\displaystyle \textbf{Fractions}
\displaystyle \text{A number of the form }\frac{a}{b},\text{ where }a\text{ and }b\text{ are natural numbers and }b\ne0,
\displaystyle \text{is called a fraction.}
\displaystyle \text{In the fraction }\frac{a}{b},\ a\text{ is called the numerator and }b\text{ is called the denominator.}
\displaystyle \textbf{Note:}
\displaystyle \text{(1) A fraction represents a part of a whole or a part of a collection.}
\displaystyle \text{(2) The numerator and denominator of a fraction are called its terms.}
\displaystyle \textbf{Example:}
\displaystyle \frac{7}{10}\text{ is a fraction where }7\text{ is the numerator and }10\text{ is the denominator.}
\displaystyle \\

\displaystyle \textbf{Types of Fractions}
\displaystyle \textbf{Decimal Fraction:}
\displaystyle \text{A fraction whose denominator is a power of }10,\text{ such as }10,\ 100,\ 1000,\ldots,
\displaystyle \text{is called a decimal fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{7}{10},\ \frac{71}{100},\ \frac{77}{1000}\text{ are all decimal fractions.}
\displaystyle \\

\displaystyle \textbf{Vulgar Fraction:}
\displaystyle \text{A fraction written in the form }\frac{a}{b},\text{ where }a\text{ and }b\text{ are integers and }b\ne0,
\displaystyle \text{is traditionally called a vulgar fraction or common fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{7}{9},\ \frac{71}{87},\ \frac{77}{876}\text{ are examples of vulgar fractions.}
\displaystyle \\

\displaystyle \textbf{Proper Fraction:}
\displaystyle \text{A fraction whose numerator is less than its denominator is called a proper fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{7}{19},\ \frac{71}{827},\ \frac{771}{876}\text{ are all proper fractions.}
\displaystyle \\

\displaystyle \textbf{Improper Fraction:}
\displaystyle \text{A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{71}{19},\ \frac{1171}{827},\ \frac{2771}{876}\text{ are all improper fractions.}
\displaystyle \\

\displaystyle \textbf{Mixed Number (or Mixed Fraction):}
\displaystyle \text{A number consisting of a whole number and a proper fraction is called a mixed number}
\displaystyle \text{or mixed fraction.}
\displaystyle \textbf{Example:}
\displaystyle 2\frac{1}{3},\ 3\frac{71}{100},\ 4\frac{77}{1000}\text{ are all mixed numbers.}
\displaystyle \\

\displaystyle \textbf{How to Convert a Mixed Fraction into an Improper Fraction?}
\displaystyle \text{Multiply the whole number by the denominator of the fractional part and add the numerator}
\displaystyle \text{of the fractional part to the product. This gives the numerator of the improper fraction.}
\displaystyle \text{The denominator remains the same as the denominator of the fractional part.}
\displaystyle \textbf{Example:}
\displaystyle 9\frac{2}{3}=\frac{9\times3+2}{3}=\frac{29}{3}.
\displaystyle \\

\displaystyle \textbf{How to Convert an Improper Fraction into a Mixed Fraction?}
\displaystyle \text{Divide the numerator of the improper fraction by its denominator.}
\displaystyle \text{The quotient forms the whole number part, the remainder forms the numerator of the}
\displaystyle \text{fractional part, and the divisor forms the denominator of the fractional part.}
\displaystyle \textbf{Example:}
\displaystyle \frac{29}{3}=9\frac{2}{3}.
\displaystyle \\

\displaystyle \textbf{Simple Fraction:}
\displaystyle \text{A fraction whose numerator and denominator are whole numbers is called a simple fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{2}{3},\ \frac{9}{5},\ \frac{7}{70}\text{ are all simple fractions.}
\displaystyle \\

\displaystyle \textbf{Complex Fraction:}
\displaystyle \text{A fraction in which the numerator, denominator, or both contain a fraction is called a complex fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{2}{\left(\frac{3}{4}\right)},\ \frac{\left(\frac{9}{23}\right)}{5},\ \frac{7}{\left(\frac{70}{87}\right)}\text{ are all complex fractions.}
\displaystyle \\

\displaystyle \textbf{Equivalent Fractions:}
\displaystyle \text{Fractions having the same value are called equivalent fractions.}
\displaystyle \textbf{Example:}
\displaystyle \frac{2}{3}=\frac{4}{6}=\frac{6}{9}.
\displaystyle \therefore \frac{2}{3},\ \frac{4}{6},\ \frac{6}{9}\text{ are equivalent fractions.}
\displaystyle \\

