A Ton of Money (or Maybe More)

One of my favorite resources from Illuminations is Too Big or Too Small, a collection of three classroom activities that develop number sense, one of which is the following problem:

Just as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2 million in a contest. The money was sent to him in two suitcases, each containing $1 million in one-dollar bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Can he make you a millionaire?

This problem is from the book Developing Number Sense in the Middle Grades by Barbara Reys and Rita Barger, published by NCTM in 1991. So it’s not new, but it’s still good.

© NCTM 2017

My first attempt to use this problem with students was dreadful (details below), but I’ve used this problem successfully many times since. Yet something about it always bothered me. I’m not opposed to fictitious scenarios if they get students interested. But this scenario, in which a friend claims to win $2,000,000 and needs a ride to the airport, seems too contrived and not adventurous enough. Luckily, I recently had food poisoning and spent an entire Saturday on the couch watching bad movies. While watching Rush Hour (1998), I found a scenario that I liked a whole lot better…

In the clip, the kidnapper asks for the following:

• $20 million in $50’s
• $20 million in $20’s
• $10 million in $10’s

Now the questions of “How much would that weigh? How big a case would you need to carry all of it?” seem a little more meaningful.

I’ll channel my inner Andrew Stadel here. For both the weight and volume:

• Give an estimate that you know is too low.
• Give an estimate that you know is too high.

Now, do the calculations, and see how close your intuition was.

When I first used this task with students, I was anticipating a great discussion about how to estimate the weight and volume of the money. I suspected that some students might estimate that you could fit 5 or 6 bills on a sheet of paper, there are 500 sheets of paper in a ream, a ream weighs about 5 pounds, yada, yada, yada. Instead, one student raised his hand and said:

A dollar bill weighs exactly 1 gram.

I asked how he knew that. “Do you collect money? Are you a numismatist?”

No. That’s how drug dealers measure cocaine. They put a dollar bill on one side of a scale, and they put the cocaine on the other.

“Oh,” I said.

Some days, your students learn something from you. And some days, you learn something from them.

After you estimate the weight and volume, check your answer by clicking over to reference.com.

If you use this video clip and activity in a classroom with students, I’d love to hear how it goes. Please post about your experience in the comments.

Constant Change

I’m frustrated.

I’m also old, cranky, and cynical. Whatever.

My frustration is not the my-flight-was-delayed-three-times-then-eventually-cancelled-and-there-are-no-more-flights-to-Cleveland-till-tomorrow-morning type. It’s not even the can’t-believe-my-boss-is-making-me-go-to-Cleveland kind of frustration. More like the why-aren’t-there-the-same-number-of-hot-dogs-and-buns-in-a-pack variety. So it’s a First World problem, to be sure, but still annoying. I’ll explain more in a moment.

But first, how ’bout a math problem to get us started?

If you make a purchase and pay with cash, what’s the probability that you’ll receive a nickel as part of your change?

Sure, if you want to get all crazy about this, then we can take all the fun out of this problem by stating the following assumptions:

• You only pay with paper currency. If you paid with coins, then the distribution of coins you’d receive as change would likely vary quite a bit.
• You never use 50¢ coins. Honestly, they’re just too obscure.
• Transaction amounts are uniformly distributed, so that you’re just as likely to receive 21¢ as 78¢ or any other amount.
• Cashiers don’t round because they dislike pennies. So, if you’re supposed to get 99¢ change, the cashier doesn’t hand you a dollar and say, “Don’t worry about it.” Instead, you actually get 99¢ change.

But stating assumptions is a form of mathematical douchebaggery, isn’t it? (As an aside, check out the definition of douchey that’s returned when you do a search. Sexist, anyone?) I prefer problems with no assumptions stated; let folks make their own assumptions to devise a model. If you and I get different answers because of different assumptions, no worries. Maybe we both learn something in the process.

Anyway, where was I? Oh, yeah…

Understanding the solution to that problem is a precursor to the issue that’s causing me frustration. I’ll give the solution in a minute, so pause here if you want to solve it on your own, but let me now allow the proverbial cat out of its bag and tell you why I’m frustrated.

At our local grocery store, there’s a coin counting machine that will count your change, sort it, and spit out a receipt that you can take to the customer service desk to exchange for paper currency. Walk in with a jar full of change, walk out with a fistful of fifties. Pretty nifty, right? Except the machine charges a ridiculous 8.9% fee to perform this service. No, thank you.

My bank used to have a similar coin counting machine, and if you deposited the amount counted by the machine into your account, there was no fee. The problem is that everyone was doing this to avoid the grocery store fee, so the machine broke often. The bank finally decided the machine wasn’t worth the maintenance fees and got rid of it. Strike two.

Which brings me to my current dilemma. One Saturday morning every month, we now spend 30 minutes counting coins and allocating them to appropriate wrappers. Which is fine. The problem, however, is that we run out of quarter and penny wrappers way faster than we run out of nickel or dime wrappers. Which brings me to the real question for the day:

Since pennies, nickels, dimes, and quarters are not uniformly distributed as change, why the hell does every package of coin wrappers contain the same number for each coin type?

