Trairāśika (Rule of Three)

Introduction

In the ancient Indian mathematical texts topics like ratio, proportion etc are dealt under the section rule of three. Ratio is used whenever comparison involving numbers.

For example: Cost of a bicycle is Rs. 10,000 and that of a motor bike is Rs. 1,00,000.

when we compare the cost of both the items. ${\frac {100000}{10000}}={\frac {10}{1}}=10:1$

Hence the cost the motorbike is ten times the cost of the bicycle. Ratio is the comparison by division. Ratio is denoted by ":" . Ratio expresses the number of times one quantity with the other. The two quantities must be in the same unit.

The two values are said to be in direct proportion when an increase/decrease in one results in an increase/decrease in the other by the same factor.

Direct proportion is seen in the following instances.

1. Cost of fuel increases as quantity of fuel increases
2. Time taken increases with increase in pages to be typed.
3. Cost of vegetable increases as weight of the vegetable increases.
4. Number of units manufactured by a machine increases with the number of hours the machine works.

Trairāśika (Rule of Three)

The Hindu name for the Rule of Three is called "trairāśika" (three terms, hence rule of three). The term trairāśika occurs in Bakshālī  manuscript, Āryabhaṭīya. Bhāskara I (c. 525) remarked on the origin of this name as "Here three quantities are needed (in the statement and calculation) so the method is called trairāśika (the rule of three terms)". A problem on the rule of three has this form : if p yields f , what will i yield ?. The three terms used are p, f , i . Hindus called the term p (pramāṇa - argument), f(phala - fruit) and i (icchā - requisition). Sometimes they are referred to simply as the first ,second and third respectively.

Āryabhaṭa II gave different names as māna, vinimaya and icchā respectively to the three terms.

Brahmagupta gives the rule as "In the rule of three pramāṇa (argument), phala(fruit) and icchā(requisition) are the (given) terms; the first and the last terms must be similar. The icchā multiplied by the phala and divided by the pramāṇa gives the fruit (of the demand) ".

Bhāskara I in his Āryabhaṭīya-bhaṣya talks about the Trairāśika

त्रयो राशयः समाहृताः त्रिराशिः । त्रिराशिः प्रयोजनमस्य गणितस्येति त्रैराशिकः । त्रैराशिके फलराशिः त्रैराशिकफलराशिः । (Āryabhaṭīya-bhaṣya by Bhāskara I on 11.26, p.116)

"Trairāśi is the three quantities assembled . It is (called) Trairāśika because of this computation with these quantities. Trairāśika -phalarāśi is the desired result in the Rule of Three. "

Trairāśika involves three known quantities and one unknown quantity. The known quantities are pramāṇa (known measure), pramāṇaphala (result related to known measure) and icchā (desired measure). The term used for the unknown quantity is icchāphala (result related to desired measure).

Example: A car covers 30 kms with 2 litres of petrol. To cover 150 kms, how many litres of petrol are required.?

Solution: For 30 kms, petrol needed = 2 litres

For 150 kms, petrol needed = 'x' litres

Here pramāṇa = 30 ; pramāṇaphala = 2 ; icchā = 150 ; icchāphala = 'x' litres

pramāṇa -> pramāṇaphala ( 30 -> 2)

icchā -> (icchā X pramāṇaphala) / pramāṇa = icchāphala

150 -> ( 150 x 2) / 30 = 300/30 = 10

x= 10 ; 10 litres of petrol are required to cover 150 kms.

Solution on Trairāśika by another mathematician Śrīdhara states: ""Of the three quantities, the pramāṇa ("argument") and icchā ("requisition") which are of the same denomination are the first and the last; the phala ("fruit") which is of a different denomination stands in the middle; the product of this and the last is to be divided by the first."

Example from -Līlāvatī vs.74,p.72 : If $2{\frac {1}{2}}$ palas (a weight measure) of saffron costs ${\frac {3}{7}}$ niṣkas (a unit of money), O expert businessman , tell me quickly what quantity of saffron can be bought for $9$ niṣkas.

Solution:

pramāṇa and pramāṇaphala - ${\frac {3}{7}}$ niṣkas and $2{\frac {1}{2}}$ palas

icchā and icchāphala - $9$ niṣkas and x

As per Rule of Three - place the quantities indicated by niṣkas in first (pramāṇa ) and third (pramāṇaphala) column. place the remaining quantity in the middle column .

First - quantity (pramāṇa)	Middle - quantity (pramāṇaphala)	Last - quantity (icchā)
${\frac {3}{7}}$	$2{\frac {1}{2}}={\frac {5}{2}}$	$9$

Result = ${\frac {Middle-quantity\ X\ Last-quantity}{First-quantity}}$

icchāphalam = ${\frac {{\frac {5}{2}}\ X\ 9}{\frac {3}{7}}}$= ${\frac {5\ X\ 9\ X\ 7}{2\ X\ 3}}={\frac {105}{2}}$ palas

Hence the quantity of saffron that can be bought for $9$ niṣkas is $52{\frac {1}{2}}$ palas.

See Also

त्रैराशिक (तीन का नियम)