Methods of Division: The meaning of division is to separate a number into two or more equal parts, classes, areas, groups, and categories. In simple words, the meaning of division is to distribute the whole number to a group in equal parts or make equal parts. Let us suppose a diagonal of a rectangle divides it into two equal triangles. The result of a division may or may not be an integer always. Sometimes, the result of division will be in the form of a fraction or decimal.

The way of dividing something is called a method of division. The methods of division are of three types according to the difficulty level. These are the chunking method or division by repeated subtraction, short division method or bus stop method and long division method. This article will cover one of the most important arithmetic operations called ‘Division or Divide’ and its methods in detail.

Division

In Mathematics, there are four basic arithmetic operations. These are addition, subtraction, multiplication, and division. Based on the problems we are going to solve, the type of arithmetic operation has to be chosen according to that. 

Example-1: To share \(50\) pencils equally among \(25\) children, we need to divide \(50\) by \(25\). So, each child will get \(2\) pencils. 

Example-2: \(20\) divided by \(4\). 

If we take \(20\) objects and put them into four equal groups, there will be \(5\) objects in each group.

Thus, \(20÷4=5\).

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Division Symbol

The division can be represented by the symbol \( \div \), slash (/) or a horizontal line ( _ ). These symbols are used as per convenience while dealing with various types of problems and calculations. Also, \(\frac{a}{b}\) can be read as ‘\(a\) by \(b\)’.

For example, the division of \(60\) by \(12\) can be expressed as:

\(60 \div 12 = 5\)

Or \(\frac{{60}}{{12}} = 5\)

Thus, the result of division is the same in all the above representations.

Terminology Used in Division

Terminology means term. There are four important terms used in division. These are dividend, divisor, quotient and remainder.

Dividend: The number to be divided by another number is called the dividend.

Divisor: The number by which we divide another number (dividend) into equal parts is called the divisor.

Quotient: The result of division is called a quotient.

Remainder: The leftover number after division is called the remainder.

The pictorial representation of the above terminology is given below.

Division Through Number Line

The division is the process of repeated subtraction.

Example: \(18 \div 3\)

In this method, start from the number (dividend) in the number line and keep subtracting the number (divisor) till we reach at \(0\) (remainder), and the number of steps we go on backward counting is the quotient(or result of division).

\(3\) is subtracted repeatedly from 18 using the number line.

When \(3\) is subtracted \(6\) times from \(18\) in the number line, then we get the remainder \(0\).

Thus, \(3\) is subtracted from \(18\), six times.

Hence, \(18÷3=6\), the quotient is \(6\).

Division Formula

\({\bf{Dividend}} = {\bf{Divisor}} \times {\bf{Quotient}} + {\bf{Remainder}}.\)

After division, we can put all the values in the formula to verify or check whether our division is correct or not.

Example: \(200÷25=8\)

Here, dividend \(=200\), Divisor \(=25\), quotient \(=8\) and remainder \(=0\)

Let us put all the above values in the formula,

\({\rm{Dividend = Divisor \times Quotient + Remainder}}\)

\(⇒200=25×8+0\)

\(⇒200÷25=8\)

Hence, our division is correct.

Define Methods of Division

Methods of division can be defined as how we perform the division operation. There are three types of methods. Let us discuss each of them.

Chunking Method

The chunking method is also known as the repeated subtraction method. This method is used for lower grades while teaching basic division. In the chunking method, children will repeatedly subtract the divisor from the dividend until we get 0 or a number less than the divisor.

Short Division Method (Bus Stop Method)

The bus stop method of division is also known as the short division method. It gets its name from the idea that the dividend is sitting inside the bus stop while the divisor waits outside.

The short division is a quick and effective method to work out the division with larger numbers. After the children become comfortable with the chunking method, they will move on to short division as it can be used to solve a division problem with a very large dividend by a one-digit divisor.

Long Division Method

Long Division Method is the most common method used to solve problems based on division. In this method, the divisor is written outside the right parenthesis or the left sidebar, while the dividend is placed within, and the quotient is written above the overbar on top of the dividend.

It is a method that is used when dividing a large number (usually two-digit, three-digits or more) by a one-digit, two-digit (or larger) number. 

Example of Methods of Division

Example of chunking method

Divide \(12÷3\).

Let us understand it by some pictures,

\(12÷3\) means we need to divide \(12\) apples into \(3\) in each group.

Step 1: Subtract or cross out \(3\) apples, now left with \(9\) apples, i.e., \(12-3=9\).

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Step 2: Subtract or cross out another \(3\) apples, now left with \(6\) apples, i.e., \(12-6=3\).

Step 3: Subtract or cross out another \(3\) apples, now left with \(3\) apples i.e. \(12-9=3\).

Step 4: Subtract or cross out another \(3\) apples, now left with \(0\) apples i.e. \(12-12=0\).

To get the result as \(0\), we followed four steps or subtracted \(3\) repeatedly for four times. So, it becomes clear that the answer is \(4\).

Hence, \(12÷3=4\).

Example of Short Division Method

Example: \(90÷6\)

Here, we need to divide \(90\) by \(6\) in the short division method.

Now, \(6×1=6\) and it leaves a remainder \(3\).

The remainder is then passed onto the next number, i.e., \(0\) to make it \(30\).

Now, \(6×5=30\), leaves the remainder \(0\).

 So, when put together, the answer will be \(15\).

Hence, the result is \(15\).

Long Division Examples

Example, \(436÷4\)

1. Here, \(4\) is the first digit of the dividend and it is equal to the divisor. So, \(4 \div 4 = 1.\) So, \(1\) is written on top.
2. Now, subtract: \(4-4=0\),
3. Bring the second digit of the dividend down and place it on the right side of \(0\).
4. Now, \(3<4\). So, we can write \(0\) as the quotient and bring down the next digit of the dividend and place it on the right side of \(3\).
5. Now, we have \(36\) as the new dividend. \(36>4\) and \(36\) is divisible by \(4\). As, \(4×9=36\).
6. Write \(9\) as the quotient. 
7. Now, subtract: \(36-36=0\).
8. Thus, the remainder is \(0,\) and the quotient is \(109.\)

Methods of Division Note

1. If a number is divided by \(1\), the quotient will be the same as the dividend. 

Example: \(98÷1=98\)

2. If the dividend and divisor are the same, then the quotient will be \(1\).

Example: \(45÷45=1\)

3. If \(0\) is divided by any number, the result is \(0\).

Example: \(0÷13=0\)

4. If a dividend is divided by \(0\), then the result is undefined.

Example, \(106÷0=∞\) (undefined)

Division of Fraction

Fractions can be divided. While dividing fractions, the division operator needs to be converted into multiplication. 

Example: Divide \(\frac{3}{5}\) by \(\frac{7}{5}\).

Here, the numerator is \(\frac{3}{5}\) and the denominator is \(\frac{7}{5}\).

Now, \(\frac{3}{5} \div \frac{7}{5}\) can be written as \(\frac{3}{5} \times \frac{5}{7}\) 

Thus, \(\frac{3}{5} \times \frac{5}{7} = \frac{3}{7}\)

Hence, the result is \(\frac{3}{7}\).

Division of Decimal

In mathematics, the division of decimals can be observed in many concepts such as algebra, geometry and other numerical concepts. The division of decimals is quite similar to that of fractions. 

Example: Divide \(0.2\) by \(0.3\).

First, let us convert the given decimal \(0.2\) by \(0.3\) in terms of the fraction.

\(0.2 = \frac{2}{{10}}\)

\(0.3 = \frac{3}{{10}}\)

Now, \(\frac{{0.2}}{{0.3}} = \frac{2}{{10}} \div \frac{3}{{10}}\)

\( = \frac{2}{{10}} \times \frac{{10}}{3}\)

\( = \frac{2}{3}\)

\(=0.666\).

