The Pharaoh's Will
Date: 05/16/2000 at 09:26:53 From: George Balalis Subject: The Pharaoh's Will As he lay dying, the first Pharaoh of Ufractia proclaimed: "I bequeath to my oldest child one third of my estate; to the next oldest child one quarter of my estate; to the next oldest child one fifth of my estate; and to each succeeding child, except the youngest, the next unit fraction of my estate; and to the youngest the remainder." When the pharaoh died and his estate was divided, the youngest child received the smallest share, which was worth 27,000 gold ingots. A) What was the value of the oldest child's share? The pharaoh's successor was so impressed with this method that he proclaimed: "In future, a Pharaoh's estate will be divided according to these rules: 1) Each surviving child, except the youngest, will be bequeathed a unit fraction of the estate. 2) The oldest child will be bequeathed the biggest fraction. 3) If a child is bequeathed 1/n of the estate, the next oldest child, except the youngest, will be bequeathed 1/(n+1) of the estate. 4) The remainder is to be at most the next unit fraction and is to be bequeathed to the youngest child." B) The second pharaoh had eight children living when he died. What fraction of his estate was bequeathed to the oldest child? C) When the third Pharaoh died, he had more surviving children than the first Pharaoh did, but less than eleven surviving children. It was found that the rules could not be followed precisely. How many children might have survived the third Pharaoh? I would like, if at all possible, to see your answer as soon as possible, as it would help me greatly in my efforts. Thank you.
Date: 05/17/2000 at 11:48:12 From: Doctor TWE Subject: Re: The Pharaoh's Will Hi George - thanks for writing to Dr. Math. This is an interesting problem. Let's take a look at what we know. The first Pharaoh of Ufractia divided his estate as follows: Child Fraction ----- -------- 1st 1/3 2nd 1/4 3rd 1/5 : : Let's work out each child's share, beginning with the oldest. The oldest child received 1/3 of the estate, leaving: 1 - 1/3 = 3/3 - 1/3 = 2/3 So 2/3 of the estate is left for the remaining children. Our information said, "to the youngest the remainder" and "the youngest child received the smallest share," so there must be at least two more children. If there were only one more child, (s)he would receive 2/3 of the estate, which is more than the oldest child received. Child Fraction Remaining ----- -------- --------- 1st 1/3 2/3 : : : The next oldest child received 1/4 of the estate, leaving: 2/3 - 1/4 = 8/12 - 3/12 = 5/12 So 5/12 of the estate is left for the remaining children. Since the 5/12 remaining is more than the 1/4 (= 3/12) that the second-oldest child received, there must be at least two more children. Child Fraction Remaining ----- -------- --------- 1st 1/3 2/3 2nd 1/4 5/12 : : : The third oldest child received 1/5 of the estate, leaving: 5/12 - 1/5 = 25/60 - 12/60 = 13/60 So 13/60 of the estate is left for the remaining children. Since the 13/60 remaining is more than the 1/5 (= 12/60) that the third-oldest child received, there must be at least two more children. Child Fraction Remaining ----- -------- --------- 1st 1/3 2/3 2nd 1/4 5/12 3rd 1/5 13/60 : : : The fourth oldest child received 1/6 of the estate, leaving: 13/60 - 1/6 = 13/60 - 10/60 = 3/60 (= 1/20) So 1/20 of the estate is left for the remaining children. That is less than the 1/6 that the fourth-oldest child received, so there could possibly be only one more child. Child Fraction Remaining ----- -------- --------- 1st 1/3 2/3 2nd 1/4 5/12 3rd 1/5 13/60 4th 1/6 1/20 : : : Could there be two more children? For the Pharaoh's pattern to continue, the 5th child should receive 1/7 of the estate, but only 1/20 (< 1/7) remains, so we can't continue and we can conclude that the Pharaoh had five children. The youngest would receive the remaining share, which is 1/20 of the estate. From this, and the fact that his share was worth 27,000 gold ingots, can you figure out what the original estate was worth? Follow a similar method to find the answers to parts B and C. You'll have to try other fractions for the oldest child's share, because we saw that 1/3 only worked for five kids. See if you can find a way to figure out the oldest child's share (based on the total number of children) other than trial-and-error. I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
Date: 05/18/2000 at 04:19:54 From: GeorgeB Subject: Re: The Pharaoh's Will Thanks for that, it helped a lot. The next thing is, I tried to come up with a way other than trial-and-error, but there isn't really another way. Well, there must be... but I can't find it. That also applies to part (C). (C) is a bit different, and I can see a relation but nothing of use. Please write back with your opinion, i.e. your way of answering it. Thank you.
Date: 05/18/2000 at 13:09:42 From: Doctor TWE Subject: Re: The Pharaoh's Will Hi again George - thanks for writing back. What I did (I guess this is really a variant on trial-and-error) was to add unit fractions 1/n + 1/(n+1) + 1/(n+2) + ... until the sum was greater than 1. I then counted the number of fractions added - this would be the number of children for which it works with 1/n as the starting fraction. Next, I subtracted the last unit fraction (so that the sum was less than 1 again) and subtracted that result from 1 to see what the youngest child's share was. (I must confess, I did this on a spreadsheet to save time.) For example, with n = 4 I got: 1/4 + 1/5 = 9/20 9/20 + 1/6 = 37/60 37/60 + 1/7 = 319/420 319/420 + 1/8 = 743/840 743/840 + 1/9 = 2509/2520 2509/2520 + 1/10 = 2761/2520 > 1 1 - 2509/2520 = 11/2520 That's 7 unit fractions, so n = 4 works for seven children, and the youngest child gets 11/2520 of the estate (about 1/229th.) With n = 3 we got five children, with n = 4 we get seven children. That means that the system won't work with six children. If we start with the oldest getting 1/3, there's not enough to give the fifth child the 1/7 as mandated by the formula. If we start with the oldest getting 1/4, the youngest child would get 97/840 - more than the 1/9 that is the next unit fraction. If you construct a table comparing n and C (the number of children that n works for), you'll see it "skips" certain values. The value that is skipped between your answer for part B and 11 is the answer to part C. I hope this helps. If you have any more questions, write back again. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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