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Dividing Fractions

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A fraction is the number expressed in form of $\frac{p}{q}$, where ‘p’ and ‘q’ are whole numbers and q $\neq$ 0. Here we are going to learn about dividing fractions, we mean to find the division of two given fractions.

Dividing fractions is one of the basic arithmetic operation in fractions. When we divide the two fractions, we need to find the inverse of the second fraction and multiply it with the first fraction. To multiply two fractions, we have to multiply the numerator with the numerator and the denominator with the denominator.
The resultant fraction we get is the division of the two given fractions. Now we will convert the resultant fraction into its lowest form.
Divide $\frac{l}{m}$ ÷ $\frac{n}{p}$

Invert the second fraction and change the sign to multiplication.
$\frac{l}{m}$ $\times$ $\frac{p}{n}$

Multiply the fraction and reduced it.

$\frac{l}{m}$ $\times$ $\frac{p}{n}$ = $\frac{lp}{mn}$

Rules for Dividing Fractions

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Rules for division of fractions is explained below:
Step 1: If there is a mixed fraction convert it into improper fraction.
Step 2: Change the divisor sign to multiplication sign.
Step 3: Multiply the first fraction by the inverted divisor fraction (i.e. reciprocal of the second fraction).
Example 1: Divide $\frac{2}{3}$ by $\frac{7}{1}$

Solution:
Keep the first fraction $\frac{2}{3}$ as it is = $\frac{2}{3}$

And then reciprocal the second fraction $\frac{7}{1}$ = $\frac{1}{7}$

Now multiply the both fractions = $\frac{2}{3}$ x $\frac{1}{7}$ = $\frac{2}{21}$
Therefore, the solution is $\frac{2}{21}$.

Dividing fractions with Mixed Numbers

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Mixed numbers consist of an integer followed by a fraction. Method to be followed for dividing fractions with mixed numbers is:
  1. The whole number and the mixed number should be written as an improper fraction.
  2. Write the reciprocal of the divisor and multiply.
  3. If possible simplify.
  4. Perform simple multiplication of numerators and the denominators.
  5. Put the answer in lowest terms.
Example 1: Divide 3$\frac{1}{2}$ and 3$\frac{5}{7}$

Solution: Convert each mixed number to an improper fraction.
3$\frac{1}{2}$ = $\frac{7}{2}$

3$\frac{5}{7}$ = $\frac{26}{7}$

Invert the divisor and multiply

$\frac{7}{2}$ * $\frac{7}{26}$

= $\frac{49}{52}$

The obtained result is a proper fraction and cannot be simplified further.

Therefore $\frac{49}{52}$ is the answer.

Dividing Fractions with Exponents and Variables

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If there are numerical coefficients in the given fraction to be divided then use the power (exponent) rule to divide the variables. When dividing the variable subtract exponents instead of adding.

Example 1: Divide $\frac{5^{18}}{5^{2}}$

Solution: $\frac{5^{18}}{5^{2}}$
As the bases are same we get,

5$^{(18-2)}$

= 5 $^{16}$

Example 2: Divide : $\frac{60m^{5}n^{6}}{15m^{2}n^{2}}$

Solution:
Given: $\frac{60m^{5}n^{6}}{15m^{2}n^{2}}$

We know that $\frac{60}{15}$ = 4

= 4m$^{(5-2)}$n$^{(6-2)}$

= 4m$^{3}$n$^{4}$

Example 3: Solve $\frac{m^{9}}{n^{4}}$ $\div$ $\frac{m^{4}}{n^{2}}$

Solution:
$\frac{m^{9}}{n^{4}}$ $\div$ $\frac{m^{4}}{n^{2}}$

The second fraction needs to be inverted, $\frac{m^{4}}{n^{2}}$

The reciprocal of second fraction is $\frac{n^{2}}{m^{4}}$

= $\frac{m^{9}}{n^{4}}$ x $\frac{n^{2}}{m^{4}}$

= $\frac{m^{9}×n^{2}}{n^{4}×m^{4}}$

To divide the variables, keep the variables and subtract the exponents (powers).

= m$^{(9-4)}$ x n$^{(2-4)}$

= m$^{5}$ x n$^{-2}$

The solution can be rewritten as $\frac{m^{5}}{n^{2}}$.

Divide Fractions with Different Denominators

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Division on whole numbers, mixed numbers and improper fractions is simple and easy. The whole numbers division may result in the whole numbers, finite numbers, and so on.
Given below is a case explained for whole numbers.
The following steps are used to solve a fraction with whole number:
  1. convert a whole number into a fraction, for example, consider 9 in the numerator and 1 as a denominator under 9, so you have $\frac{9}{1}$.
  2. Second fraction is $\frac{5}{2}$, the reciprocal is $\frac{2}{5}$.
  3. Multiply the two denominators and two numerators. The result we get is $\frac{18}{5}$
  • If we divide a fraction by its inverse, then the result will be Square of the fraction. e.g, $\frac{\frac{2}{5}}{\frac{5}{2}}$ = $\frac{2}{5}$ * $\frac{2}{5}$ = ($\frac{2}{5}$)$^{2}$
  • If the fraction is divided by 1, the result is the given fraction itself. So we say that 1 is the multiplicative identity. e.g, $\frac{\frac{9}{5}}{1}$ = $\frac{9}{5}$

Example 1: Divide 6 by $\frac{2}{5}$

Solution:
Step 1: Convert a whole number into a fraction part.

The fraction of 6 is $\frac{6}{1}$

Step 2: Find reciprocal of a newly converted fraction.

Second fraction = $\frac{2}{5}$

Reciprocal = $\frac{5}{2}$

Step 3: Multiply two denominator and two numerator

$\frac{6}{1}$ x $\frac{5}{2}$

$\frac{6*5}{1×2}$

We get, $\frac{30}{2}$ .

Therefore, the solution is 15

Example 2: Divide $\frac{\frac{2}{7}}{\frac{5}{3}}$

Solution: $\frac{2}{7}$ * $\frac{3}{5}$

= $\frac{2*3}{7*5}$

= $\frac{6}{35}$