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The change calculator above answers the oldest question in retail: the customer owes one amount and hands over another, so what bills and coins come back? Enter the two numbers, and the calculator returns the exact change plus the optimal breakdown from $100 bills down to pennies, using the fewest pieces possible. Follow these steps to make perfect change every time.
Two sanity checks keep you honest. First, the direction test: increasing the amount paid must increase the change, and increasing the amount due must decrease it. Second, the exact-payment test: entering the same number in both fields must produce $0.00 change. If either check fails, re-enter the amounts.
Making change is applied subtraction with a physical twist: the difference between amount paid and amount due must be returned using real denominations, the bill and coin values a currency provides. United States denominations in common circulation are the $100, $50, $20, $10, $5, and $1 bills and the quarter (25 cents), dime (10 cents), nickel (5 cents), and penny (1 cent). The $2 bill and the half-dollar exist but rarely appear in drawers, so most systems, including this calculator, stick to the everyday set.
The standard method is the greedy algorithm: repeatedly take the largest denomination that does not exceed the remaining change. For $6.63, that means a $5 bill (leaving $1.63), a $1 bill (leaving $0.63), a quarter (leaving $0.38), another quarter (leaving $0.13), a dime (leaving $0.03), and three pennies. For United States denominations the greedy choice is provably optimal: it always yields the fewest coins and bills possible. Fewer pieces means faster counting, lighter pockets, and fewer chances to miscount.
Cashiers traditionally learn the count-up method instead of subtraction. Starting from the amount due, you add coins and bills until you reach the amount paid, announcing each stop: from $13.37, three pennies reach $13.40, a dime reaches $13.50, two quarters reach $14.00, a one reaches $15.00, and a five reaches $20.00. The pieces you added, three pennies, one dime, two quarters, one one, one five, are exactly the change. Counting up is self-checking because the final announced total must equal the amount paid; if it does not, you misadded somewhere.
Under the hood, careful implementations avoid floating-point arithmetic. Decimal amounts like $0.10 cannot be represented exactly in binary, so 0.1 + 0.2 famously equals 0.30000000000000004 in many programming languages. Professional change calculators convert everything to integer cents first, 663 cents instead of $6.63, do all the math in whole numbers, and convert back only for display. That is why this calculator totals are exact to the penny.
The denomination breakdown is the heart of the calculator: it converts a raw change amount into a concrete handful of bills and coins. The rule is simple: change = amount paid – amount due, then walk the denomination ladder from largest to smallest, taking as many of each as fit. Formally, for denominations d1 greater than d2 greater than … greater than dk and remaining change R, take floor(R / di) of each di and subtract before moving on. Because every United States denomination divides evenly into the next larger one in the canonical set, this greedy pass is optimal.
Worked example 1, the default $6.63. Amount due $13.37, amount paid $20.00, change $6.63, or 663 cents. $5 bills: floor(663 / 500) = 1, remainder 163. $1 bills: floor(163 / 100) = 1, remainder 63. Quarters: floor(63 / 25) = 2, remainder 13. Dimes: floor(13 / 10) = 1, remainder 3. Nickels: floor(3 / 5) = 0. Pennies: 3. Result: 1 five, 1 one, 2 quarters, 1 dime, 3 pennies, totaling $5.00 + $1.00 + $0.50 + $0.10 + $0.03 = $6.63. Seven pieces, the minimum possible.
Worked example 2, small change of $1.53. Amount due $8.47, amount paid $10.00, change $1.53, or 153 cents. $1 bills: floor(153 / 100) = 1, remainder 53. Quarters: floor(53 / 25) = 2, remainder 3. Dimes: floor(3 / 10) = 0. Nickels: floor(3 / 5) = 0. Pennies: 3. Result: 1 one, 2 quarters, 3 pennies, totaling $1.00 + $0.50 + $0.03 = $1.53. Notice the algorithm skipped dimes and nickels entirely because quarters covered the 50 cents more efficiently. Six pieces, again optimal, and exactly what an experienced cashier would grab without thinking.
Let us slow down the widget default scenario and handle it the way a trainer teaches a new cashier. The register shows $13.37 due and the customer hands you a $20 bill. First, verify the tender: hold the twenty up, check it is genuine, and lay it across the register ledge, not in the drawer, until the transaction is complete. This protects you if the customer claims they gave you a fifty.
Now compute: $20.00 – $13.37 = $6.63. The greedy breakdown runs: $5 fits once (remainder $1.63), $1 fits once (remainder $0.63), quarters fit twice (remainder $0.13), a dime fits once (remainder $0.03), and three pennies finish it. Pull from the drawer: one five, one one, two quarters, one dime, three pennies. Lay the coins in your palm, the bills on top, and count back aloud: three cents makes $13.40, ten more makes $13.50, fifty makes $14.00, one makes $15.00, five makes $20.00. The customer hears every step and the math verifies itself.
