Change for a US Dollar for Dummies: Higher Lower Mathematics Content from the guide to life, the universe and everything

Change for a US Dollar for Dummies: Higher Lower Mathematics

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Shopkeepers without an electronic till
Al Zimmermann1 reports that: 'About three years ago I went to a Citibank ATM in midtown Manhattan to withdraw some cash. The machine rejected my request with the following message:
I cannot give you $130 because I only have bills in $50 and $20 denominations. Please choose another amount.
Of course $130 = $50 + 4 × $20.'
– Frank Morgan, Mathematical Association of America, '293 Ways to Make Change for a Dollar'

Once upon a time, there were no computers in US retail stores. We know: it's hard to imagine. Sometimes, there weren't even, gasp, cash registers at the checkout. There are people living today2 who can remember shopkeepers who kept their cash in a drawer, metal container, or even in a cigar box. It was not unknown for purchases to be recorded on slips of paper3. Sums were added by hand! It even happened that percentages were calculated without the use of electronics. Of course, the more savvy and modern stores in those far-off days might post a laminated card with the sales tax schedule on, for quick reference.

In those dark days, retail employees learned a special skill, one that has been largely forgotten these days. They learned how to 'make change' – count up from the amount charged to the amount given by the customer. This is such a fascinating skill that we'd like to share it with you. If you collect a few US coins, you can play along.

US Money: A Primer

20th-century US coinage consists of only six coins: a penny (one cent), a nickel (five cents), a dime (ten cents), a quarter (25 cents), a half-dollar (50 cents), and a dollar (one dollar = 100 cents).

US Coins

Not pictured are the half-dollar and dollar coin, which are rarer. Dollar coins are in great use at gambling casinos, where they are used to play slot machines. This is a very bad habit and ought not to be encouraged, even if it does mean that most people in Las Vegas probably know more about the great suffragist Susan B Anthony than they might have otherwise.

For the purpose of this exercise, we will ignore the dollar coin, which isn't really 'change' because it represents a dollar, and the half-dollar, because This Researcher didn't have one handy. Besides, most people don't want to receive half dollars in their change. They are bulky and don't fit into parking meters or vending machines.

With these coins alone, it is possible to make change for a dollar in 242 different ways4. This has been determined by mathematicians, who are good at this sort of thing. They even make up formulas and write learned papers on the topic. We don't care.

What we want to know is: If a customer hands me a dollar (coin or bill) for an item which costs less than a dollar, how do I give back the right change without molesting a computer or writing anything down on paper? It's easier than you think.

How to Give Back Change

  1. Assume you have the requisite variety of coinage in your cashbox: pennies, nickels, dimes, and quarters.
  2. Start with the amount the item costs.
  3. Use the smallest denomination coins – the pennies – to reach a level where it is possible to move up to the next convenient coin – nickel, dime, or quarter. The aim is to reach a dollar with the fewest coins possible.
  4. Count your way up until you have reached $1.00.


An item costs 69 cents ($0.69). Customer hands you a dollar bill ($1.00).

  1. Set the dollar bill on the counter for transactional transparency. Open your cash drawer.
  2. Say, '69 cents.' Count back a penny. '70.'
  3. Add a nickel. '75.' You are now at 75 cents5. Given the four coins we're working with, the next step is obvious.
  4. Add a quarter. 'And 25 makes $1.00.'
  5. Remember to put the dollar bill in the cash drawer.

You are happy. You have earned 69 cents. The customer is happy: they have 31 cents and an item worth 69 cents (they hope). Successful exchange.

'Wait!' you protest. 'I don't have any quarters in my cash drawer!' Relax.

  1. Say, '69 cents.' Count back a penny. '70.'
  2. Add 3 dimes. Say, '80, 90, a dollar.'

No dimes? Use nickels. If you only have pennies? Prepare to face an annoyed customer. Offer them penny candy instead.

Practice your change-making skills. If you don't have any coins handy, make some out of cardboard, or label some bottle caps. Or use different-coloured beads or marbles.

For each of these purchases, imagine the customer has given you a dollar bill. One, two, three… make change!

  1. Purchase = $0.27
  2. Purchase = $0.33
  3. Purchase = $0.47
  4. Purchase = $0.86
  5. Purchase = $0.426

Extra credit question: The customer owes you $0.41. She hands you a dollar bill and a penny. Don't panic. What do you do7?

What Have We Learned?

  • Life without computers was complicated.
  • Those anti-metric Americans can mess up any decimal system.
  • Mental arithmetic can be fun, if we take a hands-on approach.

1Al Zimmermann is a software designer who designs fiendishly difficult online programming contests.2In 2020.3That's what spindles (office memo spikes) were for: to save receipts.4In comparison there are 4,563 ways to make change for both one Pound and one Euro as £1 and €1 both have the same denominations of smaller amounts (1, 2, 5, 10, 20, 50 pence/cents).5Your aim here is to give back the fewest coins possible, unless the customer specifically asks for smaller change.6Answers:
  1. 3 pennies, 2 dimes, 2 quarters.
  2. 2 pennies, 1 nickel, 1 dime, 2 quarters.
  3. 3 pennies, 2 quarters.
  4. 4 pennies, 1 dime.
  5. 3 pennies, 1 nickel, 2 quarters.
Of course, other combinations are possible. Try them out: pretend your cash drawer is low on dimes, but you've just opened a roll of nickels, etc. Or that the customer has five kids and they all have new penny loafers, and want to fill up those slots on their uppers with shiny new copper coins.
7Give her a dime and two quarters. She's not into pennies. They always end up in the washing machine, anyway.

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