पाटीगणित (श्रीधराचार्य - प्राचीन संस्कृत टीका एवं प्रो. कृपाशंकर शुक्ल सविस्तार व्याख्या)

34 DETERMINATONS
took a house bearing a rent of 40 (rûpas) per month. Say,
learned man, after how much time is the debter relieved of
his debt, and what does the rich man get by the gain of bare
accommodation.¹
Rule for converting several bonds into one :
51. The sum (samâsa) of the interests (phala) (accru-
ing on the given bonds) for the elapsed months (gatakâla), being
divided by the sum (aikya) of the interests (on the same
bonds) for one month, gives the time (in months for the equi-
valent single bond); and 100 times the sum of the interests for
one month (on the bonds), being divided by the sum of the
capitals (dhanayoga) (of the bonds), gives the rate of interest
per cent (per month) (for the single bond).²
Let the capitals, rates of interest and the times elapsed for the given
n bonds be as follows :
Capitals : P₁, P₂, P₃,..., Pₙ.
Rates of interest : r₁, r₂, r₃, ... , rₙ per cent per month
Times elapsed : t₁, t₂, t₃, .. , tₙ months.
Then the time elapsed (t) and the rate of interest (r) for the equivalent
single bond are given by
P₁ t₁ r₁ / 100 + P₂ t₂ r₂ / 100 + ... + Pₙ tₙ rₙ / 100
t = —————————————————————————————————————— months,
P₁ r₁ / 100 + P₂ r₂ / 100 + P₃ r₃ / 100 +....+ Pₙ rₙ / 100
( P₁ r₁ / 100 + P₂ r₂ / 100 + P₃ r₃ / 100 +....+ Pₙ rₙ / 100 ) × 100
and r = ——————————————————————————————————————————————————————
P₁ + P₂ + P₃ + .... + Pₙ
per cent per month.
Âryabhaṭa II puts the above results in the following simplified forms : ³
P₁ t₁ r₁ + P₂ t₂ r₂ + ... + Pₙ tₙ rₙ
t = ——————————————————————————— months,
P₁ r₁ + P₂ r₂ + ... + Pₙ rₙ
P₁ r₁ + P₂ r₂ + ... + Pₙ rₙ
and r = ——————————————————————— per cent per month.
P₁ + P₂ + ... + Pₙ

¹ A similar example occurs in GK, I, p. 73, lines 8-10.
² Cf. GT, p. 86, lines 12-15. Also see GSS, vi. 77-77½.
³ See MSi, xv. 33. The reading as emended by S. Dvivedi is
incorrect.
Indological Truths

MIXTURES. SIMPLE INTEREST 35
Ex. 57-58. (There are 4 bonds on which) capitals
amounting to 100, 200, 300, and 400 are given (to some one
on interest) at the rates of 2, 3, 4, and 5 per cent (per month)
in the respective order ; and months amounting to 2, 3, 5, and
4 each multiplied by 2, have passed (since the execution of
the respective bonds). Say, how would a single bond
(eka-patra) be now made out of these.¹
Ex. 59. Also say, O learned (mathematician), how a
single bond be made (out of 4 bonds) with the same capitals
as previously stated and with rates per cent (per month) of
interest augmented by ½ (in each case) and months elapsed
increased by ¼ (in each case).²
Rule for finding the time in which a capital lent out on
interest at a given rate would become a (given) multiple of
itself :
52(i). The product of the time and the argument,
being divided by the fruit and (then) multiplied by the
multiple minus one, gives the (required) time.³
Ex. 60(i). A sum of money is put to interest at 5 per
cent per month. When will it become twice of itself ?⁴
Ex. 60(ii). And when will (another sum of money) put
to interest at 3½ per cent (per month) become 1¼ of itself ?⁵
In Ex. 60(i), time = 1 month, argument = 100, fruit = 5, and
multiple = 2. Therefore the required time
1 × 100
= ------------ × (2 — 1), i.e., 20 months.
5
Similarly in Ex. 60(ii), time = 1 month, argument = 100, fruit
= 3½, i.e. 7/2, and multiple = 1¼. Therefore the required time
1 × 100 50
= ------------ × (1¼ — 1), i.e. -- months
7/2 7
== 7 months, 4 days, 17 ghaṭikās, 8 4/7 caṣakas.

