A Mule and a Donkey
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. Can you guide me? Is there a place to find out more about this puzzle? Is it solveable? Thanks for your time and help. Justin (typed by my sister, Nicole)
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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.