# How many men, women, and children are there?

• Math100
I'm not sure what the problem is.I'm not sure what you are referring to. The problem is to find all solutions in positive integers, not just those that don't involve any fractional or negative people.f

#### Math100

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.

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##.

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##.

• Math100
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