¹ Cf. GSS, vi. 78½ ; GT. p. 86, lines 25-28 (contd. on p. 87 ),
² Cf. GT, p. 87, lines 22-23.
³ Cf. BrSpSi, xii. 14 (i); GT, p. 85, line 19. Both are literally the same.
⁴ A similar example occurs in Pṛthûdaka's comm. on BrSpSi, xii.
14. The same example occurs in GT, p. 85, lines 24-25.
⁵ A similar example occurs in Pṛthûdaka's comm. on BrSpSi, xii.
14. The same example occurs in GT, p. 86, lines 4-5.

36 DETERMINATIONS
(ii) ALLIGATION
Rule for finding the varṇa of the alloy obtained by melting
together a number of pieces of gold of given weights and
varṇas :
52(ii). The sum of the products of weight and varṇa
of the several pieces of gold, being divided by the sum of the
weights of the pieces of gold, gives the varṇa (of the alloy).¹
That is, if n pieces of gold of weights w₁, w₂, w₃, ..., wₙ and
varṇas v₁, v₂, v₃, ..., vₙ respectively are melted together, then the varṇa
v of the alloy is given by
v = (w₁ v₁ + w₂ v₂ + ... + wₙ vₙ) / (w₁ + w₂ + ... + wₙ) .
The term varṇa used above is analogous (though not equivalent) to
the modern term ‘carat’. It is used to denote the fineness of gold. Gold
of 1 varṇa contains 1 part of pure gold and 15 parts of impurities (in the
form of baser metals) ; gold of 2 varṇas contains 2 parts of pure gold and
14 parts of impurities ; and so on. Pure gold, called kalyâṇa-suvarṇa, is of
16 varṇas (or 24 carats). Thus the amount of pure gold in a piece of
gold of weight w and varṇa v is equal to wv/16.
The rationale of the above rule is as follows : The amount of pure
gold in the n pieces of gold mentioned above is equal to
(w₁ v₁)/16 + (w₂ v₂)/16 + ... + (wₙ vₙ)/16 , (1)
and the amount of pure gold in the alloy of those pieces of gold is equal to
((w₁ + w₂ + ... + wₙ) v) / 16 (2)
Since (1) and (2) are equal, therefore
((w₁ + w₂ + ... + wₙ) v) / 16 = (w₁ v₁)/16 + (w₂ v₂)/16 + ... + (wₙ vₙ)/16 .
Hence
v = (w₁ v₁ + w₂ v₂ + ... + wₙ vₙ) / (w₁ + w₂ + ... + wₙ) .
Ex. 61. Of what varṇa do 9, 5, and 17 mâṣas of
gold of 12, 10 and 11 varṇas (respectively) become when
melted together ?

¹ Cf. BM, III, H 1, 16 verso ; GSS, vi. 169 (i), 172½ ; L (ASS),
p. 99, vs. 103 (i); GK, I, p. 76, lines 5-7.

MIXTURES. ALLIGATION 37
Ex. 62. Of what varṇa do (the three pieces of gold of)
5 plus ⅓, 4 plus ⅙, and 7 plus ½ māṣas and 11 plus ½, 10, 8
minus ½ varṇas (respectively) become when mixed in one ?¹
Rule for finding the varṇa of the refined gold (obtained by
mixing up and refining a number of pieces of gold of given
weights and varṇas), when the weight of the refined gold is
known :
53(i). The sum of the products of varṇa and weight
of the several pieces of gold, being divided by (the weight of)
the refined gold, gives the varṇa (of the refined gold).²
That is, if n pieces of gold of weights w₁, w₂, ..., wₙ and varṇas
v₁, v₂, ... , vₙ respectively are mixed up and refined, then the varṇa v
of the refined gold is given by
w₁v₁ + w₂v₂ + ... + wₙvₙ,
v = -------------------------
w
where w is the weight of the refined gold.
The rationale is similar to that of rule 52(ii).
Ex. 63. 5, 8, and 6 suvarṇas of varṇas 12, 9, and 15 minus
½ respectively, being mixed up and refined, are seen to be
reduced to 16 suvarṇas in all. Quickly say the varṇa of that
(refined gold).
Rule for finding the weight of the refined gold, when the
varṇa of the refined gold is known :
53(ii). The same (sum of the products of varṇa and
weight of the several pieces of gold), being divided by the
varṇa (of the refined gold), gives the weight of the refined gold.³
That is, if n pieces of gold of weights w₁, w₂, ... , wₙ and of
varṇas v₁, v₂, .. , vₙ respectively are mixed up and refined, then the weight
w of the refined gold is given by
w₁v₁ + w₂v₂ + ... + wₙvₙ
w = --------------------------- ,
v
where v is the varṇa of the refined gold.
¹ Similar examples occur in BM, III, H 1, 16 verso; H 2, 17 recto
and 17 verso; GSS, vi. 170-171½.
² Cf. MSi, xv. 39 (ii); GK, I, p. 76, vs. 18.
³ Cf. BM, III, H3, 18 recto; GSS, vi. 175½, 176½, MSi, xv. 40;
L (ASS), p. 102, vs. 106; GK, I, pp. 76-77, vs. 19.