The Royal Sovereign Assorted Coin Preformed Wrappers is the best-selling collection of coin wrappers on Amazon, and it provides 54 wrappers for each coin type. They also offer a 360‑pack with 90 wrappers for each coin type; Minitube offers a 100‑pack with 25 wrappers for each coin type; and Coin-Tainer offers a 36‑pack with 9 wrappers for each coin type. But what no one offers, so far as I can tell, is a collection of coin wrappers with a distribution that more closely resembles the distribution of coins that are received as change.

Whew! It feels good to finally raise this issue for public consideration.

So, the question that I really wanted to ask you…

Given the distribution of quarters, dimes, nickels, and pennies that are received in change, and given the number of coins needed to fill a coin wrapper — 40 quarters, 50 dimes, 40 nickels, and 50 pennies — how many of each wrapper should be sold in a bundled collection?

To answer this question, I determined the number of coins of each type required for every amount of change from 1¢ to 99¢. The totals yield the following graph:

The number of pennies is nearly five times the number of nickels. And there are nearly twice as many quarters as dimes.

But I realize that’s a theoretical result that may not match what happens in practice, since this assumes that the amounts of change from 1¢ to 99¢ are uniformly distributed (they aren’t) and that cashiers don’t round down to avoid dealing with pennies (they do). In fact, when I made a purchase of $2.59 yesterday, instead of getting one penny, one nickel, one dime, and one quarter as change, the cashier gave me one penny, three nickels, and one quarter, in what was clearly a blatant attempt to skew my data.

So for an experimental result, I counted the pennies, nickels, dimes, and quarters in our home change jar. The results were similar:

The ratio of pennies to nickels is closer to three, but the ratio of quarters to dimes is still roughly two.

Using a hybrid of the theoretical and experimental results, and accounting for the fact that only 40 quarters and nickels are needed to fill a wrapper whereas 50 pennies and dimes are needed, it seems that an appropriate ratio of coin wrappers would be:

quarters : dimes : nickels : pennies :: 17 : 8 : 6 : 19

Okay, admittedly, that’s a weird ratio. Maybe something like 3:2:1:4, to keep it simple. Or even 2:1:1:2. All I know is that 1:1:1:1 is completely insane, and this nonsense has got to stop.

Hello, Royal Sovereign, Minitube, and Coin-Tainer? Are you listening? I’ve completed this analysis for you, free of charge. Now do the right thing, and adjust the ratio of coin wrappers in a package accordingly. Thank you.

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Wow, that was a long rant. Sorry. If you’ve made it this far, you deserve some comic relief.

How many mathematicians does it take to change a light bulb?
Just one. She gives it to a physicist, thus reducing it to a previously solved problem.

If you do not change direction, you may end up where you are heading. – Lao Tzu

The only thing that is constant is change. – Heraclitus

Turn and face the strange ch-ch-ch-changes. – David Bowie

A Buddhist monk walks into a Zen pizza parlor and says, “Make me one with everything.” The owner obliged, and when the pizza was delivered, the monk paid with a $20 bill. The owner put the money in his pocket and began to walk away. “Hey, where’s my change?” asked the monk. “Sorry,” said the owner, “change must come from within.”

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As for the “probability of a nickel” problem that started this post, here’s my solution.

For change amounts from 1¢ to 25¢, there are ten values (5‑9 and 15‑19) for which you’ll receive a nickel as part of your change.

This pattern then repeats, such that for change amounts from 25n + 1 to 25n + 25, where n is the number of quarters to be returned, you’ll receive a nickel when the amount of change is 25n + k, where k ∈ {5, 6, 7, 8, 9, 15, 16, 17, 18, 19}. For 0 ≤ n < 4, there are 40 different amounts of change that will contain a nickel, so the probability of getting a nickel as part of your change is 40/100, or 40%.

Getting Rich the Hard Way

Ask a silly question, get a silly answer.

Teacher: If you have $4, and you ask your father for another dollar, how much would you have?

Johnny: Four dollars.

Teacher: Young man, you don’t know your addition facts!

Johnny: Ma’am, you don’t know my father!

Johnny’s father and my dad seem to have a lot in common. But my dad would have been proud of me yesterday. While walking home from the local coffee shop, I noticed a corner of a dollar bill on the ground. Not the whole bill, mind you, just a corner that had been ripped off. I thought not much of it, until two feet later I saw another scrap of the dollar bill… then another… and another…

I know and understand Calculus, and I realized that a lot of little things can add up to a lot, so I spent 15 minutes scouring the area for as many pieces of the dollar bill as I could find. I took them home and asked my sons, “Wanna do a puzzle?” We spent a half-hour reconstructing the bill and taping it together. The pictures below show the before and after:

The bill was not in good enough shape to be accepted by a vending machine (too much tape, I suspect, and the missing piece on the right side surely didn’t help, either), but it was in good enough shape for my bank to give me four shiny quarters in exchange for it.

I know that a penny saved is a penny earned. But what is a dollar found?

And the bigger question: What should I do with my new-found wealth?

