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Algebra

Divide Fraction

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Division of Fractions

Fractions are a part of everything or whole. We can say that fractions are numbers which are used to denote a whole number that has been divided in equal parts. There are two parts of fractions: the first one is the numerator and the second one is the denominator. The number on the line is called Numerator and the number below the line is called Denominator. We use fractions everyday. Fractions are important because they show you the part of the whole. Fractions are significant for us because they tell us which part of the collection we want.

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How to Divide Fractions

In mathematics, divide fraction is the way of separating a fraction up into one and the same parts. When we need to do division of a fraction by second fraction, then these are the following steps for divide fraction.

Step: 1- Write down the first fraction.

Step: 2- Change ÷ with ×

Step: 3- Write down the reciprocal of the second fraction, reciprocal is a simple process of interchanging the numerator and denominator of the second fraction.

See the following picture to discuss a simple process of dividing fractions.

    \[\dfrac{a}{b} \div \dfrac{c}{d}\]

 

    \[\dfrac{a}{b} \times \dfrac{d}{c} = \dfrac{{a \times d}}{{b \times c}}\]

Dividing Fraction by Fractions

In this, we have to discuss how to divide a fraction with another fraction.

Let us see the methods of dividing a fraction with another fraction.

First Method

  • Write down the first fraction.
  • Multiplying the fraction by the reciprocal of the second fraction.
  • Reduce or simplify the fraction.

Example: 1.

    \[\dfrac{2}{4} \div \dfrac{3}{{15}}\]

 

By the rule write down the first fraction that is,

    \[\dfrac{2}{4}\]

 

Then, multiplying first fractions with reciprocal of second fraction

Reciprocal of the second fraction is a simple process of interchanging the numerator with denominator and denominator with numerator.

    \[\begin{array}{l}\dfrac{2}{4} \div \dfrac{3}{{15}}\\\dfrac{2}{4} \times \dfrac{{15}}{3} = \dfrac{{2 \times 15}}{{4 \times 3}} = \dfrac{{30}}{{12}} = \dfrac{5}{2}\end{array}\]

 

The answer is

    \[{\raise0.7ex\hbox{$2$} \!\mathord{\left/ <!-- /wp:paragraph --> <!-- wp:paragraph --> {\vphantom {2 3}}\right.\kern-\nulldelimiterspace} <!-- /wp:paragraph --> <!-- wp:paragraph --> \!\lower0.7ex\hbox{$3$}}\]

  

Second Method

  • Reduce the fractions themselves.
  • Multiplying the fraction by the reciprocal of the second fraction.

Example: 2. 

    \[\dfrac{{21}}{6} \div \dfrac{{16}}{{20}}\]

 

Reduce the fraction

    \[\dfrac{{21}}{6} = \dfrac{7}{2}\]

and

    \[\dfrac{{16}}{{20}} = \dfrac{4}{5}\]

 

Now, multiply the fractions by the reciprocal of second fraction

The reciprocal of second fraction

    \[\dfrac{4}{5}\]

is

    \[\dfrac{5}{4}\]

 

Multiplying both 

    \[\dfrac{7}{2} \times \dfrac{5}{4} = \dfrac{{7 \times 5}}{{2 \times 4}} = \dfrac{{35}}{8}\]

 

The answer will be the

    \[\dfrac{{35}}{8}\]

 

Division of Fractions with Whole Numbers

In this, we learn how to divide a fraction with the whole number.

Let us see the following steps for dividing a fraction with the whole number.

  • Write down the first fraction, simplify if needed.
  • Write the whole number with the denominator of 1.
  • Multiplying the first fraction by the reciprocal of the second fraction.

Example: 1. Solve this

    \[\dfrac{{16}}{{24}} \div 7\]

 

By the rule, write down the first fraction that is,

    \[\dfrac{{16}}{{24}} = \dfrac{2}{3}\]

 

Whole number with denominator 1,

    \[7 = \dfrac{7}{1}\]

 

Multiplying the fraction by the reciprocal of second fraction that is, 

    \[\dfrac{2}{3} \times \dfrac{1}{7} = \dfrac{{2 \times 1}}{{3 \times 7}} = \dfrac{2}{{21}}\]

 

The answer will be the

    \[\dfrac{2}{{21}}\]

 

Dividing Fraction with Decimals

In this, we learn how to dividing fraction with decimals

Let us see the following examples of dividing fractions with decimals.

