The Rule of Five in Līlāvatī: Difference between revisions

Latest revision as of 22:09, 30 August 2023

Here we will know the rule of five as mentioned 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

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

References

@@ Line 1: / Line 1: @@
-==Verse 97==
+Here we will know the rule of five as mentioned in Līlāvatī.
-प्रमाणकालेन हतं प्रमाणं विमिश्रकालेन हतं फलं च
-स्वयोगभक्ते च पृथक् स्थिते च मिश्राहते मूल कलान्तरे स्तः ॥९७॥
+==Verse 89==
+पञ्चसप्तनवराशिकादिकेऽन्योन्यपक्षनयनं  फलच्छिदाम् ।
-To compute simple interest and principal multiply the standard principal (100) by the standard period (1 month or 1 year).<ref>{{Cite book|title=Līlāvatī Of Bhāskarācārya - A Treatise of Mathematics of Vedic Tradition|publisher=Motilal Banarsidass Publishers|year=2001|isbn=81-208-1420-7|location=New Delhi|pages=89-93}}</ref> Next multiply the given period by the given rate of interest. Keep the two products a, b separately. Multiply a by the amount and divide it by (a+b) to get the principal. Similarly the amount multiplied by b and divided by (a+b) yields the interest.
+संविधाय बहुराशिजे वधे स्वल्पराशिवधभाजिते फलम् ॥ ८९ ॥
-'''Comment''' : A = amount , P = principal, I = interest, R = rate of interest , Y = period . P<sub>0</sub> = standard principal (usually 100)
+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
-Y<sub>0</sub> = standard period ( 1 year or 1 month).
+वर्षे गते भवति किं वद षोडशानाम् ।
-<math>P = \frac{A \ X \ P_0 \ X \  Y_0} { P_0Y_0 + RY} </math>
+कालं तथा कथय मूलकलान्तराभ्याम्
-<math>I = \frac{A \ X \ R \ X \  Y} { P_0Y_0 + RY} </math>
+मूलं धनं गणक कालफले विदित्वा ॥ ॥
-==Example==
-पंचकेन शतेनाब्दे मूलं स्वं सकलान्तरम् ।
-सहस्त्रं चेत्पृथक् तत्र वद मूल कलान्तरे ॥ ॥
+There are three problems in this.
-When the interest rate is 5% per month, the amount after one year is 1000 N (niṣkas). Find the principal and the interest.
+.If 100 niṣkas (N) fetch 5 N interest per month (M), find the interest on 16 N for one year (12 M).
+{| class="wikitable"
+|+
+| 100 N Principal
+| :
+|16 N Principal
+| colspan="4" |
+|Direct
+|-
+| colspan="3" |
+|::
+|5 N Interest
+|:
+|X
+|
+|-
+|1 Month
+|:
+|12 Months
+| colspan="4" |
+|Direct
+|}<math>X = \frac{5 \ X \ 16 X \ 12}{100 \ X \ 1} = \frac{48}{5} =  9 \frac{3}{5}
+</math>N
-Comment : The above stanza does not mention 'per month' but it seems that in those times the interest was calculated on monthly basis.
+. The above problem is altered at the same rate as in (1), find the period to fetch <math>9 \frac{3}{5}</math> interest on 16 N.
+{| class="wikitable"
+|+
+| 100 N
+|:
+| 16 N
+| colspan="4" |
+|Inverse
+|-
+| colspan="3" |
+|::
+|1 M
+|:
+|X
+|
+|-
+|5N
+|:
+|<math>9 \frac{3}{5}</math>N
+| colspan="4" |
+|Direct
+|}<math>X = \frac{1 \ X \ 100 X \ \frac{48}{5}}{16 \ X \ 5} = 12
+</math>M
-Here A = 1000, R = 5 , P<sub>0</sub> = 100 , Y<sub>0</sub> = 1 month Y = 1 year ( 12 months)
+. Suppose we are given the period and interest and we have to  find the principal (x).
+{| class="wikitable"
+|+
+|5N
+|:
+|<math>9 \frac{3}{5}</math>N
