# Partial Products Calculator

Created by Anna Szczepanek, PhD
Reviewed by Wojciech Sas, PhD and Jack Bowater
Last updated: Sep 30, 2022

With the help of our partial products calculator, you can master the partial products algorithm of multiplication. If you're not yet familiar with the partial products method or need a refresher, scroll down to find a short article explaining what it means to multiply using partial products and see step-by-step how to do partial products multiplication using a table or column approach. We also include examples of partial products so that you can see this algorithm in action.

## What is the method of partial products in math?

The partial products algorithm allows pupils to quickly & easily multiply two long numbers by turning a more difficult product into a bunch of smaller problems. This strategy is based on the expanded form of a number and on the distributive property. For instance, we can transform 26 × 43 as follows:

26 × 43 = (20 + 6) × (40 + 3) = (20 × 40) + (20 × 3) + (6 × 40) + (6 × 3)

As you can see, first, we use the expanded form and then apply the distributive property.

Done with multiplication? Discover more about the other arithmetical operations with Omni's calculators:

## How do I do partial products by hand?

The partial products algorithm consists of two main steps:

1. Create partial products by multiplying ones times ones, tens times the ones, ones times tens, tens times tens, etc. For instance, 26 × 43 would give you the following partial products:

• 20 × 40 = 800;
• 20 × 3 = 60;
• 6 × 40 = 240; and
• 6 × 3 = 18.
2. Add partial products together to arrive at the final answer via the partial products method of multiplication. In our example of 26 × 43, we have

26 × 43 = 800 + 60 + 240 + 18 = 1118.

## How to write down the partial products algorithm?

There're two slightly different methods of writing down the partial products method: one that uses a box (table) and another that uses columns. We'll discuss them right now. Note, that our partial products calculator can use both methods - choose the one you prefer!

#### Box approach

When we use the box approach, we prepare a table with the number of rows and columns corresponding to the number of digits in the numbers we want to multiply. That is if we want to multiply:

• A 2-digit number by a 2-digit number, we need a 2 × 2 table;
• A 3-digit number by a 2-digit number, we need a 3 × 2 table;
• A 5-digit number by a 3-digit number, we need a 5 × 3 table; etc.

In the columns of our table, we put the ones, tens, hundreds, etc., of one number, and in the rows, we put the ones, tens, hundreds, etc., of the other number we want to multiply using partial products. For instance, to multiply 43 × 26, we prepare the following table:

$\begin{matrix} \times & 20 & 6 \\ 40 \\ 3 \end{matrix}$

Then we fill the table with the successive partial products:

$\begin{matrix} \times & 20 & 6 \\ 40 & 800 & 240 \\ 3 & 60 & 18 \end{matrix}$

The result of 43 × 26 is the sum of the four fields inside the table:

43 × 26 = 800 + 240 + 60 + 18 = 1118

#### Columns approach

When we use columns to calculate the product using partial products, we start in exactly the same way as in the long multiplication algorithm:

$\begin{array}{rr} & 43 \\ \times & 26 \\ \hline = & \end{array}$

Then we write down under the vertical bar the results of consecutive partial multiplications:

💡 It's crucial you remember to put a sufficient number of trailing zeroes, depending on whether you've performed the multiplication of ones or tens or hundreds, etc.

$\begin{array}{rr} & 43 \\ \times & 26 \\ \hline = & 18 \\ & 240 \\ & 60 \\ & 800 \\ \end{array}$

If you're already familiar with the standard multiplication algorithm, you can easily spot the difference: here, we write down the result (partial product) as it is and do not carry over anything to the next column.

Once we've got all the partial products, it remains to add them all together to get the final result:

$\begin{array}{rr} & 43 \\ \times & 26 \\ \hline = & 18 \\ & 240 \\ & 60 \\ & 800 \\ \hline = & 1118 \end{array}$

As you can see, there's nothing difficult in the partial products multiplication. However, it requires a bunch of calculations. Thankfully, Omni's partial products multiplication calculator can perform them for you!

## How to use this partial products calculator?

It's high time we discussed what is the most effective way of working with Omni's partial products calculator. To master it, follow these steps:

1. Input the numbers you want to multiply with the partial products method.

2. Choose the output method. As we've explained above, there are two popular approaches:

• Table (box); and
• Columns.

Try experimenting with both to choose the one that suits you best!

3. The result of the partial products multiplication will appear at the bottom of the partial products calculator.

## Partial products example

The best way to understand the definition of partial products is to experiment with bigger and bigger numbers. Here, as an example of the partial products algorithm, we'll discuss the partial products multiplication of 3-digit by 3-digit numbers. To see examples involving numbers with more digits, use our partial products multiplication calculator shamelessly!

Example. Let's multiply 432 × 118 using the table approach.

1. Prepare the table:
$\begin{array}{rrr} \times & 100 & 10 & 8 \\ 400 &&& \\ 30 &&& \\ 2 &&& \end{array}$
1. Next, we fill in the table with partial products:
$\begin{array}{rrrr} \times & 100 & 10 & 8 \\ 400 & 40000 & 4000 & 3200\\ 30 &3000 & 300 & 240\\ 2 &200&20&16 \end{array}$
1. Now, we add all the partial products together. There are nine of them, so we have to be careful not to forget anything. Let's go column by column:

(40000 + 3000 + 200) + (4000 + 300 + 20) + (3200 + 240 + 16) = 43200 + 4320 + 3456 = 50976

That's it! We've got the final result:

432 × 118 = 50976.

## FAQ

### What is a partial product in math?

We talk about the partial product when we multiply two numbers bit by bit. That is, instead of performing the whole multiplication all at once, we use the expanded form of the numbers and separately multiply tens by tens, tens by ones, ones by ones, etc. Each of these chunks is called a partial product. We then add them all together to get the final answer.

### How do I use an array to find partial products?

When you use the array (table, box) approach to the partial products multiplication, you prepare a table with rows corresponding to the hundreds, tens, ones of one number and columns corresponding to the hundreds, tens, ones of the other numbers. Then you fill in the table with the results of respective multiplications. These are the partial products. Then you sum these partial products to arrive at the final answer to your problem.

### What are the partial products of 123 × 45?

There are 2 × 3 = 6 partial products in 123 × 45, because you multiply a 3-digit number by a 2-digit number. The partial products are the following:

• 100 × 40 = 4000;
• 100 × 5 = 500;
• 20 × 40 = 800;
• 20 × 5 = 100;
• 3 × 40 = 120; and
• 3 × 5 = 15.

### How do I calculate 28 × 12 by partial products?

To multiply two numbers using partial products, you have to:

1. Compute one-by-one the partial products. In our case, these are:

• 20 × 10 = 200;
• 20 × 2 = 40;
• 8 × 10 = 80; and
• 8 × 2 = 16.
2. Add all the partial products together:

200 + 40 + 80 + 16 = 336,

which is the final answer to 28 × 12.

Anna Szczepanek, PhD
Input two numbers
1st number
2nd number
Output format
Table
Addition via partial products:
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