History of Hindu Mathematics: A Source Book v1.c8.s13: 13. Commercial Problems

History of Hindu Mathematics: A Source Book v1.c8.s13

Language: English

13. Commercial Problems

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**Interest in Ancient India.** The custom of taking interest is a very old one. In India it can be definitely traced back to the time of Pâṇini (*c.* 700 B.C.) who in his *Grammar* lays down rules validating the use of the suffix *ka* to number names in case of "an interest, a rent, a profit, a tax or a bribe given."¹ The interest became due every month and the rate of interest was generally given per hundred,² although this was not always the case. The rate of interest varied in different localities and amongst different classes of people, but an interest of fifteen per cent per year seems to have been considered just. Thus in Kauṭilya's *Arthaśâstra*, a work of the fourth century B.C., it is laid down: "an interest of a *paṇa* and a quarter per month per cent is just. Five *paṇa* per month per cent is commercial interest. Ten *paṇa* per month per cent prevails in forests. Twenty *paṇa* per month per cent prevails among sea traders."³ The *Gotama Sûtra* states: "an interest of five *mâṣâ* per twenty (*kârṣâpaṇa*) is just."⁴

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¹ Pâṇini's *Grammar*, v. i. 22, 47, 49.

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² It has been pointed out by B. Datta that the idea of *per cent* first originated in India. See his article in the *American Mathematical Monthly*, XXXIV, p. 530.

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³ *Arthaśâstra*, edited and translated into English by R. Shamsastry, Mysore, III, ii, p. 214.

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⁴ *Gotama Sûtra*, xii. 26. Since 20 *mâṣâ* equal a *kârṣâpaṇa*, the rate is 15 per cent annually.

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**Interest in Hindu Ganita.** The ordinary problems relating to the finding out of interest, principal or time etc., the other quantities being given, occur in the section dealing with the Rule of Five. The Hindu works generally contain a section called *miśraka-vyavahāra* ("calculations relating to mixed quantities") in which occur miscellaneous problems on interest. The contents of this section vary in different works, according to their size and scope. Thus the *Āryabhaṭīya* contains only one rule relating to a problem on interest, whilst the *Gaṇita-sāra-saṁgraha* has a large number of such rules and problems.

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**Problem involving a Quadratic Equation.** Āryabhaṭa I (499) gives a rule for the solution of the following problem:

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The principal sum p (= 100) is lent for one month (interest unknown = x). This unknown interest is then lent out for t (= six) months. After this period the original interest (x) plus the interest on this interest amounts to A (= sixteen). The rate-interest (x) on the principal (p) is required.

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The above problem requires the solution of the quadratic equation

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$tx² + px - Ap = 0,$

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which gives x = (-p/2 ± √((p/2)² + Apt))/t.

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The negative value of the radical does not give a solution of the problem; so the result is

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$x = (√(Apt + (p/2)²) - p/2)/t.$

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This is stated by Āryabhaṭa I as follows:

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"Multiply the sum of the interest on the principal and the interest on this interest (A) by the time (t) and by the principal (p). Add to this result the square of half the principal (p/2)². Take the square-root of this. Subtract half the principal (p/2) and divide the remainder by the time (t). The result will be the (unknown) interest (x) on the principal."¹

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¹ *Â*, p. 41. The Sanskrit terms are: *mûla*=principal, *phala*=interest.

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Brahmagupta (628) gives a more general rule. His problem is:

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The principal (p) is lent out for t₁ months and the unknown interest on this (=x) is lent out for t₂ months at the same rate and becomes A. To find x.

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This gives the quadratic $x² + pt₁/t₂ x - Apt₁/(t_-) = 0,$ whose solution is $x = ± (√Apt₁)/t₂ + (pt₁/2t₂)² - pt₁/2t₂.$

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The negative value of the radical does not give a solution of the problem, so it is discarded.

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Brahmagupta states the formula thus:

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"Multiply the principal (p) by its time (t₁) and divide by the other time (t₂) (placing the result) at two places. Multiply the first of these by the mixture (A). Add to this the square of half the other. Take the squareroot of this (sum). From the result subtract half the other. This will be the interest (x) on the principal."²

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² *BrSpSi*, p. 183. This rule is also given by Mahâvîra, *GSS*, p. 71 (44).

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**Other Problems.** Mahâvîra (850) gives two other types of problems on "mixture" requiring the solution of simultaneous equations. As an example of the first type may be mentioned the following:³

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³ *GSS*, p. 69(32).