38 DETERMINATIONS
Ex. 64. (Three pieces of gold of weights) 10, 7, and 5
mâṣas are seen to be of 9, 8, and 6 varṇas (respectively). On
being mixed up and refined, they become of 11 varṇa. Say the
weight of the refined gold.
Rule for finding the unknown varṇa of one of the pieces of
gold, when the varṇa of the alloy of those pieces of gold is
known :
54. The sum of the weights of (all) the pieces of gold
being multiplied by the varṇa of the alloy (of those pieces
of gold), (then) diminished by the products of weight and
varṇa (of the pieces of gold of known weight and varṇa), and
(then) divided by the weight of the piece of gold of unknown
varṇa, gives the (unknown) varṇa.¹
That is, if n pieces of gold of weights w₁, w₂, ..., wₙ and varṇas
v₁, v₂, ..., vₙ respectively are mixed up with another piece of gold of
weight w and of unknown varṇa, and the varṇa of the alloy is found to
be v', then the unknown varṇa v will be given by
v = ((w₁ + w₂ + .... + wₙ + w) v' - (w₁ v₁ + w₂ v₂ + ... + wₙ vₙ)) / w .
Ex. 65. 1, 2, and 6 suvarṇas of 5, 3, and 4 kṣayas
(respectively) being mixed up with a pala of gold of unknown
varṇa become of 12 varṇa. (Find the unknown varṇa).²
Just as the varṇa of gold indicates the fineness of gold, so the kṣaya
of gold indicates the impurity of gold. Gold of 1 kṣaya contains 1 part of
impurities and 15 parts of pure gold; gold of 2 kṣayas contains 2 parts of
impurities and 14 parts of pure gold; and so on. Thus the varṇa and kṣaya
are related by the formula
varṇa + kṣaya = 16.
Rule for finding the unknown weight of one of the pieces of
gold, when the varṇa of the alloy of those pieces of gold is
known :
55. The sum of the weights of the pieces of gold multi-
plied by the varṇa of the alloy (of those pieces of gold), (then)
¹ Cf. MSi, xv. 39 (i); GK, 1, p. 76, vs. 18.
² Similar examples occur in BM, III, H3, 18 recto; GSS, vi. 178; 179;
L (ASS), p. 102, vs. 107; GK, I, p. 77, lines 4-7.
Indological Truths

MIXTURES. ALLIGATION 39
diminished by the products of weight and varṇa (of the pieces
of gold of known weights and varṇas), and (then) divided by
the varṇa of the piece of gold of unknown weight minus the
varṇa of the alloy, gives the (unknown weight of) gold.¹
That is, if n pieces of gold of weights w₁, w₂, ..., wₙ and varṇas
v₁, v₂, ..., vₙ respectively are mixed up with a piece of gold of varṇa v but
of unknown weight, and the varṇa of the alloy is found to be vʹ, then
the unknown weight w will by given by
w = [(w₁ + w₂ + ... + wₙ) vʹ − (w₁v₁ + w₂v₂ + ... + wₙvₙ)] / (v − vʹ)
Ex. 66. 2, 3, and 4 mâṣas of gold of kṣayas 7, 4, and 5
(respectively) being mixed up with some gold of kṣaya 2, the
alloy is found to be of kṣaya 4. Quickly say with how much
of that gold (of kṣaya 2 have they been mixed up).²
Rule for finding how much of inferior gold (and how much
of pure gold) will have to be taken in constructing a number
of test sticks of progressively decreasing varṇas :
56. The weight in terms of mâṣas (mâṣâtmaka mâna) of
one (test) stick being divided by the difference in terms of
yavas between the varṇas (of pure and inferior gold) and multi-
plied one time, 2 times, etc., of the progressive decrease of
varṇa in terms of yavas, gives the respective amounts of inferior
gold.³
That is, if n test sticks of weight w and varṇas 16 − k, 16 − 2k,...,
16 − nk are to be made out of pure gold and inferior gold of varṇa 16 − K,
then for the construction of those test sticks we will have to take the
following amounts of inferior gold in the respective order :
(w / K) × k, (w / K) × 2k, ......, (w / K) × nk.
¹ Cf. BM, III, H3, 18 verso; GSS, vi. 175½, 180; MSi, xv. 41; L
(ASS), p. 103, vs. 108; GK, I, p. 77, vs. 20.
² Cf. BM, III, H3, 18 verso; GSS, vi. 181; L (ASS), p. 103, vs. 109;
GK, I, p 77, lines 16-19.
³ Cf. GK, I, p. 78, lines 18-21, and p. 79, lines 1-2. Also see GSS,
vi. 192.

