What is Division?
Division is a mathematical operation denoted by the \(“\div” \) symbol. It can be represented as \(“a~\div~b” \) where a represents the number of objects that are being divided into b groups of equal size.
The following terms are related to Division
- Dividend: The number that is being divided
- Divisor: The number that we divide by
- Quotient: The result of division
They are related in the form of this equation: Dividend = Divisor × Quotient
Another term that is related to division as an operation in mathematics is the remainder.
So, if the divisor does not completely divide the dividend, the number that is left to be divided is called the remainder.
In the case of a remainder, the division operation is written as:
Dividend = Divisor × Quotient + Remainder
For example:
10 ÷ 3 = Quotient 3 – Remainder 1
In this example,
10 → Dividend
3 → Divisor
3 → Quotient
1 → Remainder
The different models for Division
1. The use of equal groups for division:
In this type of division, a number is divided into equal groups of equal size.
For example, let us divide 12 flowers equally into 4 equal-sized baskets. Now, how many flowers does each basket contain?
Solution:
The above problem can be represented as,
So, from 12 flowers, we create 4 equal groups. As you can see, each group contains 3 flowers.
This shows us that, 12 ÷ 4 = 3
2. The use of an area or a rectangle model for division:
An area or a rectangle model is a model in which the divisor and the quotient act as the breadth and length, respectively, of the rectangle.
For example:
Solve 354 ÷ 3 using an area model.
First, we draw a rectangle, whose breadth represents the divisor, which will be 3. Multiples of 3 will be subtracted from 354 until we get a remainder that is less than 354.
A multiple of the divisor represents the area of that part of the rectangle. Subsequently, the entire area of the rectangle is accounted for.
The values written inside the rectangle represent the area of that part. The sum of the areas of all the parts should add up to the area of the rectangle, which is the dividend. The sum of the lengths adds up to the quotient.
The first digit, 3 in 354, represents 3 hundreds. So, \(3\times~100~=~300 \) and therefore, \(\frac{300}{3}~=~100 \).
Hence, we write 100 for the length of the first part.
Now the remaining area is 354 – 300 = 54.
We continue the same process again.
\(3\times~10~=~30 \) , so \(\frac{30}{3}~=~10\).
Hence, we write 10 for the length of the next part. Now we are left with 54 – 30 = 24.
We know that \(3\times~8~=~24 \) , so \(\frac{24}{3}~=~8\). Therefore, we write 8 as the length of the last part, as 24 – 24 = 0.
Now we add the breadths to find the quotient.
Hence, the quotient is 100 + 10 + 8 = 118.
Solved Examples
Example 1: Find: \(244~\div~2 \)
First, we draw a rectangle, whose breadth represents the divisor, which will be 2. Multiples of 2 will be subtracted from 244 until we get a remainder less than 2.
The first digit, 2 in 244, represents 2 hundreds, so \(2\times~100~=~200 \).
Hence, \(\frac{200}{2}~=~100 \). So, we write 100 for the length of the first part.
Now the remaining area is 244 – 200 = 44.
We continue the same process again.
\(2\times~20~=~40 \), so Hence, \(\frac{40}{2}~=~20 \). Hence, we write 20 for the length of the next part. Now we are left with 44 – 40 = 4.
We know that \(2\times~2~=~4 \), so \(\frac{4}{2}~=~2 \). Therefore, we write 2 as the length for the last part, as 4 – 4 = 0.
Now we add the lengths to find the quotient.
Hence, the quotient is 100 + 20 + 2 = 122.
Example 2: Use a model to find the quotient and the remainder of the following division operation: 13 ÷ 3
Solution:
On creating 3 equal groups, we can see that we will be left with 1. Hence, the remainder is 1. The size of each of the groups, that is, the number of items in each group, is the quotient. Hence, 4 is the quotient.
So,
13 ÷ 3 = 4 Quotient – Remainder 1
Example 3: Ethan has 22 pencils. How many bundles of 5 pencils can he make?
Additionally, how many pencils will be left with him?
Solution:
We can group 22 into 5 equal groups, such that each group has 4 items each. However, we will be left with 2 pencils.
22 ÷ 5 = 4 Quotient – Remainder 2
This shows us that Ethan makes 4 bundles of 5 pencils each.
Therefore, Ethan will also be left with 2 pencils.
The division of any number by zero is undefined.
An array model of division is a method in which a number is divided into equal columns or rows. It helps students understand that division is the inverse of multiplication.
Let’s consider an example, \(2\times~5~=~10\).
\(10~\div~5~=~2\)
Or
\(10~\div~2~=~5\)
This means that 10 items can be divided into 5 rows or columns of 2 items each, or 2 rows or columns of 5 items each.
Using this method, the same number (divisor) is subtracted from the given number (dividend) repeatedly until the remainder is zero, or smaller than the number that is being subtracted.
Steps to follow for repeated subtraction division
- Subtract the divisor from the dividend
- Continue subtracting until you get to 0 or a number less than the divisor
- Count the number of times you subtracted the divisor from the dividend
- The count is the required quotient. If the difference is not equal to 0, the final difference is the remainder
For example:
42 ÷ 6 =_________
Solution:
6 is subtracted from 42 repeatedly until we get the difference as zero or a number that is less than the divisor, until this repeated subtraction can not proceed.
42 – 6 = 36
36 – 6 = 30
30 – 6 = 24
24 – 6 = 18
18 – 6 = 12
12 – 6 = 6
6 – 6 = 0
Counting the number of times you have subtracted the divisor from the dividend, will lead you to observe that the subtraction is repeated 7 times.
So, 42 ÷ 6 = 7.