Rules of five, seven and nine in Pāṭīgaṇitam

Rules of five, seven and nine is known as (pañca-sapta-nava rāśika) in Samskrita.

Verse

नीते फलेऽन्यपक्षं विभजेद् बहुराशिपक्षमितरेण ।

छेदानां व्यत्यासं कृत्वाऽभ्यासं च राशीनाम् ॥ ४५ ॥

Translation

After transposing the fruit from one side to the other, and then having transposed the denominators (in like manner) and multiplied the numbers (so obtained on either side), divide the side with larger number of quantities (numerators) by the other.

The two sides referred to in this rule are known as (i) the argument side (pramāṇarāśi-pakṣa) and (ii) the requisition side (icchārāśi-pakṣa).

Example 1

If $1{\frac {1}{2}}$ be the interest on $100{\frac {1}{2}}$ for ${\frac {1}{3}}$ of a month, what will be the interest on $60{\frac {1}{4}}$ for $7{\frac {1}{2}}$ months.

Argument Side			Requisition Side
Arguments	$100{\frac {1}{2}}={\frac {201}{2}}$		Requisitions	$60{\frac {1}{4}}={\frac {241}{4}}$
	${\frac {1}{3}}$			$7{\frac {1}{2}}={\frac {15}{2}}$
Fruit	$1{\frac {1}{2}}={\frac {3}{2}}$		Fruit	$0$ (unknown)

Here 0 is written for desired quantity on the requisition side as it is unknown.

Transposing the fruit, we get

Argument Side			Requisition Side
Arguments	$100{\frac {1}{2}}={\frac {201}{2}}$		Requisitions	$60{\frac {1}{4}}={\frac {241}{4}}$
	${\frac {1}{3}}$			$7{\frac {1}{2}}={\frac {15}{2}}$
Fruit	$0$ (unknown)		Fruit	${\frac {3}{2}}$

Transposing the denominators, we get

Argument Side			Requisition Side
Arguments	${\frac {201}{4}}$		Requisitions	${\frac {241}{2}}$
	${\frac {1}{2}}$			${\frac {15}{3}}$
Fruit	${\frac {0}{2}}$ (unknown)		Fruit	$3$

Now we see that the number of quantities (numerators) in Requisition Side is greater than that in Argument Side.

Hence the unknown quantity (interest) $={\frac {241\times 2\times 15\times 3\times 3}{201\times 4\times 1\times 2\times 2}}={\frac {65070}{3216}}={\frac {10845}{536}}=20{\frac {125}{536}}$

Example 2

If 3 laborers earn 5 rupees in 2 days, say what will 8 laborers earn in 9 days?

Argument Side			Requisition Side
Arguments	3		Requisitions	8
	2			9
Fruit	5		Fruit	$0$ (unknown)

Here 0 is written for desired quantity on the requisition side as it is unknown.

Transposing the fruit, we get

Argument Side			Requisition Side
Arguments	3		Requisitions	8
	2			9
Fruit	$0$ (unknown)		Fruit	5

Now we see that the number of quantities in Requisition Side is greater than that in Argument Side.

Hence the unknown quantity (earnings) $={\frac {8\times 9\times 5}{3\times 2}}={\frac {360}{6}}=60$ rupees

See Also

पाटीगणितम् में 'पाँच, सात और नौ के नियम'