Regrouping
There are multiplication equations that require us to place a number in another group, and this is called regrouping. We can define regrouping as a process of rearranging groups in order to carry out a mathematical operation. While carrying out multiplication you need to regroup or rearrange the numbers in terms of place value to carry out the operation. For example, 5 × 2 , which is 10, can be regrouped, while the 0 is placed in the ones group, the one or one tens is shifted to the next place value.
Vertical Multiplication
Vertical multiplication is the most common method for multiplying multi-digit whole numbers together. We’ll be able to break down our multi-digit multiplication problem into a series of single-digit multiplication problems using this method.
While performing vertical multiplication, the numbers are arranged according to their place value. It is also necessary to have a thorough understanding of place value in order to fully comprehend vertical multiplication.
1. As we all know, the place value chart begins with the ones place, if we start at the number at the right extreme of a number.
2. To get to the next digit, we simply multiply by ten as we move to the left. The ten’s place is to the left of the ones place, because 1 x 10 = 10.
3. Then we encounter hundreds place in the place value chart (10 x 10 = 100).
4. This pattern will keep repeating indefinitely. To get the next place to the left, we simply multiply the previous place by ten.
Number | Place |
|---|---|
1 | ones |
1 x 10 | 10 tens |
10 x 10 | 100 hundreds |
100 x 10 | 1000 thousands |
1000 x 10 | 10,000 ten thousands |
10,000 x 10 | 100,000 hundred thousands |
100,000 x 10 | 1,000,000 millions |
Steps in Vertical Multiplication
- Stack the factors vertically and line up the digits by place value to create the vertical multiplication equation.
Let’s consider an example, \( 15\times 22 \).
- Draw a horizontal line underneath the bottom number and a multiplication symbol \( “\times ” \) to the left of it.
- We begin our multiplication from the one’s place. And continuing the multiplication from the right to the left. We write the individual answers below the horizontal line after each step of the multiplication, working from right to left. To maintain the proper place value, the placement is important.
- We have to do regrouping (or carrying over) as needed, and ensure that the answer is written according to the correct place value.
- In the answer section, we’ll set up a vertical addition equation and find the sum of the numbers.
Area Model in Multiplication
An area model is a rectangular diagram or model used in mathematics to solve multiplication and division problems. With this method the factors or divisors define the length and width of the rectangle.
To make the calculation simple, we can divide one large area of the rectangle into several smaller boxes using number bonds. After that, we add the total area, which is the product or quotient.
For example, To find the product of 32 x 5 by using the area model.
We start with writing 32 as (30 + 2) which is then multiplied by 5.
- Make a 2 x 1 grid, or a box with two rows and one column. On the top of the grid, we write the multiplicands (30 and 2).
- Each cell has 30 and 2 on the top. Writing the terms of the other multiplicand 5 on the left of the grid.
- Inside the first cell, write the product of the numbers 30 and 5 and write the product of 2 and 5 in the second cell.
- To obtain the final product, combine all of the partial products 150 + 10 = 160.
So, 32 x 5 = 160.
Partial Model in Multiplication
The partial product procedure includes multiplying each digit of a number with each digit of another in order to keep each digit in its original position. The products obtained during the intermediate stages of a multiplication process are then added to complete the process of multiplying the factors. A product obtained by multiplying a multiplicand by one digit of a multiplier with more than one digit is known as the partial product.
When we use the partial products method of multiplication, we are really using the distributive property of multiplication to help us break the factors into smaller parts.
Steps that are followed for the Partial Product Method
Step 1: Organize your factors.
Step 2: Use the distributive property to decompose your factors vertically based on their place value.
Step 3: Add up all the partial products to find your total
For example, To find 25 x 4
We have to write as 25 x 4 = 4 x (20 + 5)
= (4 x 20) + (4 x 5) (adding the partial products)
= 80 + 20
= 100.
Examples
1) Find the product of 5296 x 4?
Solution:
5921 x 4 = ?
Multiplying the ones (4 x 1 = 4)
Multiplying the tens (4 x 2 = 8)
Multiply the hundred and then regroup (4 x 9 hundreds = 36 hundreds, now we have to regroup 36 hundreds as 3 thousands and 6 hundreds)
Now, multiply the thousands and add the regrouped thousands (4 x 5 thousands = 20 thousands, 20 thousands + 3 thousands = 23 thousands which means 2 ten thousands and 3 thousands)
Therefore, 5921 x 4 = 23684.
2) There are eight different movie theaters in a mall. There are 411 seats in each theater. What is the total number of seats in all the theaters?
Solution:
Multiplying 411 x 8 = ?