\displaystyle \textbf{Like Fractions:}
\displaystyle \text{Fractions having the same denominator are called like fractions.}
\displaystyle \textbf{Example:}
\displaystyle \frac{2}{9},\ \frac{3}{9},\ \frac{5}{9}\text{ are all like fractions.}
\displaystyle \\

\displaystyle \textbf{Unlike Fractions:}
\displaystyle \text{Fractions having different denominators are called unlike fractions.}
\displaystyle \textbf{Example:}
\displaystyle \frac{2}{9},\ \frac{3}{19},\ \frac{5}{29}\text{ are all unlike fractions.}
\displaystyle \\

\displaystyle \textbf{How to Convert Unlike Fractions into Like Fractions?}
\displaystyle \textbf{Step 1: }\text{Find the L.C.M. of the denominators of all the given unlike fractions.}
\displaystyle \textbf{Step 2: }\text{Convert each fraction into an equivalent fraction having a denominator}
\displaystyle \text{equal to the L.C.M. of the denominators.}
\displaystyle \textbf{Step 3: }\text{The resulting fractions have the same denominator and are therefore like fractions.}
\displaystyle \textbf{Example:}
\displaystyle \text{Convert }\frac{3}{8},\ \frac{11}{12}\text{ and }\frac{8}{15}\text{ into like fractions.}
\displaystyle \text{The denominators are }8,\ 12\text{ and }15.
\displaystyle 8=2^3,\qquad12=2^2\times3,\qquad15=3\times5.
\displaystyle \therefore \text{L.C.M.}(8,12,15)=2^3\times3\times5=120.
\displaystyle \frac{3}{8}=\frac{3\times15}{8\times15}=\frac{45}{120}.
\displaystyle \frac{11}{12}=\frac{11\times10}{12\times10}=\frac{110}{120}.
\displaystyle \frac{8}{15}=\frac{8\times8}{15\times8}=\frac{64}{120}.
\displaystyle \therefore \frac{45}{120},\ \frac{110}{120}\text{ and }\frac{64}{120}\text{ are the required like fractions.}
\displaystyle \\

a41

\displaystyle \text{ LCM }= 2 \times 2 \times 3 \times 2 \times \times 5

\displaystyle \text{ Now convert the fractions to 120 denominator}

\displaystyle \frac{3}{8} = \frac{3 \times 15}{120} = \frac{45}{120}

\displaystyle \frac{11}{12} = \frac{11 \times 10}{120} = \frac{110}{120}

\displaystyle \frac{8}{15} = \frac{8 \times 8}{120} = \frac{64}{120}

\displaystyle \textbf{Fractions in Lowest Terms or in Simplest Form}
\displaystyle \text{A fraction is said to be in lowest terms or simplest form if the H.C.F. of its numerator}
\displaystyle \text{and denominator is }1.
\displaystyle \text{A fraction can be reduced to its lowest form by dividing both its numerator and denominator}
\displaystyle \text{by their H.C.F.}
\displaystyle \textbf{Example:}
\displaystyle \text{Reduce }\frac{126}{162}\text{ to its lowest form.}
\displaystyle \text{H.C.F. of }126\text{ and }162=18.
\displaystyle \therefore \frac{126}{162}=\frac{126\div18}{162\div18}=\frac{7}{9}.
\displaystyle \\

\displaystyle \textbf{Comparison of Fractions}
\displaystyle \textbf{Comparison of Fractions with Like Denominators and Unlike Numerators:}
\displaystyle \text{If two fractions have the same denominator, the fraction with the larger numerator is the larger fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{5}{7}>\frac{3}{7}.
\displaystyle \\

\displaystyle \textbf{Comparison of Fractions with Like Numerators and Unlike Denominators:}
\displaystyle \text{If two fractions have the same numerator but different denominators, the fraction with the smaller}
\displaystyle \text{denominator is the larger fraction.}
\displaystyle \textbf{Example:}
\displaystyle \frac{7}{24}>\frac{7}{36}.
\displaystyle \\

\displaystyle \textbf{Comparison of Fractions with Unlike Numerators and Unlike Denominators}
\displaystyle \textbf{Method 1: L.C.M. Method}
\displaystyle \text{Step 1: Find the L.C.M. of the denominators of the given fractions.}
\displaystyle \text{Step 2: Convert the given fractions into equivalent fractions having the L.C.M. as the common denominator.}
\displaystyle \text{Step 3: Compare the numerators of the like fractions so obtained.}
\displaystyle \textbf{Example:}
\displaystyle \text{Arrange }\frac{5}{9},\ \frac{11}{18},\ \frac{13}{24}\text{ and }\frac{17}{36}\text{ in ascending order.}
\displaystyle \text{L.C.M. of }9,\ 18,\ 24\text{ and }36=72.
\displaystyle \frac{5}{9}=\frac{40}{72},\qquad\frac{11}{18}=\frac{44}{72},
\displaystyle \frac{13}{24}=\frac{39}{72},\qquad\frac{17}{36}=\frac{34}{72}.
\displaystyle \text{Since }34<39<40<44,
\displaystyle \therefore \frac{17}{36}<\frac{13}{24}<\frac{5}{9}<\frac{11}{18}.
\displaystyle \\

