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Partial Products in Maths

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Last updated date: 28th Feb 2024
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What does Partial Product Mean?

In Mathematics, partial products mean multiplying each digit of a number with each digit of another number where each digit maintains its place value. For example, the place value of 4 in 43 would be 40.


Partial products are generally used to multiply larger numbers. With this, you can split the given number into pieces to make the multiplication process easier. Then you add those pieces back together to get the product or result.


Look at the example to understand better.


Let’s try it with 3154.

3154=1003+101+15=300+10+5 (Expanded Form)

=3004+104+54=1200+40+20 Partial Products

= 1200 + 40 + 20 (Adding partial products)

= 1260 Product


Read the steps below to understand partial products in a better way.


multiply using partial product


How Partial Products are Used to Multiply One-Digit Number?

The steps given below show how partial products can be multiplied with one-digit number:


Let’s try it with 5264.


Step 1: First write 526 in expanded form.

To write 526 in expanded form, we will look at the value of each digit in 526.

Hundreds

Tens

Ones

5

2

6


Here,

The value of 5 in 526 is 5 hundreds, which is 5100=500.

The value of 2 in 526 is 2 tens, which is 210=20.

The value of 5 in 526 is 6 ones, which is 61=6.

Therefore, 526 in the expanded form will be

526 = 500 + 20 + 6.


Step 2: Multiply each of these expanded numbers with 4. This will give you partial products.

5004=2000

204=80

64=24


Step 3: Now, add the partial products. This will give you the final product.

2000 + 80 + 24 = 2104

Therefore, 5264=2104


How Partial Products are Used to Multiply Two-Digit Numbers?

The steps given below show how partial products can be multiplied with two-digit numbers:


Let’s try it with 5425.


Step 1: First write 54 and 25 in expanded form.

To write 54 in expanded form, we will look at the value of each digit in 54.

Tens

Ones

5

4


Here,

The value of 5 in 54 is 5 tens, which is 510=50.

The value of 4 in 54 is 4 ones, which is 41=4.

Therefore, 54 in the expanded form will be

54 = 50 + 4

Similarly, we will find the expanded form of 25.

The expanded form of 25=210+51

=20+5

Hence, the expanded form of 54 and 25 is:

54 = 50 + 4

25 = 20 + 5


Step 2: Now we are left with four numbers, i.e., 50, 4, 20, and 5. In this step, multiply each part of 54 by each part of 25 as shown below. This will give you partial products.

5020=1000

505=250

420=80

45=20


Step 3: Now, add the partial products. This will give you the final product.

1000 + 250 + 80 + 20 = 1350

Therefore, 5425=1350.


Similarly, you can use partial products to multiply three-digit numbers, four-digit numbers, and so on.


In short, a partial product is a three-step process of multiplying large numbers. It is a method that splits the factors in a multiplication problem down its parts on the basis of their place value enabling the readers to understand what exactly has been multiplied rather than following the step-by-step process as performed in standard logarithm.

FAQs on Partial Products in Maths

1. What are the benefits of using partial products?

The partial products are used to multiply larger numbers. The partial product is simpler to perform than traditional long digit multiplication as using the place value involves multiplying the values that end in 0, which often results in compatible numbers (the numbers that ‘go together’ and make estimating the answer and calculating problems easier). Partial products also introduce the distributive property to students. A good understanding of partial products helps students to understand distributive property easily, which is widely used throughout Mathematics.

2. What is the relationship between the partial products algorithm and the standard algorithm?

In the partial product algorithm, when multiplying the ones, the ones are written on the first line, when multiplying the tens, the tens are written on the second line, when multiplying the hundreds, the hundreds are written on the third line, and so on. For example,


  3 5 5 

     × 6

…………

      30               (5 ones × 6)

    300               (5 tens × 6)

+1800              (3 hundreds × 6)                                         

----------

  2130

………….                                                                    

In the standard algorithm, in the ones column only the ones are written, the tens are written on top or below the ones column, the hundreds are written on the hundreds column as shown below. For example, 


3 3

3 5 5 

   × 6

…………

2130

………..