Example: 1 solve this

    \[\dfrac{6}{{25}} \div 0.5\]

 

For this, firstly convert the decimal number into fraction from, like this

    \[0.2 = \dfrac{2}{{10}} = \dfrac{1}{5}\]

 

Now, multiplying fraction

    \[\dfrac{6}{{25}} \times \dfrac{5}{1} = \dfrac{{6 \times 5}}{{25 \times 1}} = \dfrac{{30}}{{25}}\]

  

Reduce or simplify the fraction. If needed.

    \[\dfrac{{30}}{{25}} = \dfrac{6}{5}\]

 

The answer will be the

    \[{\raise0.7ex\hbox{$6$} \!\mathord{\left/ <!-- /wp:paragraph --> <!-- wp:paragraph --> {\vphantom {6 5}}\right.\kern-\nulldelimiterspace} <!-- /wp:paragraph --> <!-- wp:paragraph --> \!\lower0.7ex\hbox{$5$}}\]

 

Example: 2 evaluate this

    \[\dfrac{{15}}{{16}} \div 7.25\]

 

    \[\begin{array}{l}\dfrac{{15}}{{16}} \div 7.25\\\dfrac{{725}}{{100}} = \dfrac{{29}}{4}\\\dfrac{{15}}{{16}} \times \dfrac{4}{{29}} = \dfrac{{15}}{{116}}\end{array}\]

 

The answer will be the

    \[\dfrac{{15}}{{116}}\]

 

Division of Fractions and Mixed Numbers

In this, we discuss the division of fractions and mixed numbers. Mixed numbers that contain the whole number and proper fraction are called mixed numbers. Let see the following rules for changing the mixed fraction to an improper fraction.

If we want to convert

    \[4\dfrac{1}{2}\]

into improper fraction

Rule: 1. multiple the denominators with mixed number like this,

    \[2 \times 4 = 8\]

 

Rule: 2. now adding the resulting number with numerators like this,

    \[8 + 1 = 9\]

 

Rule: 3. the fraction will be

    \[\dfrac{9}{2}\]

 

Now we explain how to divide fraction with mixed fraction, 

Let us see the example.

    \[\dfrac{6}{{21}} \div 3\dfrac{2}{5}\]

 

First convert the mixed fraction into improper fraction.

    \[3\dfrac{2}{5} = \dfrac{{5 \times 3 + 2}}{5} = \dfrac{{17}}{5}\]

 

Dividing fraction by fraction

    \[\begin{array}{l}\dfrac{6}{{21}} \div \dfrac{{17}}{5}\\\dfrac{6}{{21}} \times \dfrac{5}{{17}} = \dfrac{2}{7} \times \dfrac{5}{{17}} = \dfrac{{2 \times 5}}{{7 \times 17}} = \dfrac{{10}}{{119}}\end{array}\]

 

Whole Number ÷ Fraction

A number without a fraction is called the whole number. When we need to divide whole number by fractions following these steps:

Step: 1. Write the whole number with denominator 1. 

Step: 2. Change    with ×, write the reciprocal of the fraction.

Step: 3. Multiple numerator with numerator and denominator with denominator.

Step: 4. Simplify or reduce the fraction.

Example: Simplify this

    \[7 \div \dfrac{{35}}{6}\]

 

    \[\dfrac{7}{1} \div \dfrac{{35}}{6}\]

    \[\dfrac{7}{1} \times \dfrac{6}{{35}} = \dfrac{{7 \times 6}}{{35}} = \dfrac{{42}}{{35}} = \dfrac{6}{5}\]

 

    \[{\raise0.7ex\hbox{$6$} \!\mathord{\left/ <!-- /wp:paragraph --> <!-- wp:paragraph --> {\vphantom {6 5}}\right.\kern-\nulldelimiterspace} <!-- /wp:paragraph --> <!-- wp:paragraph --> \!\lower0.7ex\hbox{$5$}}\]

is the answer.