\displaystyle \textbf{Method 2: Short-Cut Method (Cross-Multiplication Method)}
\displaystyle \text{If }\frac{a}{b}\text{ and }\frac{c}{d}\text{ are two fractions, where }b>0\text{ and }d>0,\text{ then:}
\displaystyle \text{(i) }\frac{a}{b}>\frac{c}{d}\quad\text{if }ad>bc.

\displaystyle \text{(ii) }\frac{a}{b}=\frac{c}{d}\quad\text{if }ad=bc.

\displaystyle \text{(iii) }\frac{a}{b}<\frac{c}{d}\quad\text{if }ad<bc.

\displaystyle \\

\displaystyle \textbf{To Insert a Fraction between Two Fractions}
\displaystyle \text{If }\frac{a}{b}<\frac{c}{d},\text{ where }b>0\text{ and }d>0,\text{ then the fraction}
\displaystyle \frac{a+c}{b+d}\text{ lies between }\frac{a}{b}\text{ and }\frac{c}{d}.
\displaystyle \text{Thus, }\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}.
\displaystyle \textbf{Example:}
\displaystyle \text{Consider the fractions }\frac{5}{6}\text{ and }\frac{7}{8}.
\displaystyle \frac{5+7}{6+8}=\frac{12}{14}=\frac{6}{7}.
\displaystyle \text{Since }\frac{5}{6}<\frac{6}{7}<\frac{7}{8},\ \frac{6}{7}\text{ lies between }\frac{5}{6}\text{ and }\frac{7}{8}.
\displaystyle \\

\displaystyle \textbf{Operations on Fractions}
\displaystyle \textbf{Addition of Fractions}
\displaystyle \textbf{1. Addition of Like Fractions}
\displaystyle \text{To add like fractions, add their numerators and retain the common denominator.}
\displaystyle \text{Sum of like fractions}=\frac{\text{Sum of the numerators}}{\text{Common denominator}}.
\displaystyle \text{Thus, }\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}.
\displaystyle \\

\displaystyle \textbf{2. Addition of Unlike Fractions}
\displaystyle \text{Step 1: Find the L.C.M. of the denominators of the unlike fractions.}
\displaystyle \text{Step 2: Convert the unlike fractions into equivalent fractions having the L.C.M.}
\displaystyle \text{as their common denominator.}
\displaystyle \text{Step 3: Add the like fractions so obtained.}
\displaystyle \textbf{Example:}
\displaystyle \text{Add }\frac{5}{6}\text{ and }\frac{7}{8}.
\displaystyle \text{L.C.M. of }6\text{ and }8=24.
\displaystyle \frac{5}{6}=\frac{20}{24}\qquad\text{and}\qquad\frac{7}{8}=\frac{21}{24}.
\displaystyle \therefore \frac{5}{6}+\frac{7}{8}=\frac{20}{24}+\frac{21}{24}=\frac{41}{24}.
\displaystyle \\

\displaystyle \textbf{Subtraction of Fractions}
\displaystyle \text{The same method used for addition is followed for subtraction.}
\displaystyle \text{After obtaining a common denominator, subtract the numerators.}
\displaystyle \textbf{Example:}
\displaystyle \text{Subtract }\frac{5}{6}\text{ from }\frac{7}{8}.
\displaystyle \text{L.C.M. of }6\text{ and }8=24.
\displaystyle \frac{5}{6}=\frac{20}{24}\qquad\text{and}\qquad\frac{7}{8}=\frac{21}{24}.
\displaystyle \therefore \frac{7}{8}-\frac{5}{6}=\frac{21}{24}-\frac{20}{24}=\frac{1}{24}.
\displaystyle \\

\displaystyle \textbf{Multiplication of Fractions}
\displaystyle \text{To multiply fractions, multiply their numerators and multiply their denominators.}
\displaystyle \text{Product of fractions}=\frac{\text{Product of the numerators}}{\text{Product of the denominators}}.
\displaystyle \text{Thus, }\frac{a}{b}\times\frac{c}{d}=\frac{ac}{bd}.
\displaystyle \text{The fraction obtained should be reduced to its lowest form, if necessary.}
\displaystyle \text{Common factors in a numerator and a denominator may be cancelled before multiplication.}
\displaystyle \textbf{Example:}
\displaystyle \frac{3}{14}\times\frac{7}{18}=\frac{3\times7}{14\times18}=\frac{1}{12}.
\displaystyle \\