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"It has been ascertained that the interest for 1½ months (t=rate-time) on 60 (c=rate-capital) is 2½ (i= rate-interest). The interest (on the unknown capital P) for an unknown period (T) is 24 (=I), and 60 (=m =P+T) is the time combined with the capital lent out. What is the time (T) and what is the capital (P)?''

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The problem gives: $iPT/ct & = & I & (1) / P+T & = & m & (2)$ $∴ P-T = ±√(m² - ct/i × 4I)$

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Hence P = ½(m ± √(m² - ct/i × 4I)),

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and T = ½(m √(m² - ct/i × 4I)).

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The above result is stated by Mahâvîra thus:

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``From the square of the mixture (m) subtract the rate-capital (c) divided by the rate-interest (i) multiplied by the rate-time (t) and four times the given interest (4I). Then the operation of *saṅkramaṇa*¹ is performed in relation to the square-root of this and the mixture (m).''²

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¹ Given the numbers a and b, the process of *saṅkramaṇa* is the finding out of half their sum and difference *i.e.* (a+b)/2 and (a-b)/2.

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² *GSS*, p. 68(29). It should be noted that both the signs of the radical are used.

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The second type of problems may be illustrated by the following example:

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``The interest on 30 (P) is 5 (I) for an unknown period (T), and at an unknown rate of interest (i) per 100 (c) per 1½ month (t). The mixture (m=i+T) is 12½. Find i and T."¹

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¹ GSS, p. 69(34).

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The solution is given by $T = ½(m ± √(m² - ctI.4/P)),$ and consequently $i = ½(m √(m² - ctI.4/P)).$

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Mahâvîra states the solution thus:

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"The rate-capital (c) multiplied by its time (t) and the interest (I) and the square of two (=4) is divided by the other capital (P). Then perform the operation of *saṅkramaṇa* in relation to the square-root of the remainder (obtained as the result of subtracting the quotient so obtained) from the square of the mixture (m) and the mixture."²

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² GSS, p. 69(33).

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**Miscellaneous Problems on Interest.** Besides the problems given above various other interesting problems are found in the Hindu works on *pâtîgaṇita*. Thus Brahmagupta gives the solution of the following problem:

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*Example.* In what time will a given sum s, the interest on which for t months is r, become k times itself?

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The rule for the solution of the above is:³

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³ *BrSpSi*, p. 181.

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"The given sum⁴ multiplied by its time and divided by the interest,¹ being multiplied by the factor² less one, is the time (required).”

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⁴ The Sanskrit term used is *pramâṇa* (argument).

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¹ The Sanskrit term used is *phala* (fruit).

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² The Sanskrit term used is *guṇa* (multiple).

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The *Gaṇita-sâra-saṃgraha* (850) contains a large number of problems relating to interest. Of these may be mentioned the following:

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(1) “In this (problem), the (given) capitals are (c₁=) 40, (c₂=) 30, (c₃=) 20 and (c₄=) 50; and the months are (t₁=) 5, (t₂=) 4, (t₃=) 3 and (t₄=) 6 (respectively). The sum of the interests is (m=) 34. (Assuming the rate of interest to be the same in each case, find the amounts of interest in each case).”³

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³ *GSS*, p. 70(38).

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Here, if the rate of interest per month for 1 be r, then $r = x₁/(c₁ t₁) = x₂/(c₂ t₂) = x₃/(c₃ t₃) =$ where x₁, x₂, x₃, . . . . . . are the interests earned on the capitals c₁, c₂, c₃, . . . . . . in t₁, t₂, t₃ . . . . . . months respectively.

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Therefore, $x₁/(c₁ t₁) = x₂/(c₂ t₂) = x₃/(c₃ t₃) = = (x₁ + x₂ + x₃ + )/(c₁ t₁ + c₂ t₂ + c₃ t₃ + )$ $= m/(c₁ t₁ + c₂ t₂ + c₃ t₃ + )$ or $x₁ = (m c₁ t₁)/(c₁ t₁ + c₂ t₂ + c₃ t₃ + ), etc.$

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This formula is given by Mahāvîra for the solution of the above problem.¹

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¹ *GSS*, p. 70(37). The formula clearly shows that Mahâvîra knew the algebraic identity $a/b = c/d = e/f = = (a+c+e+)/(b+d+f+).$

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(2) "(Sums represented by) 10, 6, 3 and 15 are the (various given) amounts of interest, and 5, 4, 3 and 6 are the (corresponding) months (for which the interests have accrued); the sum of the (corresponding) capital amounts is seen to be 140. (Assuming the rate of interest to be the same in each case, find out these capital amounts)."²

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² *GSS*, p. 70(40). The solution is given by Rule 39 on the same page.

