Introduction
Place value
Every digit in a number has a place value.
The value represented by a digit in a number based on its position in the number is known as place value. Here’s an example of the relationship between the place or position of the digits in a number and their place value.
- Place value of digit 6 = 60,000
- Place value of digit 3 = 3,000
- Place value of digit 8 = 8,00
- Place value of digit 2 = 20
- Place value of digit 4 = 4
What is a Partial Product?
Partial products can be used to multiply two or more numbers. The partial product method involves multiplying each digit of a number with each digit of another number, with each digit remaining in its original position. We usually calculate the partial product by using place value. We use the partial product method to find the product of two-digit numbers.
For example, in 13 × 27, 13 can be written in expanded form as (10 + 3) and, similarly, 27 as (20 + 7). Then the product would become
(10 + 3) x ( 20 + 7)
= 10 x (20 + 7) + 3 x ( 20 + 7)
= (10 × 20)+(10 × 7)+(3 × 20)+(3 × 7) (distributive property)
Here, 10 x 20, 10 x 7, 3 x 20, and 3 x 7 are the partial products and the sum of these products, would give the result of multiplication.
How to Find the Product of Two-Digit Numbers Using the Partial Product Method?
If the result obtained by estimation is very close to actual result, then the answer is called reasonable otherwise not.
The process of multiplying a two-digit number by another two-digit number can be carried out as follows:
Step 1: Multiply the tens by tens.
Step 2: Multiply the ones by the tens.
Step 3: Multiply the tens by the ones.
Step 4: Multiply the ones by the ones.
Step 5: Add the partial products.
Example: Find the product: 46 × 21.
Answer:
Step 1: Multiply the tens by the tens.
Step 2: Multiply ones by tens.
Step 3: Multiply tens by ones.
Step 4: Multiply the ones by the ones.
Step 5: Add the partial product.
So, 46 × 21 = 966.
Order of Determining the Partial Products
When we multiply tens by tens, tens by ones, and so on, to calculate the partial products, the order in which they are calculated does not alter the result. For example, we can first determine the partial product of ones with ones, and then tens with ones, and so on. The resultant product will remain the same.
The order doesn’t affect the final product because, subsequent to determining the partial products, we just have to add these, and, as we already know, addition is a commutative operation.
Solved Multiplication of Partial Product Examples
Example 1: Use place value and partial products to find 31 x 24.
Answer:
Step 1: Multiply tens by tens
30 x 20 = 600
Step 2: Multiply tens by ones
30 x 4 = 120
Step 3: Multiply ones by tens
1 x 20 = 20
Step 4: Multiply ones by ones
1 x 4 = 4
Add the partial products
600 + 120 + 20 + 4
= 744
Hence, the product is 744.
Example 2: Finding the missing digits.
Answer: The product of tens by tens is: 4000
4000=☐0 × ☐0
We know, 40 = 10 × 4 or 40 = 8 × 5 .
40 = 10 × 4 is not possible because 10 is a two-digit number and we need only a one-digit number. So, the possible factors of 40 are 8 × 5.
Now, the products of tens by ones are 50 and 480.
There are some cases where you multiply the tens by the ones to get 50 and 480.
- 80 × 6
- 80 × 1
- 6× 50
- 50 × 1
As we can see, 80 × 6 gives 480 and 50 x 1 gives 50, so the missing digits are 5 and 8.
Hence, the missing digits are 5 and 8.
Example 3: Sam has 54 small jars of maple syrup. The capacity of each jar is 60 ml. What is the total quantity of maple syrup that Sam has?
Answer:
Total quantity of syrup = number of jars x capacity of one jar
= 54 x 60
Step 1 : Multiply tens by tens
50 x 60 = 3000
Step 2 : Multiply tens by ones
50 x 0 = 0
Step 3 : Multiply ones by tens
4 x 60 = 240
Step 4 : Multiply ones by ones
4 x 0 = 0
Add the partial products,
3000 + 0 + 240 + 0
= 3240
Hence, Sam has 3240 ml of maple syrup in total.
Example 4:
The table below shows the packets of different types of candies prepared by Mrs. Peterson in one day. How many chocolate candies will she prepare in 10 days?
Each Packet = 12 candies
Answer:
1 packet = 12 candies
The number of chocolate candies prepared in one day
= 7 packets
= 7 x 12
= 84 candies
Chocolate candies prepared in 10 days,
= 84 x 10
Step 1 : Multiply tens by tens
80 x 10 = 800
Step 2 : Multiply tens by ones
80 x 0 = 0
Step 3 : Multiply ones by tens
4 x 10 = 40
Step 4 : Multiply ones by ones
4 x 0 = 0
Add the partial products,
800 + 0 + 40 + 0
= 840
Hence, Mrs. Peterson will prepare 840 chocolate candies in 10 days.
When multiplying two numbers, mostly multi-digit numbers, the product of each digit of one number with each digit of the other number, keeping the digit’s place intact, is known as a partial product.
To multiply two 2-digit numbers using the partial product method, simply add all the partial products of the two numbers.
The partial products can be obtained by multiplying tens by tens, tens by ones, ones by tens, and ones by ones.