+| colspan="4" |
+| Direct
+|-
+| colspan="3" |
+|::
+|100N
+|:
+|X
+|
+|-
+|1 M
+|:
+|12 M
+| colspan="4" |
+|Inverse
+|}<math>X = \frac{5 \ X \ 100 X \ \frac{48}{5}}{12 \ X \ 1} = 16
+</math>N
+==Example 2==
+सत्र्यंशमासेन शतस्य चेत्स्यात्कलान्तरं पञ्च सपञ्चमांशाः ।
-<math>P = \frac{1000 \ X \ 100 \ X \  1} { 100 \ X \ 1 + 5 \ X \ 12} = 625 </math> N
+मासैस्त्रिभिः पञ्चलवाधिकैस्तैः सार्धद्विषट्कैः फलमुच्यतां किम् ॥ ॥
-<math>I = \frac{1000 \ X \ 5 \ X \  12} { 100 \ X \ 1 + 5 \ X \ 12} = 375 </math> N
+If the interest on 100 for <math>\frac{4}{3}
+</math> months is <math>5\frac{1}{5}
+</math> what will be the interest on <math>62\frac{1}{2}
+</math> for <math>3\frac{1}{5}
+</math>months ?
-Alternatively I = A - P = 1000 - 625 = 375 N
+'''Comment''': The rule of five is:
-==Verse 99==
+{| class="wikitable"
-अथ प्रमाणैर्गुणिताः स्वकाला व्यतीतकालघ्नफलोद्धृतास्ते ।
+|+
+|100
-स्वयोगभक्ताश्च विमिश्रनिघ्नाः प्रयुक्तखण्डानि पृथक् भवन्ति ॥९९॥
+|:
+|<math>62 \frac{1}{2}</math>
-If several parts of a certain principal bear different rates of interest for different periods and yet yield the same interest, to find these parts - Take the product of the standard principal and the standard period, divide this product by the product of respective periods and rates of interest, and write these quotients separately. These quotients multiplied by the given principal and divided by the sum of the quotients written separately are the desired parts of the given principal.
+| colspan="4" |
+|Direct
-'''Comment''': Using the notation of the previous example with suffixes to denote the parts (we consider three parts):
+|-
+| colspan="3" |
-<math>P_1 = \frac{(P_1+P_2+P_3) \  X \ \frac{100}{R_1Y_1}}{\frac{100}{R_1Y_1} +\frac{100}{R_2Y_2} +\frac{100}{R_3Y_3}}</math> <math>P_2 = \frac{(P_1+P_2+P_3) \  X \ \frac{100}{R_2Y_2}}{\frac{100}{R_1Y_1} +\frac{100}{R_2Y_2} +\frac{100}{R_3Y_3}}</math> <math>P_3 = \frac{(P_1+P_2+P_3) \  X \ \frac{100}{R_3Y_3}}{\frac{100}{R_1Y_1} +\frac{100}{R_2Y_2} +\frac{100}{R_3Y_3}}</math>
+|::
-==Example==
+|<math>\frac{26}{5}
-यत्पंचकत्रिकचतुष्कशतेन दत्तं
+</math>
+|:
-खंडैस्त्रिभिर्गणक निष्कशतं षडूनम् ।
+| X
+|
-मासेषु सप्तदशपंचसु तुल्यमाप्तम्
+|-
+| <math>\frac{4}{3}
-खंडत्रयेऽपि हि फलं वद खंडसंख्याम् ॥ ॥
+</math>M
+|:
-N (niṣkas) were divided into three parts and were lent at 5 per cent (per month) for 7 months, at 3 per cent for 10 months, at 4 per cent for
+|<math>\frac{16}{5}
+</math>N
-months. If the three parts yield equal interest, find them.
+| colspan="4" |
+| Direct
-'''Comment''': Here P<sub>1</sub> + P<sub>2</sub>+ P<sub>3</sub> = 94
+|}<math>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}
+</math>
-R<sub>1</sub> = 5 Y<sub>1</sub> = 7
-R<sub>2</sub> = 3 Y<sub>2</sub> = 10
-R<sub>3</sub> = 4 Y<sub>3</sub> = 5
-As per the above formula
-<math>P_1 = \frac{94 \  X \ \frac{100}{5 X 7}}{\frac{100}{5X7} +\frac{100}{3X10} +\frac{100}{4X5}} = \frac{94 \  X \ \frac{100}{35}}{\frac{100}{35} +\frac{100}{30} +\frac{100}{20}} = \frac{94 \  X \ \frac{100}{35}}{\frac{4700}{420}} = 24 </math>