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(3) "Here (in this problem) the (given) capital amounts are 40, 30, 20 and 50; and 10, 6, 3 and 15 are the (corresponding) amounts of interest; 18 is the quantity representing the mixed sum of the respective periods of time. (Find out these periods separately, assuming the rate of interest to be the same in each case)."³

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³ *GSS*, p. 70(43). The solution is given by Rule 42 on the same page.

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(4) "The interest on 80 for 3 months is unknown; 7⅘ is the mixed sum of that (unknown quantity taken as the) capital lent out and of the interest thereon for 1 year. What is the capital here and what the interest?"⁴

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⁴ *GSS*, p. 71(46). This is similar to Âryabhaṭa's problem given before (p. 217).

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(5) "The mixed sums (capital+interest) are 50, 58 and 66, and the months (during which interests have accrued) are 5, 7 and 9 (respectively). Find out what the interest is (in each case, the capital being the same)?"¹

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¹ *GSS*, p. 71 (48). The solution requires the use of the identity $a/b = c/d = (a - c)/(b - d).$

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(6) "The mixed sums of the capital and periods of interest are 21, 23, and 25; here (in this problem) the amounts of interest are 6, 10 and 14. What is the common capital?"²

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² *GSS*, p. 72 (52).

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(7) "Borrowing at the rate of 6 per cent and then lending out at the rate of 9 per cent, one obtains in the way of differential gain 81 at the end of 3 months. What is the capital (utilised here)?"³

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³ *GSS*, p. 72 (55).

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(8) "The monthly interest on 60 is exactly 5. The capital lent out is 35; the (amount of the) instalment (to be paid) is 15 in (every) 3 months. What is the time of discharge of that debt?"⁴

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⁴ *GSS*, p. 73 (59).

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(9) "The mixed sum (of the capital amounts lent out) at the rates of 2, 6 and 4 per cent per mensem is 4400. Here the capital amounts are such as have equal amounts of interest accruing after 2 months. What (are the capital amounts lent, and what is the equal interest)?"⁵

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⁵ *GSS*, p. 73 (61).

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(10) "A certain person gives once in 12 days an instalment of 2⅗, the rate of interest being 3 per cent (per mensem). What is the capital amount of the debt discharged in 10 months?"⁶

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⁶ *GSS*, p. 73 (65).

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(11) "The total capital represented by 8520 is invested (in parts) at the (respective) rates of 3, 5 and 8 per cent (per month). Then, in this investment, in 5 months the capital amounts lent out are, on being diminished by the respective amounts of interest, (found to be) equal in value. (What are the respective amounts invested thus?)"¹

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¹ *GSS*, p. 74(67).

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(12) "The total capital represented by 13740 is invested (in parts) at the (respective) rates of 2, 5 and 9 per cent (per month), then, in this investment, in 4 months the capital amounts lent out are, on being combined with the (respective) amounts of interest, (found to be) equal in value. (What are the respective amounts thus invested?)"²

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² *GSS*, p. 74(67).

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(13) "A certain man borrows a certain (unknown) sum of money at an interest of 5 per cent per month. He pays the debt in instalments, due every ⅗ of a month. The instalments begin with 7 and increase in arithmetical progression, with 7 as the common-difference. 60 is the maximum amount of instalment. He gives in the discharge of his debt the sum of a series in arithmetical progression consisting of 60/7 terms. After the payment of each instalment, interest is charged only on that part of the principal which remains to be paid. What is the total payment corresponding to the sum of the series, what is the interest (which he paid), what is the time of the discharge of the debt, (and what is the principal sum borrowed)?"³

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³ *GSS*, pp. 74f, (72-73½). The text of the problem is very obscure. The translation given here is after Rangacarya.

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**Barter and Exchange.** The Hindu name for barter is *bhāṇḍa-prati-bhāṇḍa* ("commodity for commodity"). All the Hindu works on *pāṭīgaṇita* contain problems relating to the exchange of commodities. It is pointed out in these works that problems on barter are cases of compound proportion, and can be solved by the Rule of Five, etc. A typical problem on barter is the following:

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“If three hundred mangoes be had in this market for one *dramma*, and thirty ripe pomegranates for a *paṇa*, say quickly, friend, how many (pomegranates) should be had in exchange for ten mangoes?”¹

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¹ *L*, p. 20.

