Last week, we looked at problems involving some number of people making some number of things in some amount of time. In a classic twist on this problem, we’ll now examine several variants starting with “If a hen and a half can lay an egg and a half …”. Can we make sense of half-eggs and half-hens?
A simple case: How long to get a dozen?
We’ll start with this from 1997:
A Chicken and a Half?! If a chicken and a half lays an egg and a half in a day and a half, how long does it take to get a dozen eggs?
Steve had titled his email “Word problem Dad asks all the time”.
Doctor Sonya answered:
Dear Steve, This question feels strange to answer, because dealing with half chickens and half eggs is not something one normally encounters. Because I have no intuitive understanding of half a chicken, I'll approach this problem by plugging in simple numbers for the complicated numbers (like one and a half). Let's say we have 1 chicken that lays 1 egg in 1 day. Then how many days would it take to get a dozen eggs? 12, right? Now if we have 2 chickens that lay 2 eggs in, say, 3 days, how long does it take to get a dozen eggs? Since we are getting 4 eggs every 3 days out of these two chickens, it will take 3*3 = 9 days.
This is a standard way to approach a strange-looking problem: Try changing it to simpler numbers in a more familiar setting, figure out how to solve the new problem, and then apply that method to the real problem.
But don’t misread what she said: These examples are two different problems, not two steps in solving one problem as we did last time. Also, this problem is a little different from those we looked at last week, or will be seeing below, in that the question is about the same number of chickens we are initially told about, not some new number. That makes it a little simpler.
So what exactly are we doing when we solve these chicken-egg problems? Let's see. What I am doing is seeing how many eggs are laid total per cycle. In the second situation above, each cycle is three days long. Then I find out how many cycles we'll need to go through to get 12 eggs. If we get 4 eggs per cycle, we'll need 12/4 = 3 cycles. Now I multiply the number of cycles by the length of each cycle to get the total time it will take. This gives me 3*3 = 9 days. So 2 chickens who each lay 2 eggs in three days take 9 days to lay a dozen eggs. This method should work no matter what our numbers are. Try plugging in the chicken and a half, the egg and a half, and the day and a half, and see what you get.
So, what is the answer to the actual problem? (Note how Doctor Sonya subtly chose not to do Steve’s work for him.)
Well, we are getting 1.5 eggs from our 1.5 chickens in 1.5 days; that’s the cycle. How many cycles will it take to get 12 eggs? We divide: \(12\div1.5=8\), so in 8 cycles we’ll get 12 eggs; and 8 cycles of 1.5 days is \(8\times1.5=12\) days.
A more typical version: How many eggs?
This is from 1998:
A Hen and a Half I am having a hard time trying to find the answer to this problem. Will you please help me? If a hen and a half lays an egg and a half in a day and a half, how many eggs will five hens lay in 6 days?
Here we are given a new number of hens, rather than sticking with the hen and a half; and this time, we’re being asked how many eggs we’ll get in a given time, rather than how long it will take. This is like the problems from last time, apart from the non-integer data.
I answered, taking the same approach of trying it first with nicer numbers:
Hi, Chase. This is an odd problem designed to confuse you. So let's try simplifying the problem, and then use what we learn from the simpler problem to solve the original. Suppose two hens lay four eggs in three days. That means that each hen laid two eggs during those three days, so each hen lays 2/3 of an egg per day. Now if you had five hens and six days, they would lay five times as many eggs per day, totaling 10/3 per day; multiply that by 6 days, and there would be 20 eggs.
My approach this time is to find a unit rate by making two changes, first keeping the same number of days, and then keeping the same number of hens. Then I applied that rate to the question.
Now try doing the same thing with the original numbers. If that still confuses you, maybe you can make a general formula if A hens lay B eggs in C days, and then just plug in the numbers. When you find the answer, it will look simple and surprisingly familiar.
Again, let’s carry it out:
- If 1.5 hens lay 1.5 eggs in 1.5 days, then 1 hen lays \(1.5\div1.5=1\) egg in the same 1.5 days.
