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### A Mule and a Donkey

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Date: 04/08/99 at 00:45:38
From: Justin Rittgasser
Subject: Euclid, "A mule and a donkey"

I am in Grade 5 and I am doing research on Euclid. I came across a
puzzle posted on another site.  It is supposed to be a famous puzzle
attributed to Euclid. It goes: A mule and a donkey were talking. The
mule said, "I'm carrying more than you. In fact, if you gave me one of
your sacks, then I would have twice as many as you. If I gave you a
sack, our loads would be equal." How many sacks was each animal
carrying?

I am struggling a bit in math, but I like to do these brainteasers.
puzzle? Is it solveable?

Thanks for your time and help.

Justin
(typed by my sister, Nicole)
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Date: 04/08/99 at 12:11:15
From: Doctor Peterson
Subject: Re: Euclid, "A mule and a donkey"

Hi, Justin.

This problem is pretty easy to do using algebra; but for you and
Euclid, neither of whom probably know algebra, it's a brain teaser!

Here's how I'd do it with algebra, if you can follow it:

Let's call the number of sacks on the mule and the donkey M and D
respectively. Then the two statements mean this:

If you gave me one of your sacks, then I would have twice as many
as you. (That is, if the mule had one more and the donkey one less,
the mule would have twice as many.)
M + 1 = 2 * (D - 1)

If I gave you a sack, our loads would be equal.
(That is, if the mule had one less and the donkey one more, they
would have the same amount.)
M - 1 = D + 1

From the second statement, I know that the mule has 2 more than the
donkey:

M = D + 2

I can put this fact into the first statement and rewrite it using
rules of algebra:

(D + 2) + 1 = 2 * (D - 1)

D + 3 = 2 * D - 2
D + 3 - D = 2 * D - 2 - D
3 = D - 2
5 = D

So the donkey has 5 sacks. The mule has 2 more, or 7.

Check this out: if the donkey gave one to the mule, they would have 4
and 8; if the mule gave one to the donkey, they would both have 6.

The ancient Greeks and Egyptians spent a lot of time on problems like
this, because they didn't have any easy way to work with them. In a
few years you'll be able to do things Euclid had trouble with!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Middle School Algebra
Middle School Equations
Middle School History/Biography

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