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**Other Types of Commercial Problems.** Of various other types of commercial problems found in the Hindu works may be mentioned (1) problems on partnership and proportionate division, and (2) problems relating to the calculation of the fineness of gold.² Most of these problems are essentially of an algebraic character, but they are included in *pâṭîgaṇita* (arithmetic). The formulæ giving the solution of each type of examples precede the examples. These formulæ are too numerous to be mentioned. The following examples, however, will illustrate the nature and the scope of such problems:

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² Such problems are found in the *Lîlâvatî*, the *Gaṇita-sâra-saṁgraha*, the *Triśatikâ*, etc.

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(1) A horse was purchased by (nine) dealers in partnership, whose contributions were one, etc., upto nine; and was sold by them for five less than five hundred. Tell me what was each man’s share of the sale-proceed.

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(2) Four colleges, containing an equal number of pupils, were invited to partake of a sacrificial feast. A fifth, a half, a third and a quarter (of the total number of pupils in the college) came from the respective colleges to the feast; and added to one, two, three and four, they were found to amount to eighty-seven; or, with those deducted, they were sixty-seven. Find the actual number of the pupils that came from each college.

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(3) Three (unequal) jars of liquid butter, of water and of honey, contained thirty-two, sixty and twenty-four *pala* respectively: the whole was mixed together and the jars filled again. Tell me the quantity of butter, of water and of honey in each jar.¹

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¹ This and the two previous examples are given by Pṛthudakasvâmî to illustrate Rule 16 of the *gaṇitâdhyâya* of the *Brâhma-sphuṭa-siddhânta*.

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(4) According to an agreement three merchants carried out the operations of buying and selling. The capital of the first consisted of six *purâṇa*, that of the second of eight *purâṇa*, but that of the third was unknown. The profit obtained by these men was 96 *purâṇa*. In fact the profit obtained by him (the third person) on his unknown capital happened to be 40 *purâṇa*. What was the amount thrown by him into the transaction and what was the profit of each of the other two merchants?²

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² *GSS*, p. 94(223-5).

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(5) There were four merchants. Each of them obtained from the others half of what he had with him (at the time of the respective transfers of money). Then they all became possessed of equal amounts of money. What was the measure of money each had to start with?³

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³ *GSS*, p. 99(267½).

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(6) A great man possessing powers of magical charm and medicine saw a cock fight going on, and spoke separately in confidential language to both the owners of the cocks. He said to one, "If your bird wins, then you give the stake-money to me. If, however, your bird loses then I shall give you two-thirds of that stake-money." He went to the owner of the other cock and promised to give three-fourths (of his stake-money on similar conditions). In each case the gain to him could be only 12 (gold-pieces). Tell me, O ornament on the head of mathematicians, the money each of the cock-owners had staked.¹

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¹ *GSS*, pp. 99-100(270½-2½).

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(7) The mixed price of 9 citrons and 7 fragrant wood-apples is 107; again the mixed price of 7 citrons and 9 fragrant wood-apples is 101. O arithmetician, tell me quickly the price of a citron and of a wood-apple, having distinctly separated those prices.²

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² *GSS*, p. 84(140½-2½).

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(8) Pigeons are sold at the rate of 5 for 3 (*paṇa*), *sârasa* birds at the rate of 7 for 5 (*paṇa*), swans at the rate of 9 for 7 (*paṇa*) and peacocks at the rate of 3 for 9 (*paṇa*). A certain man was told to bring at these rates 100 birds for 100 (*paṇa*) for the amusement of the king's son, and was sent to do so. What (amount) does he give for each (of the various kinds of birds that he buys)?³

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³ *GSS*, p. 85(152-3).

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(9) There are 1 part (of gold) of 1 *varṇa*, 1 part of 2 *varṇa*, 1 part of 3 *varṇa*, 2 parts of 4 *varṇa*, 4 parts of 5 *varṇa*, 7 parts of 14 *varṇa*, and 8 parts of 15 *varṇa*. Throwing these into the fire, make them all into one (mass), and then (say) what the *varṇa* of the mixed gold is. This mixed gold is distributed among the owners of the foregoing parts. What does each of them get?⁴

Linked texts (8)

Aṣṭādhyāyī — Pāṇini, Sanskrit
Gautamadharmasūtra — Gautama (dharmasūtrakāra), Sanskrit
Kauṭilīya Arthaśāstra — Kauṭilya, Sanskrit
Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit