Standard Algorithm — Definition, Formula & Examples
Standard algorithm is the traditional, step-by-step procedure taught in school for performing addition, subtraction, multiplication, or division by working through each place value column, usually from right to left.
A standard algorithm is a conventional, efficient computational procedure that operates digit by digit across place-value columns—ones, tens, hundreds, and so on—using regrouping (carrying or borrowing) to produce an exact result for any whole-number arithmetic operation.
How It Works
You line up the numbers by place value, then work one column at a time, starting from the ones place (rightmost column). When a column's result is 10 or more in addition, you regroup by carrying the extra ten to the next column. In subtraction, if the top digit is smaller than the bottom digit, you borrow from the next column. For multiplication, you multiply by each digit of the bottom number and then add the partial products. For division, you use long division to divide, multiply, subtract, and bring down, repeating until every digit has been processed.
Worked Example
Problem: Use the standard algorithm to add 367 + 485.
Step 1: Line up the numbers by place value and add the ones column: 7 + 5 = 12. Write 2 in the ones place and carry the 1 to the tens column.
Step 2: Add the tens column, including the carried 1: 6 + 8 + 1 = 15. Write 5 in the tens place and carry the 1 to the hundreds column.
Step 3: Add the hundreds column, including the carried 1: 3 + 4 + 1 = 8. Write 8 in the hundreds place.
Answer: 367 + 485 = 852
Another Example
Problem: Use the standard algorithm to subtract 602 − 347.
Step 1: Ones column: 2 − 7 is not possible, so borrow 1 ten from the tens place. But the tens digit is 0, so first borrow from the hundreds: the 6 becomes 5, the 0 becomes 10, then lend 1 ten to the ones: tens digit becomes 9, ones digit becomes 12. Now 12 − 7 = 5.
Step 2: Tens column: after borrowing, the tens digit is 9. Subtract: 9 − 4 = 5.
Step 3: Hundreds column: after borrowing, the hundreds digit is 5. Subtract: 5 − 3 = 2.
Answer: 602 − 347 = 255
Why It Matters
The standard algorithm is a core skill in elementary math (grades 3–6) and is explicitly required by Common Core standards in the Numbers and Operations in Base Ten (NBT) domain. It builds fluency that students rely on throughout middle school, pre-algebra, and everyday tasks like budgeting or calculating measurements. Understanding how regrouping works also deepens place-value knowledge, which is essential for learning decimals and more advanced arithmetic.
Common Mistakes
Mistake: Forgetting to carry (regroup) when a column sum reaches 10 or more.
Correction: After adding each column, check whether the result is 10 or greater. If so, write only the ones digit in that column and carry the tens digit to the next column to the left.
Mistake: Borrowing incorrectly when a digit is 0, such as trying to borrow from 0 without going to the next column first.
Correction: When the digit you need to borrow from is 0, move left until you find a nonzero digit. Reduce it by 1, then convert each 0 in between to 9 (or 10 if it is the column lending to the ones place). Work back to the column where you need the extra value.