How to do long multiplication by hand for three‑digit by two‑digit numbers?
Executive summary
Long multiplication (the standard or grade‑school algorithm) multiplies a three‑digit number by a two‑digit number by multiplying each digit of the multiplicand by each digit of the multiplier and then adding the shifted partial products (this is the historic “multiply every digit by every digit and add” approach) . The practical steps—stacking numbers in columns, multiplying ones then tens, writing a zero to shift the tens partial product, carrying digits, and summing the rows—are the classroom standard and are taught widely with worked examples and practice worksheets [1].
1. The standard layout: stack and align by place value
Begin by writing the larger (three‑digit) number on top and the two‑digit multiplier beneath it, aligning ones, tens and hundreds in columns; the standard algorithm depends on this column alignment so place value stays clear .
2. Multiply the ones digit of the multiplier across the top number (first partial product)
Multiply the ones digit of the two‑digit multiplier by the ones digit of the three‑digit multiplicand, proceed leftward through tens then hundreds, write each resulting digit directly below the line in the ones/tens/hundreds columns, and carry any tens to the next column as usual; this first row of results is the first partial product .
3. Multiply the tens digit of the multiplier and shift (second partial product)
Before computing with the tens digit of the multiplier, write a zero (or shift one place to the left) beneath the ones column so the tens partial product is aligned one digit to the left; then multiply the tens digit across the three digits of the top number (again carrying as needed) and place that second partial product on the next line, properly shifted by the trailing zero .
4. Add the partial products to get the final product
Draw a line under the partial products and add them columnwise, carrying when sums exceed nine; the resulting number is the final product because the algorithm effectively adds the place‑value contributions of each digit multiplication (this summing of shifted partial products is the core of long multiplication) .
5. Concrete worked example (712 × 23) to show the flow
Using the example 712 × 23, multiply 712 by 3 to get the first partial product 2136 (writing 6, carry 3, etc.), write a zero and then multiply 712 by 2 (tens place, so treat as 20) to get 14240, then add 2136 + 14240 = 16376; worked examples like this are given in many teaching resources to illustrate carries, zeros for shifts, and column addition .
6. Common pitfalls, checks and prerequisites
Students should master single‑digit multiplication facts (0–9) before attempting multi‑digit long multiplication because the algorithm multiplies many single digits in sequence; common mistakes include failing to shift the tens partial product (forgetting the zero), misplacing carries, or misaligning columns during the final addition—practice worksheets and step‑by‑step calculators can help spot these errors [2].
7. Alternative methods and educational choices
There are legitimate alternatives to the standard algorithm—grid or box methods, lattice multiplication, and mental decomposition into place‑value partial products—that teach the same principle (multiplying and adding place‑value contributions) and may be easier for some learners; many worksheet sites and educational pages present those options alongside the classic method .
8. Why this method endures and curricular notes
Long multiplication is taught because it scales: the same column method works for larger numbers and matches formal multiplication theory (multiply digits, shift by place value, add), which is why curriculum guides and educational sites continue to present the standard algorithm for multiplying three‑digit by two‑digit numbers .