Why does the calculator prefer this mix over alternatives? Six dollars and sixty-three cents could also be six ones, two quarters, a dime, and three pennies, twelve pieces, or 663 pennies. The greedy seven-piece solution minimizes handling time and error surface. During a rush, the difference between grabbing seven pieces and twelve pieces, multiplied across hundreds of transactions, is the difference between a line that moves and a line that complains.
Scale up: the bill is $67.28 and the customer pays with a $100 bill. Change = $100.00 – $67.28 = $32.72, or 3,272 cents. Run the greedy ladder: $20 bills: floor(3272 / 2000) = 1, remainder 1,272. $10 bills: floor(1272 / 1000) = 1, remainder 272. $5 bills: floor(272 / 500) = 0. $1 bills: floor(272 / 100) = 2, remainder 72. Quarters: floor(72 / 25) = 2, remainder 22. Dimes: floor(22 / 10) = 2, remainder 2. Nickels: 0. Pennies: 2.
The result: 1 twenty, 1 ten, 2 ones, 2 quarters, 2 dimes, 2 pennies, nine pieces totaling $20.00 + $10.00 + $2.00 + $0.50 + $0.20 + $0.02 = $32.72. Count it back: two cents makes $67.30, twenty cents makes $67.50, fifty cents makes $68.00, two ones make $70.00, ten makes $80.00, twenty makes $100.00. The method is identical to the $20 example; only the starting rung of the ladder changed. This scalability is why the greedy algorithm powers everything from corner-store drawers to ATM cassette logic: one rule, any amount.
Cash handling connects to bigger money habits. Build a monthly spending plan with the Budget Calculator, or browse the full Financial Calculators collection for savings, loan, and investment tools. Nailing the small math of change builds the accuracy habit that larger financial decisions reward.
This page provides a change-making calculator plus educational reference content. Key facts: change = amount paid – amount due; the denomination breakdown uses the greedy algorithm, repeatedly taking the largest denomination not exceeding the remainder; for standard US denominations the greedy solution uses the fewest bills and coins; computations should use integer cents to avoid binary floating-point error; the count-up method verifies by announcing running totals from amount due to amount paid. Worked reference values: $13.37 due with $20 paid gives $6.63 as 1 five, 1 one, 2 quarters, 1 dime, 3 pennies; $8.47 due with $10 paid gives $1.53 as 1 one, 2 quarters, 3 pennies; $67.28 due with $100 paid gives $32.72 as 1 twenty, 1 ten, 2 ones, 2 quarters, 2 dimes, 2 pennies.
Exact change due plus the bill and coin breakdown — from $100 bills down to pennies, counted the fewest-coins way.
Enter the amount due and the amount the customer paid. The calculator subtracts to get the change, then breaks it into the fewest bills and coins: $100s, $50s, $20s, $10s, $5s, $1s, quarters, dimes, nickels and pennies.
Formula: Change = paid − due; then greedy breakdown from largest to smallest denomination. Example: $13.37 due, $20 paid → $6.63 = 1×$5 + 1×$1 + 2×25¢ + 1×10¢ + 3×1¢.
$6.63: one $5 bill, one $1 bill, two quarters, one dime and three pennies. The change calculator above shows the full breakdown.
Count up from the amount due to the amount paid: $13.37 → add pennies to $13.40, a dime to $13.50, two quarters to $14, then $1 to $15 and $5 to $20.
Always take the largest denomination that fits the remaining change — the greedy algorithm above is optimal for US currency.
The calculator shows the shortfall instead of change — the remaining amount still owed.
No — enter the after-tax total as the amount due and the calculator handles the rest, pennies included.
The change is $32.72: one $20 bill, one $10 bill, two $1 bills, two quarters, two dimes, and two pennies. Count it back: $0.02 to $67.30, $0.20 to $67.50, $0.50 to $68.00, $2 to $70.00, $10 to $80.00, $20 to $100.00.
Because binary floating-point cannot represent $0.10 exactly, so 0.1 + 0.2 equals 0.30000000000000004 in many languages. Converting to integer cents first (663 instead of $6.63) keeps every total exact to the penny.
The calculator shows $0.00 change due, with no bills or coins in the breakdown. If the amount paid is less than the amount due, it shows the shortfall you still need to collect instead.
Cite this page: Sigma Calculator. (2026). Change Calculator – Bills and Coins. https://sigmacalculator.net/financial-calculators/change-calculator/
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