40 DETERMINATIONS
Subtracting these amounts severally from w, we will get the respec-
tive amounts of pure gold.
The rationale is as follows :
Let the required amounts (weights) of gold of varṇa 16-K and of
varṇa 16 be respectively
w₁ , w₂ ,......... , wₙ
and (w - w₁) , (w - w₂) , ... , (w - wₙ) .
Then we evidently have
(16 - K) wᵣ + 16 (w - wᵣ) = w(16 - rk),
where r = 1, 2, ... , n ; whence
wᵣ = (w / K) × rk .
The text does not give the relation between a māṣa and a yava,
but the commentator takes one māṣa as equivalent to 16 yavas.
Ex. 67-68. A series of varṇas, from 0 kṣaya to 6 kṣaya,
uniformly increasing by ½ kṣaya, is to be constructed of sticks
of 2 māṣas each. O mathematician, if you know then quickly
say after calculating how much gold of varṇa 16 and how
much gold of varṇa 10 will have to be taken (in each case).¹
Rule for finding the weights of two gold balls of equal value
(i.e., containing equal amounts of pure gold), when the sum
of their weights and also the varṇas of the alloys formed by
mixing each of them with given proportions of the other
are known :
57. The (given) varṇas (severally) divided by the (given)
fractions, each increased by 1, should be mutually diminished
by the given fractions as multiplied by them (i.e., the given
varṇas severally divided by the given fractions, each increased
by 1). The two (results thus obtained) should be divided by
their (own) sum, then their order should be reversed, and
then they should be multiplied by the sum of the weights of
the gold balls. This will give the weights (of the gold balls)
separately.
¹ Cf GK, I, p. 80, lines 2-5.

MIXTURES. ALLIGATION 41
That is, if w be the sum of the weights of the two gold balls,
v₁ the varṇa of the alloy formed by mixing the first gold ball with a/b part
of the second, and v₂ the varṇa of the alloy formed by mixing the second
gold ball with c/d part of the first, then the separate weights w₁ and w₂ of
the first and second gold balls are given by
w₁ = (1/k) { v₂/(1 + c/d) − ((a/b)v₁)/(1 + a/b) } w, (1)
and
w₂ = (1/k) { v₁/(1 + a/b) − ((c/d)v₂)/(1 + c/d) } w, (2)
where k = { v₂/(1 + c/d) − ((a/b)v₁)/(1 + a/b) } + { v₁/(1 + a/b) − ((c/d)v₂)/(1 + c/d) } .
The rationale is as follows :
Let V₁ and V₂ be the varṇas of the first and second gold balls respec-
tively. Then we have
w₁ + w₂ = w , (3)
v₁ = (w₁V₁ + (a/b)w₂V₂) / (w₁ + (a/b)w₂) , (4)
v₂ = ((c/d)w₁V₁ + w₂V₂) / ((c/d)w₁ + w₂) , (5)
and, since the gold balls are of equal value,
w₁V₁ = w₂V₂ . (6)
Dividing (4) by (5), and making use of (6), we have
v₁/v₂ = [(1 + a/b)/(1 + c/d)] × [((c/d)w₁ + w₂) / (w₁ + (a/b)w₂)] ,
whence
w₁ / [v₂/(1 + c/d) − ((a/b)v₁)/(1 + a/b)] = w₂ / [v₁/(1 + a/b) − ((c/d)v₂)/(1 + c/d)]
But by virtue of (3), each of these is equal to w/k. Hence we
have (1) and (2).
Rule for finding the varṇas of two gold balls of equal value,
when the sum of their weights and also the varṇas of the
alloys formed by mixing each of them with given propor-
tion of the other are known :
58. The weights of the two (gold) balls (obtained by
applying the previous rule) should each be increased by the
given fraction of the weight of the other. The two (results