I decided to buy a lottery ticket. The state gambling commission organized a raffle that boasted an infinite amout of money as the prize. To my great surprise, I won! When I showed up to claim the prize, they told me it would be disbursed as 1 dollar now, 1/2 dollar next week, 1/3 dollar the thrid week, 1/4 dollar the week after that, and so on.

But the joke’s on them. My winnings for the third week will include a one-third cent piece, and that’s gotta be worth something, right?

(Note: Almost everything above is true. I really did find the pieces of a dollar bill on the ground yesterday. As best I can tell, the bill had been on the lawn when it was cut by the blades of a power mower. And my bank really did give me four quarters in exchange for the taped-up, reconstructed version.)

Retail Tales

In tough economic times, lots of folks are counting quarters and pinching pennies. To attract new customers, retailers are offering significant discounts.

• A local bookstore is having a sale: All Math Titles, 1/3 Off. So I picked up a copy of Gödel, Escher.
• Skate Charm Insurance is offering fire-and-theft policies at rock-bottom prices. When asked how they could offer them so cheap, the actuaries responded, “Who would steal a burnt car?”
• Grocery stores in Northern Virginia are promoting lite beer as a good deal, because it has 20% fewer letters than light beer.
• A local gas station recently switched to metric, and I somehow feel better paying $1 per liter instead of $3.78 per gallon.

Nobody likes change, except a kid with a piggy bank.

What coin doubles in value when half is removed?
A half dollar.

Doc: Give me an update on the boy who swallowed four quarters.
Nurse: No change yet.

In the shameless plug department: NCTM members get a 25% discount off the retail price of Math Jokes 4 Mathy Folks at NCTM conferences, and everyone else can save 24% by buying from Amazon.

Father’s Day Reflections (and Other Transformations)

I just got a new stepladder. Don’t get me wrong, it’s a fine stepladder. I just wish I had gotten to know my real ladder.

I had the privilege of knowing my real father.

At age 15, my father faked a birth certificate and joined the Navy. When he was 18, he received a dishonorable discharge — after allowing him to fight in Japan during the last two years of World War II, the Navy finally learned that my dad was under age when he enlisted. So, what did he do? He joined the Army. Before he was 21, he had been to each of the 50 states and had traveled around the world 4 times.

My father had only a sixth-grade education, but he believed in the power of school and learning. It was not easy to get my dad to part with his money. When I was in third grade, my teacher asked me, “If you have two dollars, and you ask your father for another three dollars, how much will you have?”

“I’d have two dollars,” I told her.

“Young man,” she said, “you don’t know your arithmetic.”

“No, Mrs. Wargo,” I said, “you don’t know my father!”

But he often gave me $20 for a good report card, and I was the first kid in my school to have a Commodore 64 with a disk drive. When I graduated high school, my family was subsisting on my father’s disability pension, and I considered working for a year to save money before enrolling in college. “You’re too damned smart,” he said. “Send in the forms. We’ll make it work.”

My father passed away in December 1994. The last words he said to me were, “You’re my pride and joy.” Father’s Day is always a little rough for me, but it’s a good time to reflect. I continually ask myself, “Am I a man that my father would be proud of?”

Are you kidding? I have to believe my father is smiling down from Heaven, saying, “That’s my boy! Yeah, that geeky one there! He’s the author of a math joke book and math joke blog, ya know.”

For all you math dads (and sons, too), here’s some humor for today:

Son: Dad, can you do my homework for me?
Dad: I’m sorry, son, it wouldn’t be right.
Son: That’s okay. Can you try anyway?

I spent today with my twin four-year-old sons, hiking, doing KenKen (more on that later), playing the anagram game, and helping them figure out how the number of “cheers” we do with our glasses at the dinner table is related to the triangular numbers. What a great day. Happy Father’s Day!

Make Money with Fractions

An act of Congress on July 17, 1861, gave the Treasury Secretary the authority to print U.S. currency. For a variety of reasons, it wasn’t until several years later that the Treasury Department actually began printing; in the interim, private firms printed the notes in sheets of four, sent them to the Treasury Department where the seal was affixed by hand, and then the sheets were cut apart with scissors. (How far we’ve come!)

Did you know that the U.S. government will replace worn out or damaged money if three-fifths of it is still identifiable? Similarly, two-fifths will earn the bearer half the face value.

Perhaps the U.S. government is not terribly good with fractions. (This is not surprising. A recent government report claims that five out of four government employees do not understand fractions.) Even an elementary student knows that 3/5 + 2/5 = 1. So why is the government willing to give you 150% of a bill’s value if you divide it in the ratio 60:40?

If you want to make a quick buck (or a quick $50), here’s my suggestion: Go to the bank, get a fresh $100 bill, then cut it as shown:

As divided, the left portion is 3/5 of the original bill, and the right portion is 2/5 of the original bill. Now you can exchange the left portion for a new $100 bill, and you can exchange the right portion for $50. That’s a 50% return on your money, which is better than almost every blue-chip stock in the history of NASDAQ and the NYSE.

With policies like this, is it any wonder there’s a national deficit?

* NOTE: It is illegal to purposely mutilate U.S. currency. The above post is satirical. Do not try this at home. If you do, we at MJ4MF hereby absolve ourselves of all responsibility.