Division of Fractions: Methods, Properties & Examples

We have learned about divisions in our previous chapters. The division is dividing something into two equal parts. Fraction has a numerator and denominator. We can do Division of Fraction by multiplying the first fraction by the reciprocal of the second fraction. 

Finding the reciprocal (reversing the numerator and denominator) of the second fraction is the first step in dividing fractions. Then multiply the two numerators. This article will learn the division of fractions and their real-life applications.

Fractions and Its Types

A fraction is a number that represents a part of the whole. The whole may be a single object or a group of objects. A fraction is written as \(\frac{p}{q}\) Where \(p\) and \(q\) are whole numbers and \(q \ne 0.\) Numbers such as \(\frac{1}{2},\frac{2}{3},\frac{4}{5},\frac{{11}}{7}\) are known as the fractions. The number below the division line is called the denominator. It tells us how many equal parts a whole is divided into. The number above the line is called the numerator. It tells us how many equal parts are taken or considered.

For example: Draw a circle with any suitable radius. Then, divide the circle into three equal parts (sectors).

Now, 
Have a look at the figure again.

If two parts of the circle are shaded, we can say two-thirds \(\left( {\frac{2}{3}} \right)\) of the circle is shaded and one-third \(\left( {\frac{1}{3}} \right)\) of the circle is not.

Types of Fractions

There are various types of fractions that we have explained in detail below:

1. Proper Fraction: A fraction in which the numerator is less than the denominator is called a proper fraction.
Example, \(\frac{1}{3},\frac{7}{9},\frac{{13}}{{25}}\)

2. Improper Fraction: A fraction in which the numerator is greater than or equal to its denominator is called an improper fraction.
Example: \(\frac{8}{3},\frac{17}{9},\frac{{13}}{{4}}\)

3. Mixed Fraction: A mixture of a whole number and a proper fraction is called a mixed fraction or a mixed number.
Example: \(1\frac{1}{3},4\frac{7}{9},7\frac{{13}}{{25}}\)

4. Unit Fraction: A proper fraction having \(1\) as a numerator and a positive integer as the denominator is called a unit fraction. The unit function is the reciprocal of a positive integer.
Example: \(\frac{1}{3},\frac{1}{9},\frac{{1}}{{25}}\)

5. Like and Unlike Fractions: Fractions with the same denominator are called like fractions.
Example: \(\frac{1}{7},\frac{5}{7},\frac{{6}}{{7}}\).
Fractions with different denominators are called, unlike fractions.
Example: \(\frac{1}{7},\frac{5}{71},\frac{{6}}{{25}}\)

6. Equivalent Fractions: If two or more fractions have the same value, they are called equivalent or equal fractions.
Example: \(\frac{1}{3},\frac{3}{9},\frac{{6}}{{18}}\) and \(\frac{9}{{27}}\) are an equivalent fraction as \(\frac{1}{3} = \frac{3}{9} = \frac{6}{{18}} = \frac{9}{{27}}\).

7. Decimal Fractions:
A decimal fraction is a fraction whose denominator is a power of \(10\) or a multiple of \(10\) like \(100, 1,000, 10,000,\) etc.
For example, \(\frac{3}{{10}},\frac{4}{{100}},\frac{{13}}{{10}},\frac{9}{{1000}}\) are all decimal fractions.

Division of Fractions

Let us learn the facts of the division of fractions in detail.

Example of the division of fractions: A camp was organised in a school in which \(12\) students participated. However, the camp leader wanted to divide them into groups of \(2\) students. How many groups were there?

There were \(6\) groups which were got by the division of \(12\) by \(2.\) That is \(12÷2=6,\) which means there are six \({\rm{2’s}}\) in \(12.\)

If the camp leader distributes \({\rm{6}}\,{\rm{litres}}\) of water in \(\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{litres}}\) water bottles to the students, then how many students will get water bottles? This means finding how many \(\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{litres}}\) are there in \({\rm{6}}\,{\rm{litres}}\). For this, we need to calculate \({\rm{6}} \div \frac{1}{2}.\)

Let us describe the situation.

