Fractions

A fraction is a number that is a part of a group. Fractions are written in the form of $a/b$, where $a$ and $b$ are whole numbers and $b\ne 0$.

The number ' $a$ ' is called the numerator and the number ' $b$ ' is called the denominator. E.g. $\frac{1}{2},\frac{3}{99},\frac{34}{78}$ etc.

• Simplest form of fraction

1.0Types of Fractions

Proper Fractions

Fractions having numerator less than the denominator are called proper fractions. For e.g. $\frac{3}{45},\frac{6}{68},\frac{67}{987}$ etc.

Improper Fractions

Fractions having numerator greater than the denominator are called improper fractions. For e.g. $\frac{46}{9},\frac{3664}{200}$ etc.

Mixed Fractions

Numbers having a whole number part and a fractional part are called mixed fractions. We denote a mixed fraction in the form of $\mathrm{a}\frac{\mathrm{b}}{\mathrm{c}}$. E.g. $4\frac{1}{3},19\frac{31}{60}$ etc.

Decimal Fractions

Fractions having denominator as 10,100 or 1000 or any other higher power of 10 are called decimal fractions. For e.g. $\frac{6}{100},\frac{765}{1000},\frac{6454}{10000}$ etc. Vulgar fractions Fractions having denominator as whole number other than any power of 10 are called vulgar fractions. For e.g. $\frac{5}{31},\frac{546}{8700},\frac{376}{9000}$ etc.

Unit Fractions

The fractions having 1 as numerator are called unit fractions. For e.g. $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ etc.

Simple Fractions

Fractions having both numerator and denominator as whole numbers are called simple fractions. For e.g. $\frac{7}{98},\frac{45}{876},\frac{98}{90},\frac{77}{1000}$ etc. Complex fractions Fractions having either or both the numerator and denominator as fraction or mixed fractions are called complex fractions. For e.g. $\frac{2}{\frac{1}{3}},\frac{2}{5},\frac{1\frac{1}{3}}{3}3\frac{1}{5}$ etc.

Equivalent Fractions

Fractions representing the same value are called equivalent fractions. A few equivalent fractions for $\frac{2}{3}$ are $\frac{4}{6},\frac{6}{9},\frac{8}{12}$ etc. Like and unlike fractions Fractions having the same denominator are called like fractions, whereas fractions having different denominator are called unlike fractions.

For e.g. $\frac{3}{77},\frac{66}{77},\frac{45}{77}$ etc. are like fractions. For e.g. $\frac{7}{6},\frac{8}{5},\frac{4}{7}$ etc. are unlike fractions.

• There is a denominator of one for every number. For e.g., The number 6 can be also be written as $\frac{6}{1}$
• To convert unlike fractions to like fractions, we find the LCM of the denominators of the given fractions and convert each fraction into an equivalent like fraction with the LCM as the denominator.

2.0Addition, Subtraction, Multiplication, Reciprocal and Division of Fractions

Addition and Subtraction

To add or subtract two like fractions, we add or subtract the numerator, and denominator remaining the same.

• To add or subtract mixed fractions, we can first convert them into improper fractions and then perform the operations.
• The second method is to add or subtract the whole number part separately and fractional part separately.

• $(+)+(+)=(+)$ $(-)+(-)=(-)$

• While adding two opposite sign numbers, sign of answer depends on the sign of greater number.

Multiplication of Fractions

Multiplication of a fraction by a whole number

Let us find out $2\times \frac{1}{3}$. This problem can be represented pictorially as

$\therefore 2\times \frac{1}{3}=\frac{1}{3}+\frac{1}{3}=\frac{2}{3}(\therefore$ Multiplication is a repeated addition $)$ Now what does this picture represent?

$\frac{1}{4}+\frac{1}{4}+\frac{1}{4}=3\times \frac{1}{4}=\frac{3}{4}$ Thus, $4\times \frac{1}{5}=\frac{4\times 1}{5}=\frac{4}{5},5\times \frac{2}{3}=\frac{5\times 2}{3}=\frac{10}{3},2\times \frac{7}{5}=\frac{2\times 7}{5}=\frac{14}{5}$

Multiplication of A Fraction by A Fraction

(i) To multiply two or more fractions, convert the mixed fractions (if any) to improper fractions. (ii) The numerator of the required fraction is the product of the numerators of the given fractions and the denominator of the required fraction is the product of the denominators of the given fractions. (iii) Reduce the answer to the lowest terms or while multiplying cancel the common factors (if any) from the numerators and denominators of the given fractions.