Properties of Dividing Fractions

Property 1

If we divide non-zero function by 1, we will get the fraction number, itself

Example:

    \[\begin{array}{l}\dfrac{7}{{18}} \div 1\\ = \dfrac{7}{{18}} \times 1\\ = \dfrac{7}{{18}}\end{array}\]

 

Property 2

If we divide the fraction number by itself, then the result is always 1.

    \[\begin{array}{l}\dfrac{9}{{11}} \div \dfrac{9}{{11}}\\\dfrac{9}{{11}} \times \dfrac{{11}}{9} = 1\end{array}\]

 

Conclusion:

In this chapter, we explained dividing fraction by fraction, division of fraction with whole number, dividing fraction with decimals and division fractions of mixed numbers. In this we discussed the properties of dividing fractions

Solved Examples

  1. When

        \[\dfrac{{11}}{9}\]

    is divided by 

        \[\dfrac{{22}}{{27}}\]

    what will it be get?

    \[\begin{array}{l}\dfrac{{11}}{9} \div \dfrac{{22}}{{27}}\\\dfrac{{11}}{9} \times \dfrac{{27}}{{22}} = \dfrac{3}{2}\end{array}\]

 

We will get

    \[{\raise0.7ex\hbox{$3$} \!\mathord{\left/ <!-- /wp:paragraph --> <!-- wp:paragraph --> {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} <!-- /wp:paragraph --> <!-- wp:paragraph --> \!\lower0.7ex\hbox{$2$}}\]

 

  1. When

        \[\dfrac{5}{9}\]

    is dividing by

        \[7\]

     

    \[\begin{array}{l}\dfrac{5}{9} \div 7\\\dfrac{5}{9} \times \dfrac{1}{7} = \dfrac{{5 \times 1}}{{9 \times 7}} = \dfrac{5}{{63}}\end{array}\]

 

  1. Simplify this

        \[\dfrac{9}{5} \div 5\dfrac{3}{2}\]

     

Change the mixed fraction into improper fraction

    \[5\dfrac{3}{2} = \dfrac{{2 \times 5 + 3}}{2} = \dfrac{{13}}{2}\]

 

Multiplying fraction

    \[\dfrac{9}{5} \times \dfrac{2}{{13}} = \dfrac{{9 \times 2}}{{5 \times 13}} = \dfrac{{18}}{{65}}\]

 

  1. Simplify this

        \[4\dfrac{3}{2} \div \dfrac{3}{2}\]

     

    \[\begin{array}{l}4\dfrac{3}{2} \div \dfrac{3}{2}\\4\dfrac{3}{2} = \dfrac{{2 \times 4 + 3}}{2} = \dfrac{{11}}{2}\end{array}\]

 

    \[\begin{array}{l}\dfrac{{11}}{2} \div \dfrac{3}{2}\\\dfrac{{11}}{2} \times \dfrac{2}{3} = \dfrac{{11}}{3}\end{array}\]

 

  1. Simplify this 

    \[\begin{array}{l}5 \div \dfrac{{15}}{{11}}\\\dfrac{5}{1} \times \dfrac{{11}}{{15}} = \dfrac{{11}}{3}\end{array}\]

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FAQs

1. What is a fraction?

Ans. A fraction is a term used to determine the parts of a whole object.

2. What are the names of the parts of the fractions?

Ans: Fractions are divided into two parts namely numerator and denominator.

3. What are the steps for dividing fraction by fraction?

Ans: 1. Write down the first fraction.

2. Change into, multiple numerators with numerator and denominator with denominator.

3. Simplify or reduce the fraction.

4. What are the steps for dividing whole numbers with fractions?

Ans: 1. Write the whole number with the denominator 1.

2. Change  ÷to, multiple numerators with numerator and denominator with denominator.

3. Simplify the fraction.

5. What are the properties of dividing fractions?

Ans: 1. If we divide a non-zero fraction by 1, then we get the fraction number itself.

2. When we divide a fraction by itself, then the result is always 1.

References 

https://www.cuemath.com/numbers/division-of-fractions/

https://www.mathsisfun.com/fractions_division.html

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