It means that if you share \({\rm{6}}\,{\rm{litres}}\) of water into \({\rm{1}}\,{\rm{litre}}\) bottles, \(6\) persons can get water. But if you share it in \(\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{litre}}\) water bottles, \(12\) persons can get water. If you share it in \(\frac{{\rm{1}}}{{\rm{4}}}\,{\rm{litre}}\) water bottles, \(24\) persons can get water. That is

Generally, dividing a number by a fraction is the same as multiplying that number by the reciprocal of the fraction.

The reciprocal of a fraction is just interchanging the numerator (top number) and the denominator (bottom number). The opposite reciprocal means the negative of the reciprocal number.

Example: The reciprocal of \(\frac{{15}}{{11}}\) is \(\frac{{11}}{{15}}\).

Reciprocal of \(\frac{{4}}{{3}}\) is \(\frac{{3}}{{4}}\).

In the above, when we divide \(6\) by \(\frac{1}{2},\) it simply means that \(6\) is multiplied by the reciprocal of \(\frac{1}{2} = 2.\)

To divide a fraction by another fraction

1. Find the reciprocal of the second fraction. 
2. Multiply the first fraction by the reciprocal of the second fraction. 
3. Simplify the fractions if needed.

Example: \(\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}\)

Example: \(\frac{2}{5} \div \frac{{11}}{4} = \frac{2}{5} \times \frac{4}{{11}} = \frac{8}{{55}}\)

Division of Whole Number by Fraction

To divide a whole number by a fraction is to multiply the whole number by the reciprocal of the fraction.
Example: \(6 \div \frac{1}{9} = 6 \times \frac{9}{1} = 6 \times 9 = 72\)

Division of Fraction by Whole Number

To divide a fraction by a whole number, multiply the denominator of the fraction by the whole number. Simplify the fraction (if required).
Example: \(\frac{1}{3} \div 2 = \frac{1}{{3 \times 2}} = \frac{1}{6}\)

Properties

1. If a non-zero fraction is divided by \(1\) the result is the fractional number itself.
Example: \(\frac{3}{5} \div 1 = \frac{3}{5}\)

2. If zero is divided by a non-zero fraction, then the result is zero.
Example: \(0 \div \frac{2}{5} = 0 \times \frac{5}{2} = 0\)

3. A fraction cannot be divided by zero; hence reciprocal of \(0\) does not exist.
Example: \(\frac{2}{5} \div 0 = \frac{2}{5} \times \frac{1}{0} = \frac{2}{0} = \infty \) (not defined)

4. The product of a fraction and its reciprocal is always \(1.\)
Example: \(\frac{2}{5} \times \frac{5}{2} = \frac{{2 \times 5}}{{5 \times 2}} = \frac{{10}}{{10}} = 1\) (reciprocal of \(\frac{2}{5}\) is \(\frac{5}{2}\))

Addition and Subtraction of Fractions

Depending on the type of fractions, the following methods can be used to add or subtract fractions.

Addition and Subtraction of Like Fractions

To add or subtract like fractions, just add or subtract the numerators and keep the same denominator.
Example: (i) Add \(\frac{5}{7}\)  and  \(\frac{3}{7}\).

\(\frac{5}{7} + \frac{3}{7} = \frac{{5 + 3}}{7} = \frac{8}{7}\) 

Example: (ii) Subtract  \(\frac{9}{17}\)  and \(\frac{3}{17}\).\(\frac{9}{{17}} – \frac{3}{{17}} = \frac{{9 – 3}}{{17}} = \frac{6}{{17}}\)

Addition and Subtraction of Unlike Fractions

For adding and subtracting unlike fractions, find the LCM of the denominators and then convert the fractions into equivalent fractions with this LCM as the common denominator.