Reciprocal of A Fraction

We can obtain the reciprocal of a given fraction by interchanging the numerator and denominator of the fraction. Reciprocal of any non-zero fraction $\frac{a}{b}(a\ne 0,b\ne 0)=\frac{b}{a}$

• The product of a number and its reciprocal is always 1 . E.g., $\frac{a}{b}\times \frac{b}{a}=1$
• The reciprocal (multiplicative inverse) of a proper fraction is an improper fraction.

3.0Division of Two Fractions

If $\frac{a}{b}$ and $\frac{c}{d}$ are two fractions, where $\left(\frac{c}{d}\right)\ne 0$, then $\left(\frac{a}{b}\right)÷\left(\frac{c}{d}\right)=\left(\frac{a}{b}\right)\times \left(\frac{d}{c}\right)$ i.e. the dividend is multiplied by the reciprocal of divisor.

Divide A Fraction by A Whole Number

The rule for the division of a fractional number by a whole number is fraction. $\frac{\text{ Fraction }}{\text{ Whole number }}=$ Fraction $\times$ Reciprocal of whole number.

Division of Whole Number by Fractions

To divide a whole number by a fraction, follow the steps mentioned below. (i) Find the reciprocal of the given fraction. (ii) Multiply the reciprocal with the given whole number. The product will be required answer. The rule of the division of a whole number by a fraction number is The rule of the division of a whole number by a fraction is: $\frac{\text{ Whole number }}{\text{ Fraction }}=$ Whole number $\times$ Reciprocal of Fraction . E.g.: To divide 3 by $\frac{2}{5}$, we just need to multiply 3 by $\frac{5}{2}$ (reciprocal of $\frac{2}{5}$ ). This implies, $3\times \frac{5}{2}=\frac{15}{2}$.

