Tuesday, July 02, 2013

If a Chicken + a Half Can Lay an Egg + a Half in a Day + a Half, How do You Find a Unit Rate?

This is the article I submitted to Answers.com, which they solicited from me before going in a different direction, editorially. I've decided to start posting the ideas for future pieces on this website, until I can collect them and find a home for them somewhere at Mr. Burke Math.net. (I'm paying for that site. I might as well use it.)

They didn't give me a lot to go on, other than to explain a problem, have a hook, make it entertaining and resolve the problem in the summary. I'm not entirely happy with the summary, but then it was a total rewrite of the original. As these things work out, the final draft contains about half the material of the first draft, and the focus shifted a bit (as well as narrowed a bit).


If a Chicken and a Half Can Lay an Egg and a Half in a Day and a Half, How Do You Find a Unit Rate?
By Christopher J. Burke

An old mathematical riddle asks, “If a chicken and a half can lay an egg and a half in a day and a half, then how long does it take six chickens to lay six eggs?” The expected response is “six days”. However, the correct answer is “a day and a half” because of the concept of “unit rate”.

Let’s define a few terms: In math, a ratio is a comparison of two numbers, such as “½” or “3:1”. A rate is a comparison of two numbers with different units, such as “$.79 for 2 oranges” or “100 calories per 8-oz. serving”. A unit rate is when the second number being compared is 1, such as 50 words per (1) minute or $12 per (1) ticket. You can find the unit rate by dividing two numbers: $.39½ per orange or 12.5 calories per ounce. (On a store shelf, it’s referred to as a “unit price”.)

A car’s rate of speed is measured in “miles per hour”. Its gas mileage rate is measured in “miles per gallon”. Each of these is a unit rate. They are useful measurements because cars rarely travel for exactly one mile or one hour, using exactly one gallon of gas. They allow for comparisons and calculations. Multiply “miles per hour” by hours, and you get the number of miles traveled. Divide the number of miles by “miles per gallon”, and you get the amount of gas used.

Dave’s car gets 20 miles per gallon. He drives at a rate of 50 mph for 3-½ hours. The distance he drove is (50 mph) X (3.5 hours) = 175 miles. The amount of gas he used is (175 miles) / (20 miles per gallon) = 8.75 gallons.

Fred travelled 330 miles in 6 hours and used 15 gallons of gas. His average rate of speed was (330 miles) / (6 hours) = 55 mph. His car’s gas mileage rate was (330 miles) / (15 gallons) = 22 mpg.

Returning to the original problem, the riddle gives three values to work with, which is one more than is required to find a rate. In other words, when making a comparison, one of these numbers will not be needed. You could compare chickens and eggs, chickens and days, or eggs and days. Could all three be compared? Yes, but it gets trickier. Put aside the fact that only in mathematics can half a chicken lay half an egg. Once we accept that a chicken and a half laid an egg and a half, we can see that, in the allotted time, each chicken is laying one egg. This is because (1.5) / (1.5) = 1.

This means that for the amount of time given, two chickens would lay two eggs, three chickens would lay three eggs, four chickens would lay four eggs, and so on. The time that it took each chicken to lay an egg doesn’t change. It doesn’t take six chickens longer to lay their eggs than two chickens. So the rate is one egg per chicken in a day and a half.

So given two values, such as distance and time, you can find a unit rate by dividing. Given a unit rate, you can calculate larger values, like distance or cost, by multiplying.

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Comments are welcome. If you'd like to see more of this, please, say so. If you'd like to see less of this, type in the words, "For the love of [Insert Deity], please, STOP!" Or something.


shai hadar said...

just wanted to say you're kinda very wrong...
if you want the explanation mail me at shaihadar@gmail.com

Marc Nishimoto said...

The machine is a chicken and a half. It lays an egg and a half in a day and a half, or an egg a day. So the rate is an egg/day/chicken-and-a-half.

(x, why?) said...

Thankful for your thoughtful, yet brief, analysis.

You are wrong.

The engine is a chicken. (Chickens lay eggs, half chickens really don't.) And the question asks about six chickens, not four chickens with four half-chickens.

But using your analysis that means that for six chickens you need 4 chickens-and-halves and you will get 4 eggs ***in one single day***.

The question was how long with is take six chickens to lay six eggs? One day only gives us four. I guess that means that my original answer -- which I believed you referred to as "kinda very wrong" -- is the correct answer.

You might've come up with the same answer, but you didn't mention it.

Here's the takeaway: A unit rate is a comparison of TWO THINGS. We are comparing THREE and that's what makes it a confusing, but fun riddle.

Have a nice day.

Ellen said...

After reading this, I'm grateful that I teach first grade math. I have a lot to learn, too:) Nice job!

Annie Travers said...

We need to have a question first, like "how many chickens are you going to need if you want to lay 20 eggs a day?" The unit rate for how many per day per chicken is also important and trickier. You can work out the proportions: you get 1/1.5 which is 2/3 of an egg per chicken per day.

Anonymous said...

Wow, you guys are dumb! There is no rate! This is the simplest equation in the world! If the amount of chickens=the amount of eggs and the amount of eggs = the amount of days they're all the same number! If a=b and b=c then a=c! To get a dozen eggs, it would take 12 days! A=C!!! Draw a picture or chart and you'll see it's really quite simple! :)

(x, why?) said...

Thank you, Mr. Anonymous, for chiming in.

Your answer is, at best, incomplete, and, at worst, totally incorrect compounded by calling others dumb.

As already established by others, there is a rate and it is 1.5 days per egg per chicken or 2/3 of an egg per chicken per day (which is actually the same thing).

You specified the number of days and the number of eggs, but not the number of chickens. If you're suggest 12 chickens take 12 days to make 12 eggs, you are wildly undercounting the eggs. If you are saying that 1 chicken will lay 12 eggs in 12 days, you are off by six days.

You are drawing false conclusions which is what this problem tries to get you to do with its numerical slight of hand. You fell for it.

But at least you can hide behind anonymity.

Anonymous said...

I would like to know How many days would it take for 1.5 chickens to lay 32 eggs using the
1.5 1.5 1.5

(x, why?) said...

It would depend upon the ground rules, which aren't exactly set in stone.

It's clear that 1.5 chickens will produce 1.5 eggs in 1.5 days.
Give them another 1.5 days and they will produce another 1.5 eggs, or 3 eggs in 3 days.
Keep going and we can see that they 31.5 days to produce 31.5 eggs.

If this is an all-or-nothing task requiring the full 1.5-day interval, then we can only move in discrete 1.5 egg increments, meaning it would take 33 days to get 33 eggs, which is 32 plus one for good luck or something.

If this is somehow a continuous process, then it would happen in 32 days. However, a continuous process seems even sillier than having 1.5 chickens in the first place!

Anonymous said...

Thank you, I originally tried and got 22 72 and 52.