Unit Rates

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(C)Copyright 2020, C. Burke. "AnthroNumerics" is a trademark of Christopher J. Burke and (x, why?).

If you know what I Mina. Or, mean-a?




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Two Taylor Swifts and a Sliding Yardstick

I've come to realize that as of this month of December, I am currently exactly 2 Taylor Swifts in age. That's really not so bad when I think about it because when You Belong With Me hit the charts (in 2009), I was 2.25 Taylorswifts. So i guess it's getting better. I could continue, figure out when I'll be only 1.5 Taylorswifts, but it would start to sound like an old Abbott & Costello routine. (Not that there's anything wrong with that.)

I had the thought recently about sliding yardsticks (or sliding metrics). What if you're measuring something, and the unit that you're measuring against changes along the way. It would be a great way to obscure (or hide) data.

What brings this up, without getting political, are all the political postings I see on different social platforms by people who should be smart enough to know better. (Scary thought: they are smart enough, but they assume we're all dumb enough.)

You don't judge a book by its cover and you don't evaluate major chunks of the economy based on a graphic containing a pithy quote and carefully selected factoids(*) underlined with a "witty" applause line, wrapped up in 140 characters, meant to be repeated and reported by people using the phrase "Alls I knows is...".

(*) factoid - Not a "little fact", but something that "looks like a fact", but isn't. This could be extended to the presentation of facts in a way that they seem related but are really not.

If that's all you know, you need to be more educated. Look beyond the numbers. Check those that confirm your biases and those you dismiss out of hand as made-up lies (What other kind are there?). However, when the fact-checking doesn't happen, a low-information individual is created -- someone who doesn't know all the facts, who knows that they don't have all the facts, but believes that every fact and factoid that they do have is enough, and no other piece could possibly make a difference. Any other data is obviously wrong, and if not, needs to be ignored, you Fill-In-The-Blank-ist Bad-Noun!

Oddly, though no piece of information can sway them, they are shocked that their one piece of information hasn't yet swayed you.

What spurs this on?

Comparisons of budgets and deficits and debts and prices and any number than can (and did) fluctuate wildly, so that specific highs and lows can be picked out and averaged or compared or manipulated alongside the assignment of blame or credit, regardless of the merit of that assignment. Is the latest increase big or tiny? Is it bigger or tinier than that other guy's? And what percent increase in the load was the straw that broke the camel's back?

Are these things appropriate fodder for debate? Absolutely, as long as that debate is longer than a sentence and the rebuttal is more than "You're a butthead. Neener. Neener!"

Shouting down your opponent doesn't mean you've won the argument. It means you're rude and willing to stay ignorant of facts you don't have.

My entire class can ignore me when I say that 0! = 1. They can all insist that it should be 0, and maybe even argue why. but they'll be wrong in class and wrong on every test after.

As for me, I've given way too much thought to it. I'll be exactly 1.5 Taylorswifts old when (T + 25)/T = 1.5

Yes, I can calculate it. No, I don't want to. Unlike Costello, I know that she isn't going to catch me. The answer is that the ratio is getting smaller, but I'm still getting old.

If I Wrote the State Exams . . .

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(C)Copyright 2013, C. Burke.

Alternate Title: 'Why I'm Not Allowed to Write State Exams'

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).

Enjoy.

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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