- So 5 hens will lay 5 times as many, 5 eggs, in those 1.5 days; and in 6 days, which is 4 times as long, they will lay \(5\times4=20\) eggs.
Yes, that is “surprisingly familiar”: It’s the same answer! I probably didn’t choose my example intentionally, but it provided a way to let Chase know when the answer is right.
One more time: How many eggs?
Here’s one more take on the basic problem, which is almost identical:
Six Chickens, Five Minutes If 1 and 1/2 chickens lay 1 and 1/2 eggs in 1 and 1/2 minutes, how many eggs will 6 chickens lay in 5 minutes?
Can you see that the answer will be the same? We’ve just swapped the numbers of chickens and minutes.
Doctor Roy answered, this time not using a simpler example, but turning the actual problem into a simpler one, by our trick from last time of changing two numbers at a time:
The 1 1/2 is confusing, right? So, let's get rid of it. First, let's say that we have two groups of chickens, Group A and Group B. In each group, there are 1 1/2 chickens. Then, in 1 1/2 minutes, Group A has 1 1/2 eggs and Group B has 1 1/2 eggs. So, in total, Group A and Group B have given us 3 eggs in 1 1/2 minutes. Since we don't really need to keep them separate, let's combine Group A and Group B: 3 chickens (the total number of chickens from both groups) give us 3 eggs in 1 1/2 minutes.
We doubled the number of eggs by doubling the number of hens.
We have already gotten rid of 2 of the 1/2's. Now, we have the last one to go. 1 minute is 2/3 of 1 1/2, right? So the 3 chickens should be able to lay 2/3 the number of eggs in 1 minute that they can in 1 1/2 minutes, or: 3 chickens lay 2 eggs (2/3 * 3) in 1 minute. We've gotten rid of all the 1/2's now, we know that 3 chickens lay 2 eggs in 1 minute. Now, all we have to do is figure out how many eggs 6 chickens (or twice 3 chickens) can lay in 1 minute and then multiply by 5 (since we need the figure for 5 minutes).
In effect, we have taken the reciprocal of the rate: \(1\div\frac{3}{2}=\frac{2}{3}\), and multiplied the time by that.
Finishing the work,
- if 3 chickens lay 2 eggs in 1 minute, then
- 6 chickens will lay twice as many (4 eggs) in 1 minute, and
- in 5 minutes they will lay \(5\times4=20\) eggs.
Finding the rate
Now, let’s dig a little deeper, with this question from Erryn in 2002:
One Chicken, One Day 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?
This time we’re asked for the rate of eggs per chicken per day.
I answered this one, first referring to the first two answers above and the one we’ll see next, and then saying,
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.
Rather than give an answer at all, I gave Erryn the tools to find a way.
Erryn will not, as it turns out, be using this stepwise-change method, but I’ll be showing my work soon.
But what does that even mean?
Erryn used a formula she’d found in the answer we’ll be looking at next (in which this problem is the first step of the solution), and wrote back:
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?
Thinking about half-chickens and half-eggs makes it more complicated than just working with the numbers! And the answer itself, this time, seems to make no sense. I replied, showing my own work:
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?
As we’ll see, this sort of thing actually happens all the time.
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.
The same is true of other averages (average rainfall doesn’t mean you get the same amount every day), but we don’t always expect a whole number as we do with eggs.
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
The important thing is that we don’t need to know how the eggs are being laid; all we need to know is the average rate.
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.
For more on this aspect of averages, see Making the Mean More Meaningful.
Going over the top!
How far can you take this sort of question? Consider this, from 2001:
Laying Eggs Better by Half 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?
This will need some interpretation. I answered, referring to the first two answers above (and giving the formula we saw quoted above):
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": 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.
That’s an ugly answer. Maybe we’ve misinterpreted something.
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.
Our calculation now is $$\text{Hens}=\frac{\text{Eggs}}{\text{Rate}\times\text{Days}}=\frac{10.5\text{ eggs}}{1\text{ egg/hen-day}\times10.5\text{ days}}=1\text{ hen}$$ and we solve $$x+0.5=1$$ to get $$x=0.5$$
And somehow, “half a hen” seems like a fitting punch line.