What are Partial Products? Examples of Partial Product

What is a partial product? Partial products are a perfect way to learn the multiplication of multi-digit numbers. This step-by-step product allows students to use the concept of place values and multiply the numbers. This effective multiplication strategy enhances number sense. So, we learn how to do calculations strategically rather than memorizing a series of steps.

The following article is partial products math, the steps for multiplying two – or three-digit numbers in columns. You will also come across solved examples with explanations and practice problems for you to ace your preparation.  

What is a Partial Product?

A partial product is a product obtained when we multiply a multiplicand by one digit of a multiplier that has more than one digit. 

We break the number into parts to multiply. We then multiply the parts separately and then add them together.

Partial Products Multiplication 

The following example will help you understand the concept of partial products multiplication. 

4  8×     6  To multiply 6 × 48 using the partial products concepts, we will multiply it in parts. So, first, we will write the numbers under one another.

4  8×     64  8 We will begin by multiplying 6 and 8, and we will write down the answer completely, i.e., 48.

4  8×     64  82  4  0 Then we will multiply 6 × 40 and write down the product under the 48. Here we must note that the “4” in the number 48, is in the tens place. So, the 4 stands for 40.

4  8×    64  8+ 2  4  02  8  8Lastly, we will add up the two products.

So, the steps for partial product multiplication with one-digit numbers are as follows:

• First, we will write down the two numbers below the other.
• We will begin by multiplying the one digit of the second number with the one digit of the first number. We will write down the product. 
• Next, we will multiply the one digit of the second number with the tens digit of the first number and write down the product below the first product. 
• Here, we must keep in mind that the tens digit will have a zero in addition to the number written. For example, if the number 36 and 3 is in the tens place, then we must note that 3 stands for 30. The expanded form of 36 = 30 +6.
• Lastly, we will add up the two partial products to obtain our final answer.

Partial Products Multiplication with Three-Digit Numbers

In the previous example, we learned how to multiply two-digit numbers using the concept of partial products. Here is how to multiply three-digit numbers via partial product multiplications.

2  8 4×     2  To multiply 284 × 2 using the partial products concepts, we will multiply it in parts. So, first, we will write the numbers under one another.

2 8  4×     28 We will begin by multiplying 2 × 4, and we will write down the answer, i.e., 8.

2 8  4×     2  81 6  0 Then we will multiply 2 × 80 and write down the product under the 8. Here we must note that the “8” in the number 284 is in the tens place. So, the 8 stands for 80.

2 8  4×    2  81 6 0+4 0 05  6  8Next, we will multiply 2 × 200 and write down the product under 160. Here, the place value of 2 in the number 284 is 200. Lastly, we will add up the three partial products.

Partial Products for Two-Digit Multipliers

Up till now, we have been seeing examples of partial products for one-digit multipliers. What happens when we have numbers wherein the multiplier has more than one digit. Consider the following example to understand the concept of partial products multiplication for multi-digit multiplicands and multipliers.

8 4×     2 1  To multiply 84 × 21 using the partial products concepts, we will multiply it in parts. So, first, we will write the numbers under one another.

8  4×  2  14 We will begin by multiplying 1 × 4, and we will write down the answer, i.e., 4.

8  4×   2  1  48  0 Then we will multiply 1 × 80 and write down the product under the 4. Here we must note that the “8” in the number 84, is in the tens place. So, the 8 stands for 80.

8  4× 2  1  48 08 0 8  4×  2  1  48 08 0+1 6 0 01 7 6  4Next, we will multiply 20 × 4 and write down the product under the 80. Here, the place value of 2 in the number 21 is 20. Lastly, we will multiply 20 and 80 and write the product under 80. Add up the four partial products to get the answer.

So, the steps for partial products multiplication with two-digit numbers are as follows:

• First, we will write down the two numbers, one below the other.
• We will begin by multiplying the one digit of the second number by the one digit of the first number. We will write down the product. 
• Next, we will multiply the one digit of the second number with the tens digit of the first number and write down the product below the first product. 
• We must take note that the tens digit will have a zero in addition to the number written considering its place value. For example, if the number 26 and 2 is in the tens place, then we must note that 2 stands for 20. The expanded form of 26 = 20 +6.
• Next, we will multiply the tens digit of the multiplier with the one digit of the multiplicand. We will write the partial product below the two partial products.
• Lastly, we will multiply the tens digit of the multiplier with the tens digit of the multiplicand and write it under the previous three partial products. 
• We will add up the four partial products to obtain our final answer.

Sample Word Problem

Example: The length of a rectangle is 23 meters. The breadth of the rectangle is 12 meters. What is the area of the rectangle?

Solution: Area of a rectangle = length of the rectangle × breadth of the rectangle

So, we have to multiply 23 × 12.

2 3×     1 26

2  3×  1  264 0

2  3×   1  2  44 03 0

2  3× 1  2  44 03 0+2 0 0 2 7 4

Partial Products Division

Partial products division is similar to partial products multiplication. This division strategy is an easier alternative for the long division process. It is a mental math-based concept that enhances number sense understanding. Students have to use simple subtractions and multiplications to solve the equation until they get down to 0 or close.

This method is also called the partial quotients division method, as we get the quotient in parts. We add up the parts to get the final answer.

The following example of partial products division will help you understand the division approach.

Example: Divide 960 by 6

Steps for partial products division:

• We will begin by figuring out numbers that are easy to work with, such as multiples of 10, 100, and so on. So, we will multiply 6 with 100 and write the product below the dividend.
• Subtract the product from the dividend. So, we take away 600 from 960, and we will be left with 360.
• Now, we will look for another number and repeat the steps. This time we will multiply 6 by 20.  
• Again we will subtract 120 from 360, and we will be left with 240.
• Now that we have 240, we can multiply 6 with 40. We will get 240.
• Subtract the two, and zero remains.
• Lastly, we will add up all the quotients, i.e., 100 + 20 + 40 = 160.
• The answer will be the sum of quotients. 

Practice Problems

Question 1: Multiply the following two-digit numbers:

1. 23 × 3
2. 45 × 9

Question 2: Multiply the following three-digit numbers:

1. 123 × 4
2. 897 × 8

Question 3: Multiply the following four-digit numbers:

1. 5798 × 6
2. 2378 × 9

Question 4: Multiply the following numbers:

1. 56 × 78
2. 12 × 44
3. 567 × 12
4. 45 × 121

Question 5: Emma bought 25 pens for her younger brother. The cost of each pen was $5. Find the total cost of pens using the partial products multiplication method.

Question 6: There are 40 students in a class, and Lily has 560 chocolates to distribute. How many chocolates will each student get? Use the partial products division method.