4.0Numericals

• Write the type of fractions in the following numbers: (i) $\frac{7}{9}$ (ii) $2\frac{3}{4}$(iii) $\frac{27}{5}$ Explanation (i) $\frac{7}{9}$ is a proper fraction because denominator is greater than numerator. (ii) $2\frac{3}{4}$ is a mixed fraction because it has a whole number part and a fractional part. (iii) $\frac{27}{5}$ is an improper fraction because denominator is less than numerator.
• Calculate (i) $\frac{3}{11}+\frac{7}{11}$ (ii) $\frac{7}{9}-\frac{2}{9}$ Solution: (i) $\frac{3}{11}+\frac{7}{11}=\frac{3+7}{11}=\frac{10}{11}$ (ii) $\frac{7}{9}-\frac{2}{9}=\frac{7-2}{9}=\frac{5}{9}$ To add or subtract two unlike fractions, we may convert them into equivalent like fractions and then add or subtract.
• Calculate (i) $2\frac{1}{3}+1\frac{1}{2}$ (ii) $5\frac{1}{2}-3\frac{2}{3}$ Solution: (i) I-method Converting to improper fractions, we have $2\frac{1}{3}+1\frac{1}{2}=\frac{7}{3}+\frac{3}{2}$ LCM of 3 and $2=6$ $\therefore \frac{7}{3}+\frac{3}{2}=\frac{7\times 2+3\times 3}{6}=\frac{14+9}{6}=\frac{23}{6}=3\frac{5}{6}$. II-method $2\frac{1}{3}+1\frac{1}{2}=(2+1)+\left(\frac{1}{3}+\frac{1}{2}\right)=3+\frac{2+3}{6}[$ LCM of 2 and $3=6]$ $=3+\frac{5}{6}=3\frac{5}{6}$ III-method $2\frac{1}{3}+1\frac{1}{2}$ $\Rightarrow \frac{7}{3}+\frac{3}{2}$ (LCM of 3 and $2=6$ ) $\Rightarrow \frac{7\times 2}{3\times 2}+\frac{3\times 3}{2\times 3}$ $\Rightarrow \frac{14}{6}+\frac{9}{6}$ (Converting into like fraction) $\Rightarrow \frac{23}{6}=3\frac{5}{6}$ (ii) I-method Converting to improper fractions, we have $5\frac{1}{2}-3\frac{2}{3}=\frac{11}{2}-\frac{11}{3}$ LCM of 2 and $3=6$ $\therefore \frac{11}{2}-\frac{11}{3}=\frac{11\times 3-11\times 2}{6}=\frac{33-22}{6}=\frac{11}{6}=1\frac{5}{6}$ II-method $5\frac{1}{2}-3\frac{2}{3}=(5-3)+\left(\frac{1}{2}-\frac{2}{3}\right)$ $=2+\frac{1\times 3-2\times 2}{6}=2+\frac{3-4}{6}=2+\left(-\frac{1}{6}\right)$ $=\frac{2}{1}-\frac{1}{6}=\frac{2\times 6-1\times 1}{6}$ $=\frac{12-1}{6}=\frac{11}{6}=1\frac{5}{6}$ III-method $5\frac{1}{2}-3\frac{2}{3}=\frac{5\times 2+1}{2}-\frac{3\times 3+2}{3}$ $\Rightarrow \frac{11}{2}-\frac{11}{3}$ (Convert into improper fraction) $\Rightarrow \frac{11\times 3}{2\times 3}-\frac{11\times 2}{3\times 2}$ (LCM of $2\&3=6$ ) $\Rightarrow \frac{33}{6}-\frac{22}{6}$ (Converting into like fraction) $\Rightarrow \frac{11}{6}=1\frac{5}{6}$
• Multiply: (i) $\frac{2}{3}\times 11$ (ii) $\frac{4}{9}\times 15$ Solution: (i) $\frac{2}{3}\times 11=\frac{2\times 11}{3}=\frac{22}{3}=7\frac{1}{3}$ (ii) $\frac{4}{9}\times 15=\frac{4\times 15}{9}=\frac{60}{9}=\frac{20}{3}=6\frac{2}{3}$
• Simplify : (i) $\frac{4}{15}\times \left(\frac{1}{4}+\frac{5}{6}\right)$ (ii) $1\frac{4}{5}\times 1\frac{1}{4}-\frac{3}{4}\times \frac{3}{5}$ Solution: (i) $\frac{4}{15}\times \left(\frac{1}{4}+\frac{5}{6}\right)=\frac{4}{15}\times \left(\frac{3\times 1+5\times 2}{12}\right)=\frac{4}{15}\times \left(\frac{3+10}{12}\right)=\frac{4^1}{15}\times \frac{13}{122_3}=\frac{13}{45}$ (ii) $1\frac{4}{5}\times 1\frac{1}{4}-\frac{3}{4}\times \frac{3}{5}=\frac{9}{/\_1}\times \frac{/{\mathcal{F}}^1}{4}-\frac{3}{4}\times \frac{3}{5}=\frac{9}{4}-\frac{9}{20}=\frac{9\times 5-9\times 1}{20}=\frac{45-9}{20}=\frac{36}{20}=\frac{9}{5}=1\frac{4}{5}$
• Evaluate: (i) $\frac{1}{9}$ of 36 (ii) $\frac{3}{5}$ of 30 (iii) $\frac{1}{4}$ of $\frac{12}{15}$ (iv) $\frac{7}{8}$ of ₹ 56 Solution: (i) $\frac{1}{9}$ of $36=\frac{1}{9}\times 36=4$ (ii) $\frac{3}{5}$ of $30=\frac{3}{5}\times 30=18$ (iii) $\frac{1}{4}$ of $\frac{12}{15}=\frac{1}{4}\times \frac{12}{15}=\frac{1}{5}$ (iv) $\frac{7}{8}$ of ₹ $56=\frac{7}{8}\times ₹56=₹49$
• Find the reciprocal of : (i) 4 (ii) $\frac{2}{5}$ (iii) $7\frac{1}{3}$ (iv) $\frac{1}{7}$ Solution: (i) Reciprocal of 4, i.e., $\frac{4}{1}=\frac{1}{4}$ (ii) Reciprocal of $\frac{2}{5}=\frac{5}{2}$ (iii) Reciprocal of $7\frac{1}{3}=$ Reciprocal of $\frac{22}{3}=\frac{3}{22}$ (iv) Reciprocal of $\frac{1}{7}=\frac{7}{1}=7$
• When half pizza is divided into 3 equal parts, How much part will each person get of a whole pizza? Explanation: A Half When half pizza is divided into 3 equal parts $\frac{\frac{1}{2}}{2}$ $\frac{1}{2}÷3=\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}$

