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Mathematics in Ancient and Medieval India c5.s12: Commercial Problems

Mathematics in Ancient and Medieval India c5.s12

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Commercial Problems

c5.s12.1

The ancient Indian scholars dealt with various commercial problems. Of these, the problems on mixture (*miśraka vyavahāra*) are interesting, for it relates to many problems on capital and interest (when the amount and rate of interest given), rules for finding time (for a sum lent out at simple interest and paid back in equal monthly instalments), *varṇa* of an alloy, weights, shares of partners, wages and payments etc. besides many others. The solution of these problems sometimes require the knowledge of quadratic equations. There are also some problems which are of indeterminate types. A few may be discussed in what follows:

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*Problem 1.* "The rate of interest being 5 percent per month, a certain sum is amounted to 96 in a year. Say friend, what is the capital (*mūla*) and the interest (*vṛddhidhana*)".¹

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¹ *Gaṇitatilaka*, v. 112, p. 83; *Pāṭīgaṇita*, Ex. 52; similar examples occur in *Gaṇita Kaumudī*, 1, lines 10–11, p. 60, Pṛthudaka's comm. on *Brāhmasphuṭasiddhānta* x ii. 14.

c5.s12.4

Here argument (*pramāṇarāśi*) = p = 100 (say), time (*nija-kāla*) = t = 1 month, fruit (*phala*) = f = 5, amount = capital (*mūladhana*) + interest (*vṛddhidhana*) = a = 96, other time (*parakāla*) = n = 1 year = 12 months.

c5.s12.5

The rules for finding the capital and interest are :

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$Capital = (p. t. a)/(p. t + f.n), Interest = (f. n. a)/(p.t + f.n)$

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$∴ Capital = 60, Interest = 36.$

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*Problem 2.* "The rate of interest being 5 percent per month, the commission of the surety (*bhāvyaka*) 1 percent per month, the fee of the calculator (*vṛtti*) ½ per cent per month, the charges of the scribe ¼ percent per month, a certain sum amounts to 905 in a year. (Find the capital, interest, and the shares of the surety, calculator and the scribe )".²

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² *Gaṇitatilaka*, v. 114, p. 83; *Pātīgaṇita* Ex-54.

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Here p = 100, t = 1, f₁ = 5, f₂ = 1, f₃ = ½, f₄ = ¼, a = 905, n = 1 year = 12

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$Capital = (p. t. a)/(p.t + ( f₁ + f₂ + f₃ + f₄ ) × n.) = 500$

c5.s12.12

Interest = (f_1 n. a)/(p.t + ( f_1 + f_2 + f_3 + f_4). n) = 300 Commission of surety = (f_2. n. a)/(p.t + ( f_1 + f_2 + f_3 + f_4). n) = 60

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Similarly the fee of the calculator = 30

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and the charges of the scribe = 15

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*Problem 3*. “What in the *varṇa* when (three pieces of gold of) 1 6/(3′), 2 5/(6′), 1 5/(2′) *māṣas*, having 2 3/(2′), 10, 1 5/(2′) *varṇas* (respectively) are mixed in one ?”¹

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¹ *Pātīgonita*, Ex. 62; Similar examples occur ih *Bakhshali Ms.* 16. verso, 17 recto and 17 verso; *Gaṇitasārasaṃgraha*, vi, 170-171.

c5.s12.17

Here w₁ = 1 6/(3′), w₂ = 2 5/(6′), w₃ = 1 5/(2′) and v₁ = 2 3/(2′), v₂ = 10, v₃ = 1 5/(2′). The mixed *varṇa* v of the alloy is given by the rule as follows.²

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² C/o. *Bakhšālī Ms.* 16 verso; *Gaṇitasārasāṃgraha*, vi. 169; *Pātīganita*, Rule 52; *Gaṇita Kaumudī*, I, lines 5-7, p. 76.

c5.s12.19

v = (v_1 w_1 + v_2 w_2 + v_3 w_3 + \dots + v_n w_n)/(w_1 + w_2 + \dots + w_n)

c5.s12.20

here v = (2 3/(2^\prime). 1 6/(3^\prime) + 10. 2 5/(6^\prime) + 1 5/(2^\prime). 1 5/(2^\prime))/(1 6/(3^\prime) + 2 5/(6^\prime) + 1 5/(2^\prime))

c5.s12.21

= 1911/204 = 9 75/204 = 9 25/68.

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*Problem 4*: “There are two small balls of gold of equal worth (i e. having equal quantities of pure gold, whose combined weight is 5 *māṣas*. When they are respectively combined with ⅓ part and ½ part of the other, they become 10 and 9 *varṇas* respectively. (Find their weights and *varṇas* respectively. )”³

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³ *Pātīganita*, Ex. 69.

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Let w₁ and w₂ be the weights of the two gold balls having *varṇas* v₁ and v₂.

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here w₁ + w₂ = 5……(1) w₁ v₁ = w₂v₂……( 2 )

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v₁ = ((w₁ + ⅔ w₂))/((1 + ⅔) w₁) . 10 ( 3 )

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and v₂ = (w₂ + ½ w₁)/((1 + ½) w₂) . 9 ( 4 )

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The results (1), (2), (3) and (4), give w₁ = 2 māṣas, w₂ = 3 māṣas and v₁ = 12 and v₂ = 8.

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The results (3) and (4) were known also to other scholars.¹

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¹ C/o. *Gaṇitasārasaṃgraha*, vi, 209–12, *Pātīgaṇita*, Rule 58, *Gaṇita Kaumudī*, 1, lines 9–18, p. 81.

c5.s12.31

*Problem 5* : “Pigeons are sold at the rate of 5 for 3 (*rūpas*), cranes at the rate of 7 for 5 (*rūpas*), swans at the rate of 9 for 7 (*rūpas*), and peacocks at the rate of 3 for 9 ( *rūpas* ). Knowing the rates as stated above, bring 100 birds for 100 *rūpas* for the amusement of the prince”.²

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² A similar problem occurs in the *Bakhśālī Ms.* III, 58 verso which runs as follows : “A man earns 3 *maṇḍas* in a day, a woman 1½ *maṇḍas* in a day, and a *śūḍha* ½ *maṇḍa* in a day. If 20 of them earn 20 *maṇḍas* in a day, how many of each category are there ?”.

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This is an indeterminate problem involving large number of solutions and occurs in the *Gaṇitasārasaṃgraha* ( vi, 152–53 ), *Pātīgaṇita*, ( Ex. 78–79 ) ; *Bījagaṇita* of Bhāskara II ( Ānandāśrama Sanskrit series p. 163 ), and *Gaṇitakaumudī* of Nārāyaṇa ( p. 1, lines 2–5, p. 93). The problem leads to the solution of :

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$5x + 7y + 9z + 3t = 100$ $3x + 5y + 7z + 9t = 100$

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where (5x, 3x), (7y, 5y), (9z, 7z) and (3t, 9t) are the number and price of the pigeons, cranes, swans, and peacocks. Shukla has given 16 integral solutions of which four such solutions were given earlier by the commentator of Śrīdharācārya.¹

c5.s12.36

¹ *Pātīgaṇita*, Ed. by Shukla, p. 51.

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