View source for 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.<ref>{{Cite book|last=Shukla|first=Kripa Shankar|title=Āryabhaṭīya of Āryabhaṭa|year=1976|location=New Delhi|pages=70.}}</ref>

That is,

<math>\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}</math>

<math>\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}</math>
===Example :===
What is the sum of <math>\frac{1}{2}</math>,<math>\frac{1}{6}</math>,<math>\frac{1}{3}</math>?

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

<math>\frac{1}{2}=\frac{1}{2} \times \frac{6}{6} \times \frac{3}{3}=\frac{18}{36} </math>

<math>\frac{1}{6}=\frac{1}{6} \times \frac{2}{2} \times \frac{3}{3}=\frac{6}{36} </math>

<math>\frac{1}{3}=\frac{1}{3} \times \frac{2}{2} \times \frac{6}{6}=\frac{12}{36} </math>

Now the denominators are same for the given fractions.

We have to add <math>\frac{18}{36}</math>, <math>\frac{6}{36}</math>, <math>\frac{12}{36}</math> to get the sum

<math>\frac{18}{36}+\frac{6}{36}+\frac{12}{36}=\frac{18+6+12}{36}=\frac{36}{36}=1</math>

Hence the sum of <math>\frac{1}{2}</math>,<math>\frac{1}{6}</math>,<math>\frac{1}{3} = 1</math>
==See Also==
[[भिन्नों का सार्व हर में लघुकरण]]
==References==
<references />
[[Category:Mathematics in Āryabhaṭīyam]]
[[Category:General]]