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### Laying Eggs Better by Half

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Date: 12/05/2001 at 08:47:39
From: Kenneth Klein
Subject: Puzzle Ratio

If a hen and a half lays an egg and a half in a day and a half,  how
many and a half that lay better by half will lay half a score and a
half in a week and a half?
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Date: 12/05/2001 at 12:55:49
From: Doctor Peterson
Subject: Re: Puzzle Ratio

Hi. Kenneth.

I think this wins the prize as the most complicated version I've seen
of this old puzzle. You can find other versions by searching our site
for the phrase  and a half (that exact phrase):

A Hen and a Half
http://mathforum.org/dr.math/problems/chase9.23.98.html

A Chicken and a Half?!
http://mathforum.org/dr.math/problems/green5.7.97.html

For a quick solution to your version, we can start by finding the
laying rate in eggs per hen-day:

1.5 eggs
------------------- = 2/3 eggs/hen-day
1.5 hens * 1.5 days

Now a hen that is "better by half" will lay 1.5 times as many, or
1 egg per hen-day. A score and a half is 1.5*20 = 30, and half of that
is 15. (I could also read that phrase as "1.5 times half a score," but
I would get the same answer.) A week and a half is 1.5 * 7 days, or
10.5 days. I have to solve a rate equation of the form

Eggs = Rate * Hens * Days

for the number of hens:

Eggs              15 eggs              15
Hens = --------- = ------------------------- = ---- hens = 1 3/7 hens
Rate*Days   1 egg/hen-day * 10.5 days   10.5

Finally, what in the world does "how many and a half" mean? I think we
have to solve

1.5 X = 10/7

which gives X = 20/21 hens.

After all that work, the problem wasn't designed to give us a whole
number answer?? Ahh - maybe "half a score and a half" was meant to be
10.5. Then we can replace 15 eggs with 10.5 eggs, and the first
calculation just gives 1 hen. And in that case, "how many and a half"
probably means

X + 1/2 = 1

so the answer is half a hen: "Half a hen and a half (one hen) that lay
better by half (one egg per hen day) will lay half a score and a half
(10.5 eggs) in a week and a half (10.5 days)."

No wonder mathematicians don't speak like this! Our goal is to
communicate rather than to confuse.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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