Operations on Fractions

Definition of Fractions

A fraction is a number expressed in the form $\frac{a}{b}$ where $a$, $b$ are integers and $b \ne 0$. The number $a$ is called the numerator and $b$ is called the denominator.

Forms of Fractions

Fraction numbers can be expressed in several forms, namely common fractions, decimal fractions, mixed fractions, percents, and percents.

Ordinary Fraction

Ordinary fractions are expressed in the form $\frac{a}{b}$ where $a$, $b$ are integers and $b \ne 0$. If $a < b$ is called a pure fraction, while if $a > b$ is called a impure fraction.

Mixed Fraction

A mixed number is a fraction that consists of two parts, namely an integer and a fraction. The general form is $a\frac{b}{c}$ with $c \ne 0$. Fractions can be converted to impure fractions or vice versa.

Examples

• $2\frac{3}{4} = \frac{{2 \times 4 + 3}}{4} = \frac{{11}}{4}$
• $3\frac{1}{6} = \frac{{3 \times 6 + 1}}{6} = \frac{{19}}{6}$
• $\frac{{12}}{5} = 2\frac{2}{5}$
• $\frac{{10}}{3} = 3\frac{1}{3}$

Decimal Fraction

In decimal fractions use a comma to mark the fractional number. The first digit after the comma is $\frac{1}{{10}}$, the second number after the comma is $\frac{1}{{100}}$, the third number after the comma is $\frac{1}{ {1000}}$, and so on.

Examples

• $\frac{1}{2} = \frac{{1 \times 5}}{{2 \times 5}} = \frac{5}{{10}} = 0.5$
• $\frac{3}{8} = \frac{{3 \times 125}}{{8 \times 125}} = \frac{{375}}{{1000}} = 0.375$
• $0.4 = \frac{4}{{10}} = \frac{{4 \div 2}}{{10 \div 2}} = \frac{2}{5}$
• $0.75 = \frac{{75}}{{100}} = \frac{{75 \div 25}}{{100 \div 25}} = \frac{3}{4}$

Percent and Permile

The meaning of percent is a number or comparison (ratio) to express a fraction of a hundred which is indicated by the symbol $\% $. In other words, the percentage is the part of the whole expressed in hundredths. While permil is a number or comparison (ratio) to express a fraction of a thousand indicated by the symbol $‰$.

Examples

$\large{\begin{array}{l} 5\% = \frac{5}{{100}} = \frac{1}{{20}}\\ 3,5\% = \frac{{3,5}}{{100}} = \frac{{3,5 \times 2}}{{100 \times 2}} = \frac{7}{{200}}\\ \frac{2}{5}\% = \frac{{\frac{2}{5}}}{{100}} = \frac{2}{{5 \times 100}} = \frac{2}{{500}} = \frac{1}{{250}}\\ 10 ‰ = \frac{{10}}{{1.000}} = \frac{1}{{100}}\\ 4,5 ‰ = \frac{{4,5}}{{1.000}} = \frac{{4,5 \times 2}}{{1.000 \times 2}} = \frac{9}{{2.000}}\\ \frac{5}{8} ‰ = \frac{{\frac{5}{8}}}{{1.000}} = \frac{5}{{8 \times 1.000}} = \frac{5}{{8.000}} = \frac{1}{{1.600}} \end{array}}$

Operations on Fractions

Add and Subtract

In addition and subtraction of fractions, you must pay attention to the denominator of the fraction to be operated. If the denominators are the same then just add or subtract the numerators, but if the denominators are different then it is necessary to equalize the denominators or you can use the formula below. In addition and subtraction of fractional numbers, you can use the following formula:

$\large{\begin{array}{l} \boxed{\frac{a}{b} \pm \frac{c}{b} = \frac{{a \pm c}}{b}}\\ \boxed{\frac{a}{b} \pm \frac{c}{d} = \frac{{\left( {a \times d} \right) \pm \left( {b \times c} \right)}}{{\left( {b \times d} \right)}}} \end{array}}$

Examples

$\large{\begin{array}{l} a.\frac{3}{5} + \frac{1}{5} - \frac{2}{5} = \frac{{3 + 1 - 2}}{5} = \frac{2}{5}\\ b.\frac{2}{4} + \frac{1}{3} - \frac{1}{6} = \frac{6}{{12}} + \frac{4}{{12}} - \frac{2}{{12}} = \frac{{6 + 4 - 2}}{{12}} = \frac{8}{{12}} = \frac{2}{3}\\ c.\frac{2}{7} + \frac{3}{4} = \frac{{2 \times 4 + 7 \times 3}}{{7 \times 4}} = \frac{{8 + 21}}{{28}} = \frac{{29}}{{28}} = 1\frac{1}{{28}} \end{array}}$

Multiplication

The formula that applies to multiplication of fractions is as follows:

$\boxed{\large{\frac{a}{b} \times \frac{c}{d} = \frac{{a \times c}}{{b \times d}} = \frac{{ac}}{{bd}}}}$

Examples

$\begin{array}{l} a.\frac{2}{3} \times \frac{4}{5} = \frac{{2 \times 4}}{{3 \times 5}} = \frac{8}{{15}}\\ b.2\frac{2}{5} \times 1\frac{1}{3} = \frac{{12}}{5} \times \frac{4}{3} = \frac{{12 \times 4}}{{5 \times 3}} = \frac{{48}}{{15}} = 3\frac{3}{{15}} = 3\frac{1}{5}\\ c.5 \times \frac{3}{4} = \frac{{5 \times 3}}{4} = \frac{{15}}{4} = 3\frac{3}{4} \end{array}$

Division of Fractions

The formula for dividing fractions is as follows:

$\boxed{\large{\frac{a}{b} \div \frac{c}{d} = \frac{{a \times d}}{{b \times c}} = \frac{{ad}}{{bc}}}}$

Examples

$\begin{array}{l} a.\frac{2}{7} \div \frac{3}{5} = \frac{{2 \times 5}}{{7 \times 3}} = \frac{{10}}{{21}}\\ b.3\frac{1}{2} \div 2\frac{3}{4} = \frac{7}{2} \div \frac{{11}}{4} = \frac{{7 \times 4}}{{2 \times 11}} = \frac{{28}}{{22}} = 1\frac{6}{{22}} = 1\frac{3}{{11}}\\ c.3 \div \frac{3}{4} = \frac{3}{1} \div \frac{3}{4} = \frac{{3 \times 4}}{{1 \times 3}} = \frac{{12}}{3} = 4 \end{array}$

Exponents of Fractions

The formula that applies to the power of fractions is as follows:

$\boxed{\large{{\left( {\frac{a}{b}} \right)^n} = \frac{{{a^n}}}{{{b^n}}}}}$

Examples

$\begin{array}{l} a.{\left( {\frac{2}{3}} \right)^3} = \frac{{{2^3}}}{{{3^3}}} = \frac{8}{{27}}\\ b.{\left( {1\frac{1}{4}} \right)^2} = {\left( {\frac{5}{4}} \right)^2} = \frac{{{5^2}}}{{{4^2}}} = \frac{{25}}{{16}} = 1\frac{9}{{16}} \end{array}$

Thus the discussion and examples of operations on fractions. Hope it's useful