Now, add or subtract the equivalent fractions
Example: (i) Add \(\frac{5}{4}\)  and  \(\frac{1}{6}\).
LCM of \(4\) and \(6=12\)

Now, \(\frac{5}{4} = \frac{{5 \times 3}}{{4 \times 3}} = \frac{{15}}{{12}}\) and \(\frac{1}{6} = \frac{{1 \times 2}}{{6 \times 2}} = \frac{2}{{12}}\)

So, \(\frac{{15}}{{12}} + \frac{2}{{12}} = \frac{{17}}{{12}}\)

Example: (ii) Subtract \(\frac{2}{3}\) from  \(\frac{4}{5}\).

LCM of \(3\) and \(5=3×5=15\)

Now, \(\frac{2}{3} = \frac{{2 \times 5}}{{3 \times 5}} = \frac{{10}}{{15}}\) and \(\frac{4}{5} = \frac{{4 \times 3}}{{5 \times 3}} = \frac{{12}}{{15}}\)

So, \(\frac{{12}}{{15}} – \frac{{10}}{{15}} = \frac{2}{{15}}\)

Addition and Subtraction of Mixed Fractions

For adding and subtracting mixed fractions, follow these steps.
1. Convert the mixed fractions into improper fractions.
2. Convert the improper fractions obtained into like fractions.
3. Keep the denominator the same and add or subtract the numerators of the equivalent fractions to obtain a single fraction.
4. Reduce the fraction into its lowest terms, if required, and then convert it again into a mixed fraction.
Example: Solve \(1\frac{3}{7} – \frac{1}{6}\)

\( = \frac{{10}}{7} – \frac{1}{6}\)

\( = \frac{{10 \times 6}}{{7 \times 6}} – \frac{{1 \times 7}}{{6 \times 7}} = \frac{{60}}{{42}} – \frac{7}{{42}}\)

 \( = \frac{{60 – 7}}{{42}} = \frac{{53}}{{42}} = 1\frac{{11}}{{42}}\)

Multiplication of Fractions

When two fractions are multiplied, the numerator and denominators are multiplied separately. The numerator of the first is multiplied with the numerator of the second, and the denominator of the first is multiplied by the denominator of the second.For example: \(\frac{1}{2} \times \frac{3}{4} = \frac{{1 \times 3}}{{2 \times 4}} = \frac{3}{8}\)

Multiplication of a Fraction by a Whole Number

Multiplying a fraction by a whole number means multiplying the whole number by the numerator of the fraction and keeping the denominator the same. After multiplication, simplify the fraction if required to get the product in the simplest form.
For example: \(\frac{4}{7} \times 14 = \frac{{4 \times 14}}{7} = \frac{{56}}{7} = \frac{8}{1}\)

Multiplication of a Fraction Using the Operator ‘of’

The word ‘of’ denotes multiplication. 
For example, \(\frac{2}{4}\) of \(4\) cakes mean \(2\) cakes i.e. \(\frac{2}{4} \times 4 = 2.\)

Solved Examples

Q.1. Solve \(\frac{{20}}{{30}} \div \frac{9}{{30}}.\)
Ans: Given \(\frac{{20}}{{30}} \div \frac{9}{{30}}\)
\(\frac{{20}}{{30}} \div \frac{9}{{30}} = \frac{{20}}{{30}} \times \frac{{30}}{9}\)
\( = \frac{{20}}{9}\).