Hence, the result is \(0.666\)

Division of Polynomial

Unlike the numbers and the fractions, polynomials can also be divided by another polynomial. Polynomial division can be performed in two methods. One is the polynomial long division, which is similar to the division of numbers, but polynomial expressions will appear instead of numbers. Another method of dividing polynomials is synthetic division method.

Solved Examples – Methods of Division

Q.1. Divide \(125\) by \(5\) in the short division method.
Ans: Here, we need to divide \(125\) by \(5\) in the short division method.

Now, \(5×2=10\) and it leaves a remainder \(2\).
The remainder is then passed onto the next number, i.e., \(5\) to make it \(25\).
Now, \(5 × 5 = 25\), leaves the remainder \(0\).
So, when put together, the answer will be \(25\).
Hence, the result is \(25\).

Q.2. Find the value of: \([32 + 2 \times 17 + ( – 6)] \div 15\)
Ans:
Given, \([32 + 2 \times 17 + ( – 6)] \div 15\)
Now, \([32 + 34 + ( – 6)] \div 15\)
\( = (66 – 6) \div 15\)
\( = 60 \div 15 = \frac{{60}}{{15}} = 4\)
Hence, the required answer is \(4\).

Q.3. Divide \(0.214\) by \(0.02\).
Ans: First, let us convert the given decimal \(0.214\) by \(0.02\) in terms of the fraction.
\(0.214 = \frac{{214}}{{1000}}\)
\(0.02 = \frac{2}{{100}}\)
Now, \(\frac{{0.214}}{{0.02}} = \frac{{214}}{{1000}} \div \frac{2}{{100}}\)
\( = \frac{{214}}{{1000}} \times \frac{{100}}{2}\)
\( = \frac{{214}}{{20}} = \frac{{107}}{{10}}\)
\(= 10.7\)
Hence, the result is \( 10.7.\)

Q.4. Divide \(\frac{2}{5}\) by \(\frac{5}{7}\)
Ans: Here, the numerator is \(\frac{2}{5}\) and the denominator is \(\frac{5}{7}\).
Now, \(\frac{2}{5} \div \frac{5}{7}\) can be written as \(\frac{2}{5} \times \frac{7}{5}\) 
Thus, \(\frac{2}{5} \times \frac{7}{5} = \frac{{2 \times 7}}{{5 \times 5}} = \frac{{14}}{{25}}\)
Hence, the result is \(\frac{{14}}{{25}} \cdot \).

Q.5. Divide the polynomial \(2{x^2} + 3x + 1\) by \(x + 2\) and verify whether the division is correct or not.
Ans:

So, here the quotient is \(2x – 1\) and the remainder is \(3\).
Also, \((2x – 1)(x + 2) + 3 = 2{x^2} + 3x – 2 + 3 = 2{x^2} + 3x + 1\)
i.e \(2{x^2} + 3x + 1 = (x + 2)(2x – 1) + 3\)
Therefore, \({\rm{Dividend = Divisor \times Quotient + Remainder}}\)
Hence, the answer is correct

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Summary

This article covered what is division, different methods of division, formulas to verify whether the division is correct or not, terminology used in the division, examples of different methods, division through number line, some solved examples etc. It will clear all your division related doubts.

FAQs on Methods of Division

Q.1. How many types of division methods are there?
Ans: There are three types of division methods according to the difficulty level.
(i) Chunking method or division by repeated subtraction method
(ii) Short division method or bus stop method
(iii) Long division method

Q.2. What method of division is preferred the most?
Ans: The long division method is preferred for better understanding to divide any number, especially when the result of the quotient is not an integer.

Q.3. What is the short division method?
Ans: In short division, the steps are performed mentally and are not written down. The short division is normally used with division problems having one-digit divisors.

Q.4. How do you do simple division step by step?
Ans: Let us follow some trick to remember all the steps used in long division. Here is a trick to mastering long division. Use the acronym DMSB, 
D stands for Divide
M stands for Multiply
S stands for Subtraction
B stands for Bring Down
These are all the steps to perform simple division.

Q.5. What are the \(4\) terminologies of division?
Ans: There are \(4\) main terminologies of division. These are dividend, divisor, quotient and remainder.