One Chicken, One DayDate: 04/19/2002 at 10:20:39 From: Erryn Bard Subject: Fractions It's the same old chicken and egg question, but with a twist ending: If a chicken and a half lays an egg and a half in a day and a half, how many eggs can one chicken lay in one day? Date: 04/19/2002 at 12:25:12 From: Doctor Peterson Subject: Re: Fractions Hi, Erryn. Have you seen our answers to related questions? A Chicken and a Half?! http://mathforum.org/library/drmath/view/58675.html A Hen and a Half http://mathforum.org/library/drmath/view/57464.html Laying Eggs Better by Half http://mathforum.org/library/drmath/view/56844.html (I found these by searching our site for the phrase "and a half.") The second of these is the simplest. I like to just transform the problem one step at a time into something simpler: How many eggs will a chicken and a half lay in three days? In one day? How many will three chickens lay in one day? How about one chicken? There are many ways to think about this, suited for different levels of experience. I would like to see what ideas you came up with, so I can help you do it in your own way. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/19/2002 at 14:31:32 From: Erryn Bard Subject: Fractions We found this via the URL's you gave: 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 so the answer to our question is 2/3 egg. However, after drawing it out on paper we surmise that if the half chicken lays 1/2 egg in 1 1/2 days, the whole chicken will lay 1 egg in 1 1/2 days. Therefore, how much will that one chicken lay in one day? We could split the one and half days into 3rd's and thus get 2/3, but how do you get 2/3 egg per chicken? Are we supposed to think of 1 1/2 chickens as a whole, and the 1 1/2 eggs as a whole, rather than splitting the eggs up between the 1 1/2 chickens? Date: 04/19/2002 at 15:19:15 From: Doctor Peterson Subject: Re: Fractions Hi, Erryn. You have the right answer. Using my approach, 1 1/2 chickens lay 3 eggs in 3 days 1 1/2 chickens lay 1 egg in 1 day 3 chickens lay 2 eggs in 1 day 1 chicken lays 2/3 egg in 1 day So what does that mean? How can there be a fraction of an egg? When we talk about a fractional rate, such as 2/3 egg per day, we have to visualize it in whatever way is appropriate to the problem. Obviously if each hen lays once a day at this rate, there will be a lot of broken eggs lying around. Instead, the rate has to be thought of as an average. It may be that every day 2/3 of the chickens lay an egg, so that if you have 3 chickens you get 2 eggs every day. It may be that each chicken lays 2 eggs every 3 days, either by taking every third day off, or by laying one egg every 36 hours. So it might look like day 1 day 2 day 3 +---------------+---------------+---------------+ hen 1 o o hen 2 o o hen 3 o o ----- total oo =2 oo =2 oo =2 or day 1 day 2 day 3 +---------------+---------------+---------------+ hen 1 o o hen 2 o o hen 3 o o ----- total o o =2 o o =2 o o =2 or even day 1 day 2 day 3 +---------------+---------------+---------------+ hen 1 o o hen 2 o o hen 3 o o ----- total ooo =3 ooo =3 =0 where the number differs from day to day, but in any 3 days you get 6 eggs. All of these fit. Or maybe 1/3 of the chickens are roosters and never lay an egg, but for every 3 chickens, 2 eggs are laid every day! day 1 day 2 day 3 +---------------+---------------+---------------+ hen 1 o o o hen 2 o o o rooster ----- total oo =2 oo =2 oo =2 Talking about averages always involves pretending things are more uniform than they really are; rather than talking about the number of whole eggs each individual hen lays, and when, we just spread around the eggs that are laid, as if each hen laid 2/3 of an egg. The total number of eggs would be the same, and that's all the farmer cares about (unless he'd like to stop feeding some of those roosters and increase his profits). The same thing happens if we say that the average family has 2.5 children. That doesn't mean there are a lot of pieces of children running around. Rather, one family may have 2 and the next has 3, so that there are 5 children for every 2 families. To average it out, we divide them equally and pretend there are 5/2, or 2.5, children in each family. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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