- What is Multiplication?
- What is Place Value?
- How can we state the Value of each Digit in a Number using the Place Value System?
- What are partial products?
- Multiplication using an area model and partial products
- Multiplication using place value and partial products
- Solved Examples
- Frequently Asked Questions
What is Multiplication ?
Multiplication is the process of calculating the result when a number is multiplied several times. The product is the result of multiplication, and each of the numbers that are multiplied is referred to as a factor of the product.
What is Place Value?
Every digit in a number has a place value. The value of a digit in a number based on its position in the number is known as its “place value.”
For example, let’s find the place value of each digit in the number 19.
The number 19 can be written as “10 + 9”, or “1 tens + 9 ones.” So, the place value of 9 is “ones”, and the place value of 1 is “tens.”
How can we state the Value of each Digit in a Number using the Place Value System?
A place value chart usually assists us in determining and comparing the place value of digits in any number. One shift to the left, increases the value of a digit by 10 times. For instance, we move from the ones to the tens, or from the tens to the hundreds, or from the hundreds to the thousands. The indication here is that as we move from one place to the next towards the left in the chart, the value increases ten times.
Here’s an example of how a place value chart can aid in determining a number’s place value in the millions.
In another example, let’s look at the number 4762158.
4 is in the millions place, so its place value is 4000000.
7 is in the hundred thousand place, so its place value is 700,000.
6 is in the ten thousands place, so its place value is 60,000.
2 is in the thousands place, so its place value is 2000.
1 is in the hundreds place, so its place value is 100.
5 is in the tens place, so its place value is 50.
8 is in the ones place, so its place value is 8.
What are partial products?
In a multiplication operation, partial products are found by breaking down a factor based on place value into ones, tens, hundreds, and so on, and multiplying each of these by the other factor.
Multiplication using an area model and partial products
The area model method is based on a simple equation used for calculating the area of a rectangle. We use the formula length times width to obtain the total area, and this is represented as, ‘Length × Width = Area’.
The steps for drawing an area model are:
Step 1: Write the numbers in their expanded form.
Step 2: Now, draw the rectangular blocks to represent the expanded form.
Step 3: Write the multiplication of each number in its own block.
Step 4: Add all the products that are in the block, then write the result.
The area model, along with partial products, can be used to determine the product of a multiplication equation.
For example, let’s find the multiplication result of 194 x 5 using area models and partial products.
Write the number 194 in its expanded form and draw the area model to represent the products:
1 9 4
× 5
_______
5 0 0 (5 × 100)
4 5 0 (5 × 90)
+ 2 0 (5 × 4)
_________
9 7 0
So, the product of 194 and 5 is 970.
Multiplication using place value and partial products
The place value, along with partial products, can be used to determine the product of a multiplication equation.
Solved Examples
Example 1: Determine the place value of each digit in the number 39.
Solution: The number 39 can be written as 30 + 9 or, 3 tens + 9 ones. So, the place value of 9 is ones, and the place value of 3 is tens.
Example 2: Find the multiplication result of 324 x 5 using an area model and partial products.
Write the number 324 in its expanded form and draw the area model to represent the partial products:
3 2 4
× 5
_______
1 5 0 0 (5 × 300)
1 0 0 (5 × 20)
+ 2 0 (5 × 4)
_________
1 6 2 0
So, the product of 325 and 5 is 1620.
Example 3: Find the product of the given multiplication: 55 × 2
Solution:
Step 1:
Tens Ones
5 5
× 2
__________
1 0 0 <———- partial product
Multiply the ones by the tens: 2 ones × 5 tens = 2 × 50 = 100
Step 2:
Tens Ones
5 5
× 2
__________
1 0 <———- partial product
Multiply the ones by the ones: 5 ones x 2 ones = 5 x 2 = 10.
5 5
× 2
__________
1 0 0
1 0
_________
1 1 0
We have two partial products here, 100 and 10.
Just add the two partial products to get the answer: 100 + 10 = 110
So, 2 × 55 = 110.
Example 4: There are 7 boxes of candy in a car. Each box contains a dozen candies. How many candies are there in total in the car?
Solution:
1 dozen = 12 candies
Total candies = 12 × 7
Step 1:
Tens Ones
1 2
× 7
__________
7 0 <———- partial product
Multiply the tens by the ones: 1 tens × 7 ones = 10 × 7 = 70
Step 2:
Tens Ones
1 2
× 7
__________
1 4 <———- partial product
Multiply the ones by the ones: 2 ones x 7 ones = 2 x 7 = 14
Just add the two partial products to get the answer: 70 + 14 = 84
So, the product is 12 × 7 = 84
Example 5: Find the product of the given multiplication equation – 555 x 2
Solution:
Step 1:
Hundreds Tens Ones
5 5 5
× 2
___________
1 0 0 0 <———- partial product
Multiply the ones by the hundreds: 2 ones × 5 hundreds = 2 x 500 = 1000
Step 2:
Hundreds Tens Ones
5 5 5
× 2
___________
1 0 0 <———- partial product
Multiply the ones by the tens: 2 ones x 5 tens = 2 x 50 = 100.
Step 3:
Hundreds Tens Ones
5 5 5
× 2
___________
1 0 <———- partial product
Multiply the ones by the ones: 5 ones x 2 ones = 5 x 2 = 10
Just add the three partial products to get the product: 1000 + 100 + 10 = 1110.
So, 2 x 555 = 1110.
No, changing the order does not affect the result because the partial product will be the same in every scenario.
The area model for multiplication is a visual model that shows the area of a suitable rectangle as the product of two numbers.