Math CentralQuandaries & Queries


My name is Robby and I need help with a problem.

I have found at least 4 correct answers, but I can't figure out how I got the answer or if there is a formula that I can use. I got my answers by trial and error.

Problem: One hundred bushels of corn are to be divided among 100 men, women, and children. Men get 3 bushels each. Women get two bushels each. Children get 1/2 bushel each. How can the bushels be distributed with no leftovers? Is there more than one solution?

So far, I have 4 answers: 8 men,20 women, and 72 children,
2 men, 30 women, and 68 children, 5 men, 25 women, and 70 children, and 11 men, l5 women, and 74 children.

I need to know if there is a formula to use, or if it's just trial and error, or if a pattern is involved. Help! I'm stumped!

Robby (7th grade)

Hi Robby. This is a kind of hard problem for 7th grade - good for you for trying to figure it out!

Let's start with some algebra:
Let m = the number of men, w = the number of women and c = the number of children.

Then m + w + c = 100, because there are 100 people in all.

Also, 3m + 2w + (1/2)c = 100, because we want the 100 bushels to divide evenly according to the proportions given.

We can re-write these both. The first one can be written like this:

100 - m - w = c

and the second can be written:

100 - 3m - 2w = (1/2)c.

Double the last line and you get:

200 - 6m - 4w = c.

Notice that we have two equations ending in "= c". Now, remember that two things that both equal a third thing have to equal each other, so that means:

100 - m - w = 200 - 6m - 4w

and this simplifies to

3w = 100 - 5m.

There is a lot of information in this equation. First of all the right side is divisible by 5 so the left side is divisible by 5. Since 3 is not divisible by 5, w must be divisible by 5. Hence the only possibilities for w are 0, 5, 10, ..., 100.

But w certainly can't be 100 since 3 times w would be 300. So how large can w be? If w = 35 then 3 times w = 105 which is too large so w can only be 0, 5, 10, 15, 20, 25 or 30. That's only 7 cases so you can try them all.

For each possible w find m using 3w = 100 - 5m and then c using m + w + c = 100 and check to see if
3m + 2w + (1/2)c = 100

Hope this helps,
Stephen and Penny.

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