Simplification of Quotients of Fractions

Here we will be knowing to determine the quotient when the numerator and the denominator are fractions as mentioned in Āryabhaṭīyam.

Verse

छेदाः परस्परहता भवन्ति गुणकारभागहाराणाम् ।

Translation

The numerators and denominators of the multipliers and divisors should be multiplied by one another.

For Example:

${\frac {\frac {a}{b}}{\frac {c}{d}}}={\frac {Dividend}{Divisor}}={\frac {ad}{bc}}=Quotient$

${\frac {{\frac {a}{b}}\times {\frac {c}{d}}}{{\frac {e}{f}}\times {\frac {g}{h}}}}={\frac {Dividend}{Divisor}}={\frac {(ac)(fh)}{(bd)(eg)}}=Quotient$

If the numerator and denominator are both fractions, this verse gives the formula to simplify them. The numerator of the dividend should be multiplied by the denominator of the divisor and the denominator of the dividend should be multiplied by the numerator of the divisor.

Example 1:

${\frac {\frac {3}{4}}{\frac {5}{7}}}={\frac {Dividend}{Divisor}}={\frac {3\times 7}{4\times 5}}={\frac {21}{20}}=Quotient$

Example 2:

${\frac {{\frac {3}{4}}\times {\frac {1}{6}}}{{\frac {5}{7}}\times {\frac {2}{3}}}}={\frac {Dividend}{Divisor}}={\frac {(3\times 1)(7\times 3)}{(4\times 6)(5\times 2)}}={\frac {63}{240}}=Quotient$

See Also

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