Partial Products Calculator (Old Method)
See exactly how the partial-products (expanded-form) multiplication method works: the calculator breaks each factor into its place-value parts (hundreds, tens, ones), shows every individual multiplication, and adds them together to give the final product.
Sum of all partial products
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Sum all partial products
How does this calculator work?
Break each factor into hundreds, tens, and ones; multiply every pair of parts; sum the results. For example, 456 × 78 = 400×70 + 400×8 + 50×70 + 50×8 + 6×70 + 6×8 = 28,000 + 3,200 + 3,500 + 400 + 420 + 48 = 35,568.
Formula
How this is calculated
The partial-products method — sometimes called the "old" or "expanded" multiplication method — works by decomposing each factor into its place-value components and then multiplying every combination. For example, 456 is split into 400, 50, and 6; 78 is split into 70 and 8. You then compute 3 × 2 = 6 individual products (400×70, 400×8, 50×70, 50×8, 6×70, 6×8) and sum them all. This makes each multiplication simpler because you are always multiplying rounded, easy numbers.
The method is mathematically identical to the standard long-multiplication algorithm — it simply makes the distributive property explicit. For an m-digit number multiplied by an n-digit number (counting only non-zero leading place values), you get at most m × n partial products. The sum of those products equals the standard result, which you can verify by comparing with the direct product shown.
This approach is widely taught in primary school to build intuition about place value and the distributive law before introducing the more compact long-multiplication notation. The calculator supports integers up to 999,999 × 999,999 — for larger numbers, the number of steps becomes unwieldy and a different algorithm (Karatsuba, for instance) is more efficient.
Frequently asked questions
It breaks each factor into its place-value parts (hundreds, tens, ones), multiplies every pair of parts, then adds all those partial products together. It makes the distributive property explicit and is the same as long multiplication, just laid out differently.
Both apply the distributive property of multiplication over addition. Long multiplication does this compactly with carries; partial products does it explicitly. The total is always identical.
This calculator handles whole integers only, since the place-value decomposition relies on integer digits. For decimals, multiply both numbers by a power of 10 to make them integers, use the tool, then divide the result by the same power of 10.
Also known as
TG we-Calculate Editorial Team. (2026). Partial Products Calculator (Old Method) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/partial-products-old-calculator
TG we-Calculate Editorial Team. "Partial Products Calculator (Old Method)." TG we-Calculate. 2026. https://we-calculate.com/calculator/partial-products-old-calculator.
TG we-Calculate Editorial Team, "Partial Products Calculator (Old Method)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/partial-products-old-calculator
@misc{wecalculate_partial_products_old_calculator, title = {Partial Products Calculator (Old Method)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/partial-products-old-calculator}}, year = {2026}, note = {TG we-Calculate} }
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