• Divide $\frac{1}{9}$ by 7 . Solution: To divide $\frac{1}{9}$ by 7 , we will find the reciprocal of 7 . Reciprocal of 7 is $\frac{1}{7}$. So we get $\frac{1}{9}\times \frac{1}{7}=\frac{1}{63}$. Therefore, $\frac{1}{9}÷7=\frac{1}{63}$.
• Sergio wants to cut the $\frac{5}{3}$ rd portion of a strip into 8 equal parts. What will be the fraction of each piece after this division? Solution If Sergio will cut $\frac{5}{3}$ portion into 8, it means we need to divide $\frac{5}{3}$ by 8 . Reciprocal of 8 is $\frac{1}{8}$. $\Rightarrow \frac{5}{3}÷8$ $\Rightarrow \frac{5}{3}÷\frac{1}{8}$ $\Rightarrow \frac{5}{24}$ Therefore, the fraction of each piece will be $\frac{5}{24}$
• Simplify: $24÷1\frac{3}{5}$ Explanation: $=24÷\frac{(1\times 5+3)}{5}$ $=24÷\frac{8}{5}$ Reciprocal of fraction of ' $\frac{8}{5}$ ' is ' $\frac{5}{8}$, $=24\times \frac{5}{8}$ $=\frac{24\times 5}{1\times 8}$ $=3\times 5$ $=15$
• Simplify : $49÷\frac{7}{3}$ Solution: Reciprocal of fraction of $\frac{7}{3}$ ' is ' $\frac{3}{7}$, $=49\times \frac{3}{7}$ $=\frac{49\times 3}{1\times 7}$ $=7\times 3$ $=21$
• Solve: (i) $\frac{2}{5}÷\frac{1}{2}$ (ii) $2\frac{3}{4}÷\frac{4}{5}$ (iii) $1\frac{1}{2}÷1\frac{3}{5}÷1\frac{2}{3}$ Explanation: (i) $\frac{2}{5}÷\frac{1}{2}=\frac{2}{5}\times \frac{2}{1}=\frac{4}{5}$ (ii) $2\frac{3}{4}÷\frac{4}{5}=\frac{11}{4}\times \frac{5}{4}=\frac{55}{16}$ (iii) $1\frac{1}{2}÷1\frac{3}{5}÷1\frac{2}{3}=\frac{3}{2}÷\frac{8}{5}÷\frac{5}{3}=\frac{3}{2}\times \frac{5}{8}\times \frac{3}{5}=\frac{9}{16}$
• A man spends $\frac{2}{5}$ of his money and has ₹ 90 left. How much did he have initially? Solution: Let the total amount, he initially had be ₹ 1 . Money spent by him $=\frac{2}{5}$ of $₹1=₹\left(\frac{2}{5}\times 1\right)=₹\frac{2}{5}$ $\therefore$ Money left with him=₹ $\left(1-\frac{2}{5}\right)=₹\frac{3}{5}$ But, it is given that he has ₹ 90 left. $\therefore \frac{3}{5}$ of the whole amount $=₹90$ $\therefore$ Whole amount $=₹\left(90\times \frac{5}{3}\right)=₹150$ Therefore, he had initially ₹ 150 with him.
• A carpenter cuts off from a plank $\frac{5}{12}$ of its length and then $\frac{6}{7}$ of what remains. If the remaining piece is $2\frac{1}{2}\mathrm{m}$ long, find the original length of the plank. Solution: $\therefore$ Let us take the total length of the plank as 1 m . portion of the plank cut off $=\frac{5}{12}\mathrm{m}$ $\therefore$ The portion of the plank left $=1-\frac{5}{12}=\frac{12-5}{12}=\frac{7}{12}\mathrm{m}$ $\therefore$ The portion of the plank cut next $=\frac{6}{7}$ of $\frac{7}{12}\mathrm{m}=\frac{6}{7}\times \frac{7}{12}\mathrm{m}=\frac{1}{2}\mathrm{m}$ The remaining portion of the plank $=1-\left(\frac{5}{12}+\frac{1}{2}\right)$ $=1-\frac{5}{12}-\frac{1}{2}=\frac{12-5-6}{12}=\frac{12-11}{12}=\frac{1}{12}\mathrm{m}$ Given, $\frac{1}{12}$ of the length of the plank $=2\frac{1}{2}\mathrm{m}=\frac{5}{2}\mathrm{m}$. $\therefore$ Length of the plank $=\frac{5}{2}\times \frac{12}{1}=30\mathrm{m}$
• A man buys a box of fruits containing 286 fruits. Out of these $\frac{1}{2}$ of the fruits are apples and the rest are pears. $\frac{4}{13}$ of the pears are rotten. He sells the good pears at ₹ $4\frac{1}{11}$ each. How much money does he receive on selling the good pears? Solution: $\Rightarrow$ Number of apples $=\frac{1}{2}$ of total fruits $=\frac{1}{2}\times 286=143$ $\Rightarrow$ Number of pears $=\left(1-\frac{1}{2}\right)$ of total fruits $=\frac{1}{2}\times 286=143$ $\Rightarrow$ Number of rotten pears $=\frac{4}{13}$ of $143=\frac{4}{13}\times 143=4\times 11=44$ $\therefore$ Good pears $=143-44=99$ $\therefore$ Amount of money received on selling good pears at $₹4\frac{1}{11}$ each $=\left(4\frac{1}{11}\times 99\right)=\left(\frac{45}{11}\times 99\right)=₹(45\times 9)=₹405$.