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Making Change
Roland Piquepaille writes "There are mostly four kinds of coins in circulation in the U.S: 1 cent, 5 cents, 10 cents, and 25 cents. But is it the most efficient way to give back change? This Science News article says that a computer scientist has found an answer. "For the current four-denomination system, [Jeffrey Shallit of the University of Waterloo] found that, on average, a change-maker must return 4.70 coins with every transaction. He discovered two sets of four denominations that minimize the transaction cost. The combination of 1 cent, 5 cents, 18 cents, and 25 cents requires only 3.89 coins in change per transaction, as does the combination of 1 cent, 5 cents, 18 cents, and 29 cents." He also found that change could be done more efficiently in Canada with the introduction of an 83-cent coin and in Europe with the addition of a 1.33- or 1.37-Euro coin. Check this column for more details and references." The paper (postscript) is online.
Why not just get rid of silly prices like 99.99 and 4.37 and 1.49. ?
Why not round prices to dimes ? Or even quarters ?
for the last time people, I am "frodo from middle eaRTH", not "middle eaST".
That's part of the process that doesn't seem to have been looked at well. While the model of 'giving change' may include a number of coins, it also includes the entire process including a cashier translating the change amount into coinage, then counting it back to the customer, and much of the time, the customer counting and checking it's the right amount.
For the pure maths side of it it's pretty neat, all the same - just not completely useful when it comes to Real World Stuff
You think it's bad enough when the cashier has to use the machine to figure out how much change to give, and in what denominations? You think it's bad when the little old lady in front of you in line starts counting, and then double-counting to make sure, the change she's going to hand over?
You give them a 29 cent piece and see how fast things get.
I'm willing to bet that most of the "coin cost" or whatever you want to call it comes from pennies, anyway -- if the dollar amounts are random, every 5 transactions are going to involve (0 + 1 + 2 + 3 + 4 = ) 10 pennies, or 2 pennies per transaction. Rounding prices to the nickel would be simpler, easier, and more efficient.
employee theft. .99, or 1.95, or 7.53, they gotta make change by opening the register.
If the price is 1.00$, the person working the regster can just take the buck, or five, or whatever, and slide it into their pocket. If its
All Troll + "offtopic" mods are meta moderated as "Unfair", because you abused the system.
- Some egghead thinks "optimal" means "fewest coins returned in change, on average."
- He recommends introducing 18 and 83 cent coins.
- The people who actually use coins laugh at this idiocy.
Sheesh, "optimal" coinage denominations are those that make using coins easiest. That means quick mental calculations of change, manipulating them with your fingers, and passing them back and forth.The ivory tower academics are certainly earning their reputation for foolishness.
taken! (by Davidleeroth) Thanks Bingo Foo!
Beyond that, it also means the cashier would have to figure out what the most efficient combination of change is . . for 0.36 don't give a quater two nickels and a penny, just give two 0.18 coins.
The logic for determining change is really easy for a cashier. start with the largest coin and work your way down until it all adds up.
When was the last time a cashier counted back your change. They usually give whatever the computer tells them. Counting back change is a lost art. There is no question that these unusual coin values would be more difficult. Ease in making change isn't simply number of coins given back.
It's funny (in a VERY sad way) that to him, the cash register was this magic oracle that told him what to do, and that it didn't occur to him that what he was doing was even knowable without its use.
I am a High School math teacher, and I can't agree with this statement enough. Somedays I laugh, somedays I cry, but it is always sad when I see a student need the calculator for the most BASIC of operations (And I am not even counting the OP's example as "basic", that would be "basic+")
I think it all comes from the fact that students are allowed to use calculators at such an earlier point in thier schooling. I am only 29, but I was not allowed to use a calculator in school until somewhere around 11th grade. It really hones (sp?) those basic math skills. I'll step off my soap-box now Sorry ;-}
Yes, yes, congratulations on completing first year computer science (or math, for that matter).
Don't you think that the current denomination systems are designed specifically so that the greedy change-making algorithm will work?
The poster you were replying to seemed aware of that; They were merely saying that since the current denominational system has this property, it is easy to use. The problem with adding 32- and/or 18-cent coins is that the greedy approach may no longer make the most optimal change.
The Deivation assumes two falacies
first,
that the price of goods is not partly determined by the demoninations of coins. for example, the reason why a candy bar is 50 cents or 65 cents and not say 48 cents is because we have nickels dimes and quarters. or that the reason a price is 5.95 cents and not 5.96 cents. etc..
Second,
this assumes your change purse is stocked with all denominations. that's true at the cash register but not in my pocket. When I reach in my pocket and I pull out some change there are a myraid of ways I can make 25 cents. 5 nickels, 2 dimes and a nickel, 5 pennies+ etc...,
not so with his optimal set. if I'm nissing any of the denominations its hard to make it up with the others.
third, entropy
again reaching for change in my pocket the goal is not to find the minumum number of coins but rather to be able to pay the bill without thinking too much. that is the more ways I can add up to the same value the more likely I will on a random grab find the right coins to make it. I dont care how many coins.
Some drink at the fountain of knowledge. Others just gargle.
I don't blame people for not being able to do math in their head. I know some smart people that have trouble doing calculations without paper. My wife was a math major in college, and she sometimes has problems doing calculations in her head.
... he wasn't even aware that there was something that he could do to arrive at the answer. It's as if he didn't know that math even existed.
The thing that struck me about this guy was that it wasn't even that he couldn't do the math in his head
Close, but I dont think the multiples-of-five is the reason why it works. As someone else said, it has to do with greedy algorithms, and our coin system was specifically designed so that you'd be able to start from the biggest coin and work your way down. We went over it briefly in one of my CS courses. I think it may have been that each coin must be at least twice the value of the previous coin. Something like that.
And as many people have mentioned, the current system is probably the best because of the ease of addition/subtraction. An 18-cent coin would be a nightmare for most minimum-wage cashiers. The only problem with our current monetary system is that inflation has made pennies freakin worthless.
This article is a complete waste of time. This might be a fun paper for a discussion about coinage, but it fails horribly when taken as practical advice.
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The US does not need another coin. Indeed, the *opposite* is true. If you get rid of the penny, you can increase efficiency tremendously, to only 2.75 coins per transaction, and a whopping 45% of transactions would require 2 or fewer coins!
Many people oppose the elimination of the penny, but bear with me for a moment. Consider the following issues:
- Pennies cannot be used in vending machines, and therefore are not as "spendable" as all the other coins.
- Prices will not rise as people think they will; they will fall instead! Everything that is priced at $n.99 will now be $n.95 instead (marketers HATE to price in round dollars because it makes their prices look higher). All other numbers will be rounded to the nearest $n.n5.
- The US government makes 12 billion pennies at a cost of $100 million each year (http://www.retirethepenny.org/), which could be put to better use than filling up my coin jar.
- Half of these pennies will disappear from circulation within a year! (http://www.shepherd-express.com/shepherd/19/41/n
- Counting out pennies costs the economy an estimated $20 billion in productivity annually (http://www.retirethepenny.org/)
- The U.S. Mint loses $8 million a year manufacturing pennies. (http://www.shepherd-express.com/shepherd/19/41/n
Think about it - do you *really* want another coin in your pocket? Thank God that politicians don't listen to us all the time!
-Mark
It doesn't matter what the denomination is. As long as change has to be made, some patrons will receive the wrong change.
Lots of cashiers don't know how to make change. Many have been trained to do it wrong. The most common error is the cashier puts the large bill the customer just handed them into the drawer before giving the customer change and watching them count it. There used to be a little slot between the plastic guard and the metal cash register enclosure that was perfect for temporarily storing that large bill in customer sight. When the customer looks at you after counting his money, pause to see whether he questions it, then put the large bill in the drawer and close it.
Adding this momentary delay before putting the customer's large bill in the drawer and closing it, protects the cashier and the customer from being short changed.
I've seen managers put large bills in the drawer before I counted my change. One gave me change for $10 instead of change for a $20. I'm a creature of habit. When I hand a cashier a large bill, I always say, "outta twenty" or whatever the bill is. I'm sure I did that with this one. But she'd already put the bill in the drawer and insisted upon a recount of the drawer and by the time she did, my food was cold. That is not the way to do things. When I pointed out her mistake, she lost her temper. Then I lost mine.
I was trained on older cash registers to do things this way by a store manager who was very particular about this. He's been in business for more than 30 years and says he's never had a dispute with a customer over incorrect change. Way back then, you had to actually count the coin change. Many of the newer cash registers do this for you. I wonder how many of today's cashiers could make change in their heads.
What's my point? Most point of sale problems concerning change making are due to lack of skill and/or poor training of the cashier. Using more efficient denominations or pricing items to the nearest buck won't fix this.
Wansu, th' chinese sailor
Life gets more complicated when coming originally from Germany, where VAT is always included in the advertized price (for end-users, not for businesses) and going to Japan, where 5% tax is mostly added. About 90% of all times I need to add 5% at the cashier. In the other cases I don't need to. Now that makes calculation the price to pay complicated.