Partial Product — Definition, Formula & Examples
A partial product is the result you get when you multiply one number by just one part (digit or place value) of another number. You add all the partial products together to find the final product.
In multi-digit multiplication, a partial product is the product obtained by multiplying the entire multiplicand by a single digit of the multiplier, taken at its respective place value. The sum of all partial products yields the complete product of the two factors.
How It Works
When you multiply multi-digit numbers, you break one factor apart by place value and multiply each part separately. Each of those separate multiplications gives you a partial product. After finding all the partial products, you add them together to get the final answer. This is exactly what happens in the standard multiplication algorithm — each row you write down is a partial product.
Worked Example
Problem: Find 23 × 14 using partial products.
Break apart by place value: Write 14 as 10 + 4. You will multiply 23 by each part.
First partial product: Multiply 23 by 4 (the ones digit of 14).
Second partial product: Multiply 23 by 10 (the tens digit of 14).
Add the partial products: Add both partial products to get the final product.
Answer: 23 × 14 = 322. The two partial products are 92 and 230.
Why It Matters
Understanding partial products helps you see why the standard multiplication algorithm works, not just how to follow it. This foundation makes it easier to multiply mentally and to later work with the distributive property in algebra.
Common Mistakes
Mistake: Forgetting to account for place value when writing partial products, such as writing 23 × 1 = 23 instead of 23 × 10 = 230.
Correction: Always consider what place the digit occupies. The 1 in 14 is in the tens place, so it represents 10, not 1.