\displaystyle \textbf{Division of Fractions}
\displaystyle \text{To divide one fraction by another, multiply the dividend by the reciprocal of the divisor.}
\displaystyle \text{Thus, }\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}=\frac{ad}{bc},\quad c\ne0.
\displaystyle \\

\displaystyle \textbf{Reciprocal of a Non-Zero Fraction}
\displaystyle \text{Let }\frac{a}{b}\text{ be a non-zero fraction, where }a\ne0\text{ and }b\ne0.
\displaystyle \text{Then }\frac{b}{a}\text{ is called the reciprocal of }\frac{a}{b}.
\displaystyle \frac{a}{b}\times\frac{b}{a}=1.
\displaystyle \\

\displaystyle \textbf{Simplification of Expressions Involving Fractions}
\displaystyle \textbf{Use of BODMAS Rule}
\displaystyle \text{We simplify expressions by applying the operations in the following order:}
\displaystyle \text{(1) B - Brackets}

\displaystyle \text{(2) O - Of}

\displaystyle \text{(3) D - Division}

\displaystyle \text{(4) M - Multiplication}

\displaystyle \text{(5) A - Addition}

\displaystyle \text{(6) S - Subtraction}

\displaystyle \textbf{Removal of Brackets:}
\displaystyle \text{When brackets are nested, simplify them from the innermost to the outermost.}
\displaystyle \text{The commonly used grouping symbols are:}
\displaystyle \text{(1) Bar or Vinculum }\overline{\phantom{abc}}

\displaystyle \text{(2) Parentheses }(\ )

\displaystyle \text{(3) Curly Brackets }\{\ \}

\displaystyle \text{(4) Square Brackets }[\ ]

\displaystyle \\

\displaystyle \textbf{Example 1: Simplify the following:}
\displaystyle 7\frac{1}{2}+\frac{1}{2}\div\frac{1}{2}\text{ of }\frac{1}{4}-\frac{2}{5}\times2\frac{1}{3}\div1\frac{7}{8}\text{ of }\left(1\frac{2}{5}-1\frac{1}{3}\right)
\displaystyle =\frac{15}{2}+\frac{1}{2}\div\frac{1}{2}\text{ of }\frac{1}{4}-\frac{2}{5}\times\frac{7}{3}\div\frac{15}{8}\text{ of }\left(\frac{7}{5}-\frac{4}{3}\right)
\displaystyle =\frac{15}{2}+\frac{1}{2}\div\frac{1}{2}\text{ of }\frac{1}{4}-\frac{2}{5}\times\frac{7}{3}\div\frac{15}{8}\text{ of }\left(\frac{21-20}{15}\right)
\displaystyle =\frac{15}{2}+\frac{1}{2}\div\frac{1}{2}\text{ of }\frac{1}{4}-\frac{2}{5}\times\frac{7}{3}\div\frac{15}{8}\text{ of }\frac{1}{15}
\displaystyle =\frac{15}{2}+\frac{1}{2}\div\frac{1}{8}-\frac{2}{5}\times\frac{7}{3}\div\frac{1}{8}
\displaystyle =\frac{15}{2}+\frac{1}{2}\times\frac{8}{1}-\frac{2}{5}\times\frac{7}{3}\times\frac{8}{1}
\displaystyle =\frac{15}{2}+4-\frac{112}{15}
\displaystyle =\frac{225+120-224}{30}
\displaystyle =\frac{121}{30}
\displaystyle =4\frac{1}{30}.
\displaystyle \\

\displaystyle \textbf{Example 2: Simplify the following:}
\displaystyle \left[3\frac{1}{4}\div\left\{1\frac{1}{4}-\frac{1}{2}\left(2\frac{1}{2}-\left(\frac{1}{4}-\frac{1}{6}\right)\right)\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{5}{4}-\frac{1}{2}\left(\frac{5}{2}-\left(\frac{1}{4}-\frac{1}{6}\right)\right)\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{5}{4}-\frac{1}{2}\left(\frac{5}{2}-\frac{3-2}{12}\right)\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{5}{4}-\frac{1}{2}\left(\frac{5}{2}-\frac{1}{12}\right)\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{5}{4}-\frac{1}{2}\left(\frac{30-1}{12}\right)\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{5}{4}-\frac{1}{2}\times\frac{29}{12}\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{5}{4}-\frac{29}{24}\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\left\{\frac{30-29}{24}\right\}\right]
\displaystyle =\left[\frac{13}{4}\div\frac{1}{24}\right]
\displaystyle =\frac{13}{4}\times\frac{24}{1}
\displaystyle =78.
\displaystyle \\

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