The Rule of Five in Līlāvatī

Verse 89

पञ्चसप्तनवराशिकादिकेऽन्योन्यपक्षनयनं फलच्छिदाम् ।

संविधाय बहुराशिजे वधे स्वल्पराशिवधभाजिते फलम् ॥ ८९ ॥

In the case of examples on the rules of five, seven, nine, etc. keep the antecedents of all proportions in the numerator. All the other terms, except the desired result, should be kept in the denominator. The product of the numerators divided by the product of the denominators is the required result.

Example 1

मासे शतस्य यदि पञ्चकलान्तरं स्यात्

वर्षे गते भवति किं वद षोडशानाम् ।

कालं तथा कथय मूलकलान्तराभ्याम्

मूलं धनं गणक कालफले विदित्वा ॥ ॥

There are three problems in this.

1.If 100 niṣkas (N) fetch 5 N interest per month (M), find the interest on 16 N for one year (12 M).

100 N Principal	:	16 N Principal					Direct
			::	5 N Interest	:	X
1 Month	:	12 Months					Direct

$X={\frac {5\ X\ 16X\ 12}{100\ X\ 1}}={\frac {48}{5}}=9{\frac {3}{5}}$N

2. The above problem is altered at the same rate as in (1), find the period to fetch $9{\frac {3}{5}}$ interest on 16 N.

100 N	:	16 N					Inverse
			::	1 M	:	X
5N	:	$9{\frac {3}{5}}$N					Direct

$X={\frac {1\ X\ 100X\ {\frac {48}{5}}}{16\ X\ 5}}=12$M

3. Suppose we are given the period and interest and we have to find the principal (x).

5N	:	$9{\frac {3}{5}}$N					Direct
			::	100N	:	X
1 M	:	12 M					Inverse

$X={\frac {5\ X\ 100X\ {\frac {48}{5}}}{12\ X\ 1}}=16$N

Example 2

सत्र्यंशमासेन शतस्य चेत्स्यात्कलान्तरं पञ्च सपञ्चमांशाः ।

मासैस्त्रिभिः पञ्चलवाधिकैस्तैः सार्धद्विषट्कैः फलमुच्यतां किम् ॥ ॥

If the interest on 100 for ${\frac {4}{3}}$ months is $5{\frac {1}{5}}$ what will be the interest on $62{\frac {1}{2}}$ for $3{\frac {1}{5}}$months ?

Comment: The rule of five is:

100	:	$62{\frac {1}{2}}$					Direct
			::	${\frac {26}{5}}$	:	X
${\frac {4}{3}}$M	:	${\frac {16}{5}}$N					Direct

$X={\frac {26}{5}}\ X\ {\frac {125}{2}}\ X\ {\frac {16}{5}}\ X\ {\frac {1}{100}}\ X\ {\frac {3}{4}}={\frac {39}{5}}=7{\frac {4}{5}}$

See Also

लीलावती में 'पाँच का नियम'