Multiplying the ones (8 x 1 = 8)
Multiplying the tens (8 x 1 = 8)
Multiplying the hundred and then regroup (8 x 4 hundreds = 32 hundreds, now we have to regroup 32 hundreds as 3 thousands and 2 hundreds)
Therefore, 3288 seats are there in all the theaters in the multiplex.
3) An interstellar object travels 85,700 miles in one hour. In two hours, What is the distance covered by the same object in two hours?
Solution:
Multiplying 85,700 x 2 = ?
Multiplying the ones (2 x 0 = 0)
Multiplying the tens (2 x 0 = 0)
Multiplying the hundred and then regroup (2 x 7 hundreds = 14 hundreds, now we’ve to regroup 14 hundreds as 1 thousands and 4 hundreds)
Multiplying the thousands and adding the regrouped thousands (2 x 5 thousands = 10 thousands, 10 thousands + 1 thousands = 11 thousands which means 1 ten thousands and 1 thousands)
Now, multiplying the ten thousands and adding the regrouped ten thousands (8 x 2 thousands = 16 ten thousands, 16 ten thousands + 1 ten thousands = 17 ten thousands which means 1 hundred thousands and 7 ten thousands)
Therefore, the interstellar object travels 171400 miles in two hours.
4) There are 16 drones that are available for purchase in an electronics store. Each drone costs 236 dollars. What is the total amount of money collected by the store after selling all the drones?
Solution:
Multiplying 236 x 16 = ?
Multiplying 236 by 6, ones. Regrouping as necessary. (6 ones x 236 = 1416 ones)
Multiplying 236 by 1 ten. Regrouping as necessary. (1 ten x 236 = 236 tens = 2360 ones)
Now adding the partial products. 1416 + 2360 = 3776.
Therefore, The store collected $3776 after selling the 16 drones.
5) What is the area of the soccer field that has a length of 140 yd and is 80 yd wide?
Solution:
Multiplying 140 x 80 = ?
Multiplying 140 by 0, ones. Regrouping as necessary. (0 ones x 140 = 0 ones)
Multiplying 140 by 8 ten. Regrouping as necessary. (8 tens x 140 = 1120 tens = 11200 ones)
Now adding the partial products. 0 + 11200 = 11200.
Therefore, the area of the soccer field is 11200 square yards.
6) Each day, drivers from a particular warehouse must deliver 50,000 packages. So today there are 152 trucks in the warehouse, and 360 packages in each truck. The drivers load all the packages onto the trucks and head out to deliver them. Is there a sufficient number of packages delivered by the warehouse drivers for the day?
Solution:
Multiply 152 x 360 = ?
Multiplying 152 by 0, ones. Regrouping as necessary. (0 ones x 152 = 0 ones)
Multiplying 152 by 6 tens. Regrouping as necessary. (6 tens x 152 = 912 tens = 9120 ones)
Multiplying 152 by 3 hundreds. Regrouping as necessary. (3 hundreds x 152 = 456 hundreds = 45600 ones)
Now adding the partial products. 0 + 9120 + 45600 = 54720.
Therefore, The drivers deliver 54720 packages for the day. Hence they drivers have delivered a sufficient number of packages.
7) An employee cleans the rectangular surface of an ice rink that measures 9.6 meters by 8.2 meters. What is the area of the surface of the ice rink?
Solution:
Multiplying 9.6 x 8.2 = ?
Multiplying the tenths by the tenths. (0.6 x 0.2 = 0.12)
Multiplying the ones by the tenths. (9 x 0.2 = 1.8)
Multiplying the tenths by the ones. (0.6 x 8 = 4.8)
Multiplying the ones by the ones. (9 x 8 = 72)
Now, adding the partial products = 0.12 + 1.8 + 4.8 + 72 = 78.72
Therefore, the area of the ice rink = 78.72 sq. meters.
(or)
Using the area model, (9 + 0.6) + (8 + 0.2) = ?
Therefore, the area of the ice rink = 72 + 1.8 + 4.8 + 0.12 = 78.72 sq. meters.
When performing operations like addition and subtraction with two-digit numbers or larger, regrouping is defined as the process of forming groups of tens. Regrouping is the process of rearranging groups in order to carry out an operation. For example, 10 tens can be combined to form 1 hundred, and so on.
A product obtained by multiplying a multiplicand by one digit of a multiplier with more than one digit is known as a partial product.
An area model is a rectangular diagram or model used in mathematics to solve multiplication and division problems. The factors and divisors define the rectangle’s length and width. To make the calculation easier, we can divide one large area of the rectangle into several smaller boxes using number bonds. After that, we add the total area, which is the product or quotient.