math100 said:
I'm sorry for this, I totally forgot that I've already posted this problem before, but may I know how you got the Diophantine equation 3x+2y=100? Because as you can observe from above, I've got 5x+3y=100.
That tells you something important about ##y##.
 
Math100 said:
but may I know how you got the Diophantine equation 3x+2y=100?
That was a typo. It's ##5x + 3y = 100##. Or, ##3x + 2y + 0.5z = 100##.

In any case, ##5x + 3y = 100## tells you something about ##y##.
 
From the other thread
PeroK said:
There are lots of solutions.

Good, I thought I was missing something here. I get seven different solutions assuming no fractional or negative people.
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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