Q.2. Simplify \(5 \div \left( {\frac{8}{{11}} \div \frac{6}{{11}}} \right).\)
Ans: Given, \(5 \div \left( {\frac{8}{{11}} \div \frac{6}{{11}}} \right)\)
\(5 \div \left( {\frac{8}{{11}} \div \frac{6}{{11}}} \right) = 5 \div \left( {\frac{8}{{11}} \times \frac{{11}}{6}} \right)\)
\( = 5 \div \frac{4}{3}\)
\( = 5 \times \frac{3}{4} = \frac{{15}}{4}\)

Q.3. An oil tin contains \({\rm{7}}\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{litres}}\) of oil which is poured in \({\rm{2}}\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{litres}}\) bottles. How many bottles are required to fill \({\rm{7}}\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{litres}}\) of oil?
Ans: The number of bottles required \(7\frac{1}{2} \div 2\frac{1}{2} = \frac{{7 \times 2 + 1}}{7} \div \frac{{2 \times 2 + 1}}{2} = \frac{{15}}{2} \div \frac{5}{2} = \frac{{15}}{2} \times \frac{2}{5}\) (reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\) )
\(=3\) bottles

Q.4. Solve \(3\frac{8}{9} \div 8\frac{3}{4}.\)
Ans: \(3\frac{8}{9} \div 8\frac{3}{4} = \frac{{3 \times 9 + 8}}{9} \div \frac{{8 \times 4 + 3}}{4}\)
= \( = \frac{{35}}{9} \div \frac{{35}}{4}\)
\( = \frac{{35}}{9} \times \frac{4}{{35}} = \frac{4}{9}\)

Q.5. A rod of length \({\rm{6}}\,{\rm{m}}\) is cut into small rods of length \({\rm{1}}\frac{{\rm{1}}}{{\rm{2}}}\,{\rm{m}}\) each. How many small rods can be cut?
Ans: The number of small rods \( = 6 \div 1\frac{1}{2} = 6 \div \frac{{1 \times 2 + 1}}{2}\)
\( = 6 \div \frac{3}{2}\)
\( = 6 \times \frac{2}{3}\) (Reciprocal of \(\frac{3}{2}\) is \(\frac{3}{2}\))
\(=4\) rods

Summary

In this article, we learned about fractions, the different types of fractions, how to do the fundamental operations on fractions like addition, subtraction, and multiplication by taking a wide range of examples. We have discussed the division of fractions in detail and different cases to divide the fractions. For example, division of a fraction by another fraction, division of the whole number by fraction, and division of a fraction by the whole number.

FAQs

Q.1. How to multiply fractions?
Ans: When two fractions are multiplied, the numerator and denominators are multiplied separately. The numerator of the first is multiplied with the numerator of the second, and the denominator of the first is multiplied by the denominator of the second.
For example: \(\frac{1}{2} \times \frac{3}{4} = \frac{{1 \times 3}}{{2 \times 4}} = \frac{3}{8}\)

Q.2. How to divide fractions with mixed numbers?
Ans: First, convert the mixed number to an improper fraction and then multiply the first fraction with the reciprocal of the second fraction.
Example: \(3\frac{8}{9} \div 8\frac{3}{4} = \frac{{3 \times 9 + 8}}{9} \div \frac{{8 \times 4 + 3}}{4}\)
\( = \frac{{35}}{9} \div \frac{{35}}{4}\)
\( = \frac{{35}}{9} \times \frac{4}{{35}} = \frac{4}{9}\)

Q.3. How to divide a number by a fraction?
Ans: To divide a whole number by a fraction, multiply the whole number by the reciprocal of the fraction.
Example: \(6 \div \frac{1}{9} = 6 \times \frac{9}{1} = 6 \times 9 = 54\)

Q.4. What divided by \(3\) give you \(4\)?
Ans: Let \(x\) divided by \(3\) gives \(4\)
Mathematically, we can write it as \(\frac{x}{3} = 4\)
\(x=4×3=12\)
Hence, when we divide \(12\) by \(3\) then we will get \(4.\)

Q.5. How to divide fractions with whole numbers?
Ans: To divide a fraction by a whole number, multiply the denominator of the fraction by the whole number. Simplify the fraction (if required).
Example: \(\frac{1}{3} \div 2 = \frac{1}{{3 \times 2}} = \frac{1}{6}\)

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