Reduction of Fractions to a Common Denominator

Here we will be knowing how to reduce fractions to a common denominator as mentioned in Āryabhaṭīyam.

Verse

छेदगुणं सच्छेदं परस्परं तत् सवर्णत्वम् ।

Translation

Multiply the numerator as also the denominator of each fraction by the denominator of the other fraction; then the (given) fractions are reduced to a common denominator.

That is,

${\frac {a}{b}}+{\frac {c}{d}}={\frac {a\times d}{b\times d}}+{\frac {c\times b}{d\times b}}={\frac {a\times d}{b\times d}}+{\frac {c\times b}{b\times d}}={\frac {ad}{bd}}+{\frac {cb}{bd}}={\frac {ad+cb}{bd}}$

${\frac {a}{b}}-{\frac {c}{d}}={\frac {a\times d}{b\times d}}-{\frac {c\times b}{d\times b}}={\frac {a\times d}{b\times d}}-{\frac {c\times b}{b\times d}}={\frac {ad}{bd}}-{\frac {cb}{bd}}={\frac {ad-cb}{bd}}$

Example :

What is the sum of ${\frac {1}{2}}$,${\frac {1}{6}}$,${\frac {1}{3}}$?

Multiply the numerator and denominator of each fraction with the denominator of other fractions.

${\frac {1}{2}}={\frac {1}{2}}\times {\frac {6}{6}}\times {\frac {3}{3}}={\frac {18}{36}}$

${\frac {1}{6}}={\frac {1}{6}}\times {\frac {2}{2}}\times {\frac {3}{3}}={\frac {6}{36}}$

${\frac {1}{3}}={\frac {1}{3}}\times {\frac {2}{2}}\times {\frac {6}{6}}={\frac {12}{36}}$

Now the denominators are same for the given fractions.

We have to add ${\frac {18}{36}}$, ${\frac {6}{36}}$, ${\frac {12}{36}}$ to get the sum

${\frac {18}{36}}+{\frac {6}{36}}+{\frac {12}{36}}={\frac {18+6+12}{36}}={\frac {36}{36}}=1$

Hence the sum of ${\frac {1}{2}}$,${\frac {1}{6}}$,${\frac {1}{3}}=1$

See Also

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