Answer
The quotient of two rational numbers of the form \[\frac{a}{b}\]and \[\frac{c}{d}\] can be written as a product of one rational number with the reciprocal of another. This can be shown as follows:
\[\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}\]
The quotient of the two rational numbers thus gives:
\[\begin{align}
& \frac{169}{30}\div \frac{13}{15}=\frac{169}{30}\times \frac{15}{13} \\
& =\frac{13\times 13\times 15}{15\times 2\times 13} \\
& =\frac{13}{2}
\end{align}\]
Two rational numbers \[\frac{a}{b}\]and \[\frac{c}{d}\] can be added by first finding the least common multiple of their denominators also known as least common denominator. The rational numbers are then multiplied by a rational number of the form \[\frac{e}{e}\] so that the denominator of both the rational numbers becomes the least common denominators as found earlier.
The least common denominator of the rational numbers \[\frac{169}{30},\frac{13}{15}\]is \[30\].
Thus, the addition of two rational numbers can be shown as,
\[\begin{align}
& \frac{169}{30}+\frac{13}{15}=\frac{169}{30}\times \frac{1}{1}+\frac{13}{15}\times \frac{2}{2} \\
& =\frac{169+13\times 2}{30} \\
& =\frac{195}{30} \\
& =\frac{13}{2}
\end{align}\]
We conclude that \[\frac{169}{30}+\frac{13}{15}\] and \[\frac{169}{30}\div \frac{13}{15}\]give the same answer.
