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

• Math100
In summary: 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.
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.

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##.

Math100
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.

Math100 and PeroK

## 1. How is the population of men, women, and children distributed globally?

The distribution of men, women, and children varies greatly depending on the region and country. However, globally, there are slightly more men than women, with an estimated 102 men for every 100 women. The population of children (under 18 years old) makes up around 26% of the world's population.

## 2. Are there more men or women in the world?

As mentioned before, globally, there are slightly more men than women. This is due to a number of factors, including higher mortality rates for men, especially in certain regions and age groups.

## 3. What is the ratio of men to women in different age groups?

The ratio of men to women varies across different age groups. In the younger age groups (under 15), there are usually slightly more boys than girls. However, in the older age groups (65 and above), there are usually more women than men due to the higher life expectancy of women.

## 4. How has the population of men, women, and children changed over time?

The population of men, women, and children has changed over time, with significant increases in the total population and changes in the distribution of these groups. For example, the global population of children has increased significantly in the past few decades, while the population of men has remained relatively stable.

## 5. What are the main factors influencing the population of men, women, and children?

There are many factors that influence the population of men, women, and children, including birth rates, mortality rates, migration, and societal and cultural norms. These factors can vary greatly between regions and can also change over time, leading to shifts in the population of these groups.

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