Change Calculator

Making change looks like simple subtraction, and the amount is. The skill is in the second half: choosing which bills and coins to hand back, and counting them so that both you and the customer can see the change is right. This change calculator does both, shows how the answer changes when pennies are not used, and lets new cashiers practice on random sales until the count-back becomes automatic.

For the other side of the register, the Cash Counter Calculator totals a whole drawer, rolls loose coins and finds the over/short at the end of a shift. The Tip Calculator splits a restaurant bill before anyone reaches for cash.

How to count back change like a cashier

The count-back, or counting up, method avoids subtraction in your head. You start at the price and add pieces until you reach the amount the customer gave you, saying the running total out loud. For a $7.63 sale paid with a $20 bill:

1. Say the price: "$7.63."
2. Pennies to the next nickel: "64, 65."
3. Dimes and nickels to the next quarter: one dime, "75."
4. Quarters to the next dollar: one quarter, "$8."
5. Ones to the next five or ten: "$9, $10."
6. Larger bills to the amount paid: a $10 bill, "and $20."

The customer hears the total climb to exactly what they handed over, which is why the method catches mistakes before anyone leaves the counter. The calculator writes out this script for every sale in its "Count it back" card, and the pieces it uses are always the fewest possible.

The method scales to large bills without getting harder. For a $64.17 sale paid with a $100 bill, the count runs "$64.17... 18, 19, 20" with three pennies, "25" with a nickel, "50, 75, $65" with three quarters, then "$70" with a $5, "$80" with a $10 and "$100" with a $20. Ten pieces, $35.83 in all, and at no point did anyone subtract 64.17 from 100.

The change formula and the fewest-pieces rule

The amount of change is one subtraction. Picking the pieces is a short loop:

Run the last two lines from the largest denomination down to the smallest. In words: take as many of the biggest bill as fit into the change, subtract them, and move down one denomination. Computer scientists call this the greedy algorithm, because it grabs the largest piece it can at every step without looking ahead. For U.S. money it always produces the smallest possible number of bills and coins.

Worked example: a $7.63 purchase paid with a $20 bill

This is the sale loaded in the calculator. The change is $20.00 − $7.63 = $12.37. Working from the largest piece down:

• $12.37 holds one $10 bill, leaving $2.37. A $5 does not fit.
• $2.37 holds two $1 bills, leaving $0.37.
• $0.37 holds one quarter, leaving $0.12.
• $0.12 holds one dime, leaving $0.02. A nickel does not fit.
• $0.02 is two pennies.

That is seven pieces: three bills and four coins. Counted back, the cashier says "$7.63... 64, 65, 75, $8, $9, $10, and $20," handing over two pennies, a dime, a quarter, two $1 bills and the $10. No other combination of U.S. bills and coins makes $12.37 in fewer than seven pieces.

Why largest-first always works with U.S. money

Greedy change-making does not work for every set of coins, so it is worth seeing why it works here. A set of denominations is called canonical when the greedy method is always optimal, and the U.S. coins (1, 5, 10 and 25 cents) are canonical for a reason that fits in a few lines. A fewest-coins handful never contains five pennies (one nickel is fewer coins), two nickels (one dime), three dimes (a quarter and a nickel) or two dimes with a nickel (one quarter). Those rules cap the coins below a quarter at 24 cents, so any amount of 25 cents or more must use as many quarters as fit, which is exactly what greedy takes first. The same argument repeats for dimes, nickels and pennies, and again for the bills.

It is not automatic. The U.S. Mint once made a 20-cent coin, in the 1870s. With a 20-cent piece in the till, greedy would make 40 cents as a quarter, a dime and a nickel, three coins, when two 20-cent pieces would do. Dexter Kozen and Shmuel Zaks proved in 1994 that if greedy ever fails for a coin system, it already fails below the sum of the two largest values, so a finite check settles the question. This calculator's method was checked against an exhaustive search of every amount up to $300, for every combination of the optional pieces ($2 bills, half dollars, dollar coins, $50 and $100 bills, with and without pennies), and greedy matched the true minimum every time.

Change breakdowns for common transactions

Every row below was computed with the calculator's own method. The left half is exact change with pennies; the right half is the same sale when the store rounds the cash total to the nearest nickel. In the breakdowns, Q is a quarter, D a dime, N a nickel and P a penny.