-<math>P_2 = \frac{94 \  X \ \frac{100}{3 X10}}{\frac{100}{5X7} +\frac{100}{3X10} +\frac{100}{4X5}} = \frac{94 \  X \ \frac{100}{30}}{\frac{100}{35} +\frac{100}{30} +\frac{100}{20}} = \frac{94 \  X \ \frac{100}{30}}{\frac{4700}{420}} = 28 </math>
-<math>P_3 = \frac{94 \  X \ \frac{100}{4X5}}{\frac{100}{5X7} +\frac{100}{3X10} +\frac{100}{4X5}} = \frac{94 \  X \ \frac{100}{20}}{\frac{100}{35} +\frac{100}{30} +\frac{100}{20}} = \frac{94 \  X \ \frac{100}{20}}{\frac{4700}{420}} = 42 </math>
-==Compute shares of profit==
-Here we will compute the shares of profit when total profit and individual investments are given.
-प्रक्षेपका मिश्रहता विभक्ता प्रक्षेपयोगेन पृथक् फलानि ॥ ॥
-An individual's share (after business) is the individual's investment multiplied by the total output and divided by the total investment.
-'''Comment''' :If a, b and c are investments and x is the output, then the shares are <math>\frac{ax}{a+b+c}</math> , <math>\frac{bx}{a+b+c}</math> , <math>\frac{cx}{a+b+c}</math> respectively.
-==Example==
-पंचाशदेकसहिता गणकाष्टषष्टिः पंचोनिता नवतिरादिधनानि येषाम् ।
-प्राप्ता विमिश्रितधनैस्त्रिशती त्रिभिस्तैः वाणिज्यतो वद विभज्य धनानि तेषाम् ॥ ॥
-Three grocers invested 51, 68 , 85 N (niṣkas) respectively. Skillfully they increased their total assets to 300 N. Find the share of each .
-'''Comment''' : Total investment = <math>51+68+85 = 204</math> N.
-Here a = 51; b = 68 ; c = 85 ; x = 204
-Using the above formula
-Their shares are <math>\frac{51\  X  \ 300} {204} = 75</math> N <math>\frac{68\  X  \ 300} {204} = 100</math> N <math>\frac{85\  X  \ 300} {204} = 125</math> N
-Their profits are <math>75-51 = 24</math> N ; <math>100-68 = 32</math> N ; <math>125-85 = 40</math> N
-==Formula for filling up of tanks==
-Here we will know the formula for filling of reservoirs (pools, lakes, tanks).
-भजेच्छिदोंऽशैरथ तैर्विमिश्रै रूपं भजेत् स्यात् परिपूर्तिकालः ॥॥
-One divided by the sum of the reciprocals (of the times taken by the sources to fill up a pool) is the time of filling (the pool when the sources are used simultaneously).
-'''Comment''': Suppose the sources take times t<sub>1</sub>, t<sub>2</sub>,... to fill up a reservoir. If they are simultaneously used then
-the time taken = <math>\frac{1}{\frac{1}{t_1}+\frac{1}{t_2}+ ...}</math>
-==Example==
-ये निर्झरा दिनदिनार्धतृतीयषष्ठैः सम्पूर्णयन्ति हि पृथग्पृथगेव मुक्ताः ।
-वापीं यदा युगपदेव सखे विमुक्ताः ते केन वासरलवेन तदा वदाऽशु ॥ ॥
-Four streams flow into a pool and separately they take <math>1</math>, <math>\frac{1}{2}</math>, <math>\frac{1}{3}</math> , <math>\frac{1}{6}</math> days respectively. If all the four are used simultaneously, find the time required to fill up the pool.
-'''Comment''': As explained in the previous stanza.
-Time = <math>\frac{1}{1+2+3+6} =\frac{1}{12}</math>th day
-The four streams can fill up 1, 2, 3, 6 pools in one day and so together
-they can fill up 12 pools in one day. So time for one pool = <math>\frac{1}{12}</math>day.
 ==See Also==
 [[लीलावती में 'पाँच का नियम']]
 ==References==
-<references />
 [[Category:Mathematics in Līlāvatī]]
+[[Category:General]]