Division is an mathematical operation where the given quantity is split into equal parts.
Mathematically it is given as Division = $\frac{a}{b}$
where a is the numerator and b is the denominator.

Example: If there are 20 ice-creams and 5 friends along. How will you share?
Solution: $\frac{20}{5}$ = 4.
The result is 4. Each person will get 4 ice-creams.

Fraction is a number that represents part of a whole and consists of numerator and denominator. Numerator gives the number of equal parts where as denominator represents the total numbers of parts that make up a whole.

To divide fractions the steps are as follows:
  1. Turn the second fraction upside-down.
  2. Multiply the first fraction by that reciprocal.
  3. Simplify the fraction.
  4. Verify the answer by using the formula given below :
$\frac{a}{b}$ / $\frac{c}{d}$ = $\frac{ad}{bc}$
Long division is a good method for handling complex divisions and is the preferred method when dividing by a number with two or more digits. It works from left to right.

Given below are the steps to be considered while doing long division.
Step 1: Observe the dividend (numerator) and divisor (denominator).

Step 2: Divide the dividend by divisor.
Division = $\frac{Dividend}{Divisor}$

Step 3: Verify the answer by using the following formula:

Dividend = Divisor $\times$ Quotient + Remainder.
A mixed fraction is a whole number and a proper fraction combined. Given below are the steps to divide mixed fractions.
  1. Convert each mixed number to an improper fraction.
  2. Invert the second improper fraction.
  3. Multiply the two numerators as well as two denominators together.
  4. If it is an improper fraction convert the result to a mixed number.
  5. Simplify the mixed number.
The part of the fraction which is obtained when the numerator of the fraction is divided by the denominator of the fraction is expressed in decimals.
Steps are as follows:
  1. Divide it without decimals.
  2. Put the decimal point in the appropriate position and give the solution.
Polynomial is an expression of finite length containing variables and constants and uses only basic arithmetic operators. The steps are as follows:
  1. The first term of the numerator should be divided by the first term of the denominator and put it in the answer
  2. Multiply the denominator by the above answer and place it below the numerator.
  3. Subtract to create a polynomial
  4. Repeat using the new polynomial.
Integer is an number which does not contain fractional or decimal component and includes a set of positive numbers, zero and negative numbers. Sign of an integer is either positive (+) or negative (-), except zero which has no sign.

To divide integers follow this two simple steps:

1. Divide the given pair of integers

$\frac{Numerator}{Denominator}$

  • If a positive integer is divided by a negative integer then the sign of the integer is negative.
  • If two positive integers or two negative integers are divided the sign of the integer will be positive.
2. Verify the answer by using the following formula:

Dividend = Divisor $\times$ Quotient + Remainder.

For example, Divide: $\frac{20}{-4}$
Solution:
Step 1: From the given problem we see that Numerator = 20 and Denominator = -4.

So, $\frac{20}{-4}$ = -5. Remainder is zero.

Step 2: Verify using the above formula

Dividend = Divisor $\times$ Quotient + Remainder.

20 = - 4 $\times$ - 5

20 = 20. Hence proved.

Therefore, $\frac{20}{-4}$ = - 5.
Radicals are in the form of root which can either be a square root, cubic root or any root expressed with the $\sqrt{}$ symbol.

Radical
A radical:
n - The index is a positive integer greater than one.
k - The radicand is a real number.

When dividing radicals divide the numbers outside (O) the radicals and then divide the numbers inside (I) the radicals.

$\frac{O_{1}\sqrt{I_{1}}}{O_{2}\sqrt{I_{2}}}$ = $\frac{o_{1}}{o_{2}} \times \sqrt{\frac{I_{1}}{I_{2}}}$

While simplifying an expression to remove the radical from the denominator, multiply the numerator and the denominator of the fraction by that same radical to create a rational number known as rationalizing the denominator.

Example: Simplify $\frac{9\sqrt{16}}{3\sqrt{4}}$
$\frac{9\sqrt{16}}{3\sqrt{4}}=\frac{9}{3} \times \sqrt{\frac{16}{4}}$

= $3\sqrt{4}$

= 6
Hence $\frac{9\sqrt{16}}{3\sqrt{4}}$ = 6

Solved Examples

Question 1: Divide  $\frac{x^{2} -3x -10}{x +2}$ using polynomials.
Solution:
Step 1:  
How to Divide Polynomials
 

Step 2:  
sdfsdf
 

Question 2: Solve : 6 $\frac{2}{8}$ / 3 $\frac{5}{9}$
Solution:
 
Convert each mixed number to an improper fraction.
$\frac{50}{8}$ / $\frac{32}{9}$

Invert the second improper fraction.
$\frac{9}{32}$

Multiplying the two numerators as well as two denominators together
$\frac{50}{8}$ $\times$ $\frac{9}{32}$ = $\frac{450}{256}$

Convert $\frac{450}{256}$ to a mixed fraction.

$\frac{450}{256}$ = 1$\frac{194}{256}$

= 1 $\frac{97}{128}$  (By simplifying)
 

Question 3: Divide 0.45 by 0.15
Solution:
 
0.45 = $\frac{45}{100}$

           0.15 = $\frac{15}{100}$

Division = $\frac{\frac{45}{100}}{\frac{15}{100}}$

$\Rightarrow$  $\frac{45}{15}$ = 3
 

Question 4: Solve the following radical: $\frac{\sqrt{2}}{\sqrt{5}-\sqrt{3}}$
Solution:
 
Given: $\frac{\sqrt{2}}{\sqrt{5}-\sqrt{3}}$

Multiply and divide the given expression by the conjugate of denominator, i.e., $\sqrt{5} + \sqrt{3}$.

$\frac{\sqrt{2}}{\sqrt{5}-\sqrt{3}} \times \frac{\sqrt{5}+\sqrt{3}`}{\sqrt{5}+\sqrt{3}}$

Simplify

= $\frac{\sqrt{2}\sqrt{5}+\sqrt{2}\sqrt{3}}{(\sqrt{5})^{2}-(\sqrt{3})^{2}}$

= $\frac{\sqrt{10}+\sqrt{6}}{5-3}$

= $\frac{\sqrt{10}+\sqrt{6}}{2}$


Therefore, $\frac{\sqrt{2}}{\sqrt{5}-\sqrt{3}}$ = $\frac{\sqrt{10}+\sqrt{6}}{2}$.