Introduction:

A fraction consists of two parts: the numerator and the denominator.

Numerator: The number above the line.

Denominator: The number below the line.

It can be expressed as a division of two integers \frac{a}{b} , where b\ne 0 .

\frac{a}{b} where  ‘a’ is numerator,  ‘b’ is denominator.

Example: For the fraction number \frac{9}{11} , 9 is the numerator and 11 is the denominator.

Types of fractions:

There are many types of fractions in mathematics, some of the main ones are as follows:

1) Unit fractions:

These are the fractions with numerator 1.

Example: \frac{1}{2},\:\frac{1}{3},\:\frac{1}{4},\:\frac{1}{5}, etc.

2) Proper fractions:

When the numerator is less than the denominator, the fraction is called the Proper fraction.

Example: \frac{1}{2},\:\frac{1}{3},\:\frac{2}{3},\:\frac{4}{5}, etc.

3) Improper fractions:

When the numerator is greater than the denominator, the fraction is called the Improper fraction.

An Improper fraction can be written as a mixed number also.

Example: \frac{2}{1},\:\frac{3}{2},\:\frac{5}{2},\:\frac{6}{2}, etc.

4) Mixed number:

A mixed number is a mixture of a natural number and a proper fraction.

Example: 5\frac{3}{4}=5+\frac{3}{4},   here 5 is a natural number and \frac{3}{4} is a proper fraction.

5) Equivalent fractions:

Fractions which have the same value after simplification are called equivalent fractions.

Examples: \frac{1}{2} and \frac{2}{4} are equivalent fractions. because when you simplify \frac{2}{4} , you get \frac{1}{2} .

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Conversion of  fractions:

Mixed fraction to improper fraction:   

a\frac{b}{c}=\frac{a\times c+b}{c}

Let’s see one example:

2\frac{3}{4}=\frac{2\times 4+3}{4}\\
2\frac{3}{4}=\frac{8+3}{4}\\
2\frac{3}{4}=\frac{11}{4}

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Improper fraction to mixed fraction:   

\frac{a}{b}=\text{Quotient}\frac{\text{Remainder}}{\text{Divisor}}

Let’s see one example:

Example: \frac{23}{7}=3\frac{2}{7}

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Properties of fractions:

While adding or subtracting fractions, we must ensure that the denominator of both fractions is equal:    \frac{a}{b}\pm\frac{c}{b}=\frac{a\pm c}{b}

Let’s see one example:

\frac{2}{3}+\frac{4}{3}=\frac{2+4}{3}\\
\frac{2}{3}+\frac{4}{3}=\frac{6}{3}

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If the denominator of both fractions is not equal then we can use cross multiplication or the LCM method. After that, we simplify the fraction if possible:    \frac{a}{b}\pm \frac{c}{d}=\frac{ad\pm bc}{bd}

Let’s see one example:

\frac{2}{3}+\frac{4}{5}=\frac{2\cdot 5+4\cdot 3}{3\cdot 5}\\
\frac{2}{3}+\frac{4}{5}=\frac{10+12}{15}\\
\frac{2}{3}+\frac{4}{5}=\frac{22}{15}

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To multiply two fractions, multiply both numerators at the top and both denominators at the bottom. After that, we simplify the fraction if possible:    \frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}

Let’s see one example:

\frac{2}{3}\cdot \frac{4}{5}=\frac{2\cdot 4}{3\cdot 5}\\
\frac{2}{3}\cdot \frac{4}{5}=\frac{8}{15}

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To divide two fractions, we need to multiply the reciprocal of the bottom fraction with the top fraction. After that, we simplify the fraction if possible:    \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}

Let’s see one example:

\frac{\frac{2}{3}}{\frac{4}{5}}=\frac{2}{3}\cdot \frac{5}{4}\\
\frac{\frac{2}{3}}{\frac{4}{5}}=\frac{2}{3}\cdot \frac{5}{2\cdot 2}

Here we can see that, the number 2 can be cancelled out.

\frac{\frac{2}{3}}{\frac{4}{5}}=\frac{1}{3}\cdot \frac{5}{2}\\
\frac{\frac{2}{3}}{\frac{4}{5}}=\frac{1\cdot 5}{3\cdot 2}\\
\frac{\frac{2}{3}}{\frac{4}{5}}=\frac{5}{6}\\

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Practice Questions on Fractions:

Simplify the following fractions:

a) \frac{4}{8}=\frac{4}{4\cdot 2}=\frac{1}{2}

b) \frac{10}{15}=\frac{5\cdot 2}{5\cdot 3}=\frac{2}{3}

​c) \frac{18}{24}=\frac{6\cdot 3}{6\cdot 4}=\frac{3}{4}

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​Add the following fractions and simplify the result:   a) \frac{4}{5}+\frac{1}{5}

\frac{4}{5}+\frac{1}{5}=\frac{4+1}{5}\\
\frac{4}{5}+\frac{1}{5}=\frac{5}{5}\\
\frac{4}{5}+\frac{1}{5}=1

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​Add the following fractions and simplify the result:   b) \frac{2}{3}+\frac{4}{6}

Here the denominator of both fractions is not equal.

So to make them equal, multiply and divide the first fraction by 2.

\frac{2}{3}+\frac{4}{6}=\frac{2}{3}\cdot \frac{2}{2}+\frac{4}{6}\\
\frac{2}{3}+\frac{4}{6}=\frac{2\cdot 2}{3\cdot 2}+\frac{4}{6}\\
\frac{2}{3}+\frac{4}{6}=\frac{4}{6}+\frac{4}{6}\\
\frac{2}{3}+\frac{4}{6}=\frac{4+4}{6}\\
\frac{2}{3}+\frac{4}{6}=\frac{8}{6}\\
\frac{2}{3}+\frac{4}{6}=\frac{4\cdot 2}{3\cdot 2}\\
\frac{2}{3}+\frac{4}{6}=\frac{4}{3}

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​Add the following fractions and simplify the result:   c) \frac{5}{18}+\frac{7}{24}

Now here we can use the cross-multiplication method but it will be a little bit complex because we have to multiply large numbers 18 and 24.

So we can use the LCM method.

Take the LCM of both denominators.

Prime factors of 18: 2 \times 3 × 3 = 2^1 \times 3^2

Prime factors of 24: 2 \times 2 \times 2 \times 3= 2^3 \times 3^1

Now, we take the highest power of each prime factor:

The highest power of 2 is 3. \rightarrow 2^3

The highest power of 3 is 2. \rightarrow 3^2

LCM= 2^3\times 3^2

LCM= 8\times 9

LCM=72

LCM of 18 and 24 is 72.

So multiply and divide the first fraction by 4 because \frac{72}{18} is 4.

And then Multiply and divide the second fraction by 3 because \frac{72}{24} is 3.

\frac{5}{18}+\frac{7}{24}=\frac{5}{18}\cdot \frac{4}{4}+\frac{7}{24}\cdot \frac{3}{3}\\
\frac{5}{18}+\frac{7}{24}=\frac{20}{72}+\frac{21}{72}\\
\frac{5}{18}+\frac{7}{24}=\frac{20+21}{72}\\
\frac{5}{18}+\frac{7}{24}=\frac{41}{72}