Rational Numbers, Properties, Operations
Operations on rational numbers
Addition : There are two possibilities
1. Rational numbers with same denominator
2. Rational numbers with different denominators.
Case1: Rational numbers having same denominator.
If \(
\frac{p}
{q}
\) and \(
\frac{r}
{q}\left( {q > 0} \right)
\) are two rational numbers then \(
\frac{p}
{q} + \frac{r}
{q} = \frac{{\left( {p + r} \right)}}
{q}
\)
Ex: Add \(
\frac{7}
{{ - 11}}
\) and \(
\frac{3}
{{11}}
\)
\(
\therefore \frac{7}
{{ - 11}} + \frac{3}
{{11}} = \frac{{ - 7}}
{{11}} + \frac{3}
{{11}} = \frac{{\left( { - 7} \right) + 3}}
{{11}} = \frac{{ - 4}}
{{11}}
\)
Case 2 : Rational numbers having different denominators:
1. Convert the denominator as common denominator.
2. Find LCM of their denominators and express each rational number with this LCM as their denominator, then add them as in case 1
Ex : Add \(
\frac{{ - 3}}
{5}and\frac{2}
{3}
\)
\(
= \frac{{ - 3 \times 3}}
{{5 \times 3}} + \frac{{2 \times 5}}
{{3 \times 5}}
\)(LCM of 5 & 3 is 15)
\(
= \frac{{ - 9}}
{{15}} + \frac{{10}}
{{15}} = \frac{{ - 9 + 10}}
{{15}} = \frac{1}
{{15}}
\)
Subtraction :
To subtract one rational number from another, subtract their numerators and keep the common denominator.
Ex 1: Subtract \(
\frac{5}
{7}
\) from \(
\frac{{ - 3}}
{{14}}
\)
\(
\frac{{ - 3}}
{{14}} - \left( {\frac{5}
{7}} \right) = \frac{{ - 3}}
{{14}} + \left( {\frac{{ - 5}}
{7}} \right)
\)
\(
= \frac{{ - 3}}
{{14}} + \left( {\frac{{ - 5 \times 2}}
{{7 \times 2}}} \right)
\) [LCM of 14, 7 is 14]
\(
\begin{gathered}
= \frac{{ - 3}}
{{14}} + \frac{{ - 10}}
{{14}} \hfill \\
= \frac{{ - 3 + \left( { - 10} \right)}}
{{14}} = \frac{{ - 13}}
{{14}} \hfill \\
\end{gathered}
\)
Ex 2: Evaluate \(
\frac{3}
{5} + \frac{{10}}
{3} - \frac{{11}}
{5} + \frac{3}
{5} - \frac{4}
{3}
\)
\(
\begin{gathered}
\frac{3}
{5} + \frac{{10}}
{3} - \frac{{11}}
{{15}} + \frac{3}
{5} - \frac{4}
{3} = \frac{{\left( {3 \times 3} \right) + \left( {10 \times 5} \right) - 11 + \left( {3 \times 3} \right) - \left( {4 \times 5} \right)}}
{{15}} \hfill \\
= \frac{{9 + 50 - 11 + 9 - 20}}
{{15}} = \frac{{37}}
{{15}} \hfill \\
\end{gathered}
\)