Exact change for these eleven sales takes 78 pieces, 28 of them pennies. The rounded versions take 46. The last row shows the biggest swing: 99 cents in nine coins becomes a single $1 bill once the $99.01 total rounds down to $99.00.

Two patterns are worth knowing at the register. Totals ending in 0 or 5 cents never need pennies in the change when the customer pays with bills. The coin-heaviest change comes from totals ending in 01 or 06 cents: paid with whole dollars, they leave change ending in 99 or 94 cents, the two amounts that need nine coins. And because every row uses the fewest pieces, any handful that uses more for the same amount contains a swap you could make, such as three dimes for a quarter and a nickel.

Rounding to the nearest nickel when pennies run out

The U.S. Mint struck its final circulating penny on November 12, 2025. Pennies remain legal tender, and the Treasury asks retailers to keep giving penny change while they have it. When they cannot, the Treasury's December 2025 guidance allows the final amount of a cash transaction to be rounded to the nearest five cents with symmetrical rounding. The whole rule fits in one table, shown here on a $4.30 base:

Add up the last column and the ups and downs cancel exactly: four endings round down by a total of 6 cents and four round up by a total of 6 cents. That balance is why the method is called symmetrical, and why the Treasury does not expect it to change prices overall. Three details matter at the register. Rounding applies only to cash: cards, checks and other electronic payments are charged to the exact cent. It applies to the final total, after sales tax, and the tax itself is still calculated on the exact amount. And a total of exactly one or two cents may be rounded up to five cents rather than down to nothing.

For now this rests on guidance and state law rather than a federal mandate. Washington's law (Chapter 138, Laws of 2026, in effect since June 11, 2026) allows the same rounding on in-person cash sales and lets a customer with exact change pay the unrounded total. Congress passed the Common Cents Act (H.R. 10167) in September 2026, the House on September 14 and the Senate on September 28. Its text lets a business round either the cash total or the change due by the same rule when exact change cannot be provided, allows rounding in the customer's favor, and leaves noncash payments at the exact amount. As of October 12, 2026, Congress.gov did not yet list it as signed into law, so check its status on congress.gov and your own state's rules before you change how you round.

Worked example: the same sale without pennies

Turn on No pennies and run the $7.63 sale again. The total ends in 3, so it rounds up two cents to $7.65, and the change from $20 is $12.35: a $10 bill, two $1 bills, a quarter and a dime. That is five pieces instead of seven, and the count-back shortens to "$7.65... 75, $8, $9, $10, and $20."

Now suppose the customer, wanting round change, hands over $20.03. The exact change, $12.40, needs no pennies at all, so nothing is rounded and the customer pays the exact $7.63: you return a $10, two $1s, a quarter, a dime and a nickel. If the customer offered $20.01 instead, the exact change would be $12.38, which cannot be made without pennies, so the calculator rounds the change itself to $12.40. Whenever the payment is a multiple of five cents, rounding the total and rounding the change give the same result, apart from the special one- and two-cent totals, so a cashier never has to choose between them.

How many coins change takes

If the cents part of the change is equally likely to be anything from 0 to 99, the greedy method hands out 4.70 coins on average with pennies and never more than nine. Two of those coins, on average, are pennies. Rounding to the nearest nickel removes exactly those two, cutting the average to 2.70 coins and the maximum to five, and half dollars trim each figure further:

In the rounded rows, 98 and 99 cents count as zero coins because they round up to one more dollar. The spread matters as much as the average. With pennies, the most common handful is five coins (19 of the 100 possible cent amounts) and only 5 amounts need one coin or none. Without pennies, two coins and three coins are the most common (25 amounts each), and 20 amounts need one coin or none. Real cash totals are not spread perfectly evenly, so read these as reference figures for comparing coin sets rather than as a forecast for any one store.

How to use the calculator and the practice mode

1. Enter the amount due, after tax.
2. Enter what the customer paid, or tap a quick button: Exact, the next whole dollar, $5, $10, $20, $50 or $100.
3. Choose the pieces you hand out. Turn on No pennies to round cash, add $2 bills, half dollars or dollar coins if your drawer has them, and switch off $50 and $100 bills if your register never gives them as change.
4. Read the result: the change due, the list of bills and coins, and the count-back script.

Cashier practice deals a random sale and payment each time you press New sale. Enter how many of each bill and coin you would hand back and press Check. You get credit for the right amount, a note when you used more pieces than necessary, and a running first-try score; Show answer reveals the fewest pieces. Sales run from 25 cents to $100, and about one in five includes a customer adding coins to get round change, a situation worth practicing because the change no longer comes from a simple bill.

Who this calculator is for

New cashiers learning a register, and the trainers who teach them. Teachers and parents working on money skills, where the count-back doubles as a mental-math exercise. Small businesses, market stalls and event volunteers making change from a cash box with no register to do the arithmetic. Store managers deciding whether to round now that pennies are scarce, who can compare exact and rounded change side by side. And anyone who wants to check that the change in their hand is right.

Key terms

• Amount due: the total the customer owes, after sales tax.
• Amount tendered: the cash the customer hands over.
• Change due: amount tendered minus amount due.
• Denomination: the face value of a bill or coin.
• Count-back (counting up): handing over change while counting up from the price to the amount tendered.
• Greedy method: always taking the largest piece that still fits.
• Canonical coin system: a set of denominations for which the greedy method always gives the fewest pieces.
• Symmetrical rounding: rounding endings of 1, 2, 6 and 7 cents down and 3, 4, 8 and 9 up to the nearest five cents, so the adjustments balance.

Situations that change the answer

• The customer adds coins. $20.13 for the $7.63 sale turns $12.37 into $12.50 (five pieces), and $20.63 turns it into $13.00 (four pieces: a $10 and three $1s). Always enter the full amount handed over.
• Large bills are off. A $9.99 sale paid with a $100 bill leaves $90.01: a $50, two $20s and a penny, four pieces. With $50 and $100 bills switched off it becomes four $20s, a $10 and a penny, six pieces.
• $2 bills and half dollars are on. $2.50 in change drops from four pieces (two $1s and two quarters) to two (a $2 bill and a half dollar).
• A compartment runs empty. The calculator assumes a full drawer. If you are out of $5 bills, replace each $5 with five $1s: $7.51 then takes ten pieces instead of six.
• Pennies are scarce. With No pennies on, totals round once, either the total or the change, never both.
• The money flows the other way. A cash refund is change in reverse: pay back the exact amount with the same fewest-pieces method. The Treasury's FAQ leaves refund policy to each business and asks that cash refunds made without pennies be rounded fairly and transparently.

Two habits prevent most disputes: leave the customer's bill on the register ledge until the change is counted, and count the change into the customer's hand rather than onto the counter.

Limitations and related calculators

This calculator assumes an unlimited supply of every denomination you enable and a correct amount due. It does not compute tax, so figure the total first with the Sales Tax Calculator, and it does not apply discounts, which the Discount Calculator handles. It also leaves out card cash-back, split payments and foreign currency. The rounding follows the Treasury guidance and the bill text described above; your state or city may add rules, and some businesses choose not to round at all. To total a whole drawer after a shift, use the Cash Counter Calculator; for a restaurant bill and tip, the Tip Calculator; and for rounding numbers to any place value, the Rounding Calculator.

Sources

The change amount and the greedy breakdown are arithmetic. The rounding rules, the status of the penny and the history in the examples come from these primary sources:

• Penny Production Cessation FAQs, U.S. Department of the Treasury, December 23, 2025: penny change, symmetrical rounding, cash only, sales tax.
• H.R. 10167, Common Cents Act, GovInfo (bill text, section 3) and its status on Congress.gov.
• Substitute House Bill 2334, Washington State Legislature (Chapter 138, Laws of 2026).
• Ceremonial strike for the final circulating one-cent coin, U.S. Mint, November 12, 2025.
• Circulating Coins, U.S. Mint: today's circulating coins and past denominations such as the 20-cent piece (1875 to 1878).
• The Seven Denominations, U.S. Currency Education Program: the $1, $2, $5, $10, $20, $50 and $100 notes in issue.
• Kozen, D. and Zaks, S., Optimal bounds for the change-making problem, Theoretical Computer Science 123 (1994), 377-388: when the greedy method is optimal.