42 DETERMINATIONS
thus obtained) being divided by the (same) fractions, each
increased by 1, then multiplied by the (respective) varṇas,
and then divided by the (respective) weights of the gold
balls, give the varṇas (of the two gold balls).¹
That is, if _w_₁ and _w_₂ be the weights of two gold balls whose
varṇas are _V_₁ and _V_₂ respectively, _v_₁ the varṇa of the alloy formed by mixing
the first gold ball with a/b part of the second, and _v_₂ the varṇa of the
alloy formed by mixing the second gold ball with c/d part of the first, then
_V_₁ = [_w_₁ + (a/b) _w_₂] / [(1 + a/b) _w_₁] · _v_₁,
_V_₂ = [_w_₂ + (c/d) _w_₁] / [(1 + c/d) _w_₂] · _v_₂.
Of these results, the first follows from equations (4) and (6) of the
previous rule, and the second from equations (5) and (6).
Ex. 69. There are two small balls of gold of equal
worth (i.e., having equal quantities of pure gold), whose com-
bined weight is 5 māṣas. When they are respectively combined
with ⅔ part and ½ part of the other, they become of 10 and 9
varṇas respectively. (Find their weights and varṇas separa-
tely).
Ex. 70. O learned one, there are two small balls of
gold of equal value. When taken together, their weight is 12
māṣas ; and when they are respectively combined with ⁵/₇ part
and ⅕ part of the other, they become of 12 and 10½ varṇas
respectively. (Find their weights and varṇas separately).
(iii) PARTNERSHIP
Rule for finding the shares of the partners in the produce :
59(i). To obtain the individual shares (of the partners)
in the produce (phalâvâptyai), the seeds (contributed by the
partners) (prakṣepa), as divided by their sum, should be severally
multiplied by the produce (phala).²

¹ See also GSS, vi 209-212; and GK, I, p. 81, lines 9-18.
² Cf. BrSpSi, xii. 16 (i); SiŚe, xiii. 19 (i); L (ASS), p. 90, vs. 94; GK,
I, p. 54, lines 11-12. Also see GSS, vi, 79½ ; MSi, xv. 36.

MIXTURES. PURCHASE AND SALE 43
The prakṣepa, according to the commentator, is "that which is thrown,
scattered, or sown as seed" and therefore "seed", and the phala is "what is
produced out of that".
Ex. 71. Two, three, five, and four prasthas of seeds (are
the contributions of the partners) and 210 (prasthas of grain)
is the produce; what are the shares (of the partners)
separately ?
Ex. 72. ½ prastha is (the contribution) of one, ⅓ (prastha)
of another, ⅑ (prastha) of still another, and 1700 (prasthas) is
the produce. Say what are their shares (in the produce)
separately.
(iv) PURCHASE AND SALE
Rule for obtaining the quantities and prices of each of a
number of commodities separately, when they are purchased
in a specified proportion for a given amount of money :
59(ii). Having divided the (given) rate-prices (of the
commodities) by the (corresponding) rate-quantities (of those
commodities) and (then) having multiplied them by the
corresponding proportions (in which those commodities are
purchased), apply the previous rule.¹
Âryabhaṭa II states the rule more explicitly. He writes :²
"Set down the rate-prices as multiplied by the respective propor-
tional parts and divided by the rate-quantities of the respective commodi-
ties in two places. Multiply the results set down in one place as well as the
proportional parts by the mixed price and divide by the sum of the results
set down in the other place. These give the prices (paid for the respective
commodities) and the quantities (of the respective commodities that are
obtained) respectively."
Ex. 73-74. 7 kuḍavas of mudga ("seeds of Phaseolus
mungo") are obtained for 9 paṇas, and ½ kuḍava of rice is
obtained for one paṇa.

¹ Cf. SiŚe, xiii. 19 (ii); GK, I, p. 57, vs. 2. Also see GSS, vi. 87½-89½.
² MSi, xi. 37-38 (i). Also see L (ASS), p. 92, vs. 98.

44 DETERMINATIONS
Then, O merchant, take 3 paṇas and a half and quickly
give me one part of rice and two parts of mudga.
Solution. Dividing the rate-prices of mudga and rice by the corres-
ponding rate-quantities of those commodities, we get
9/7 and 2/1.
Multiplying these by the corresponding proportions in which they
are purchased (i.e., by 2 and 1), we get
18/7 and 2.
Now we apply the previous rule. Thus dividing 18/7 and 2 by their
sum, and multiplying them by the total price (i.e., 3½ paṇas), we get
63/32 and 49/32 paṇas
These are the prices of mudga and rice respectively.
The quantities of mudga and rice obtained for 63/32 paṇas and 49/32 paṇas
respectively can now be obtained by the rule of three. These come out to
be 49/32 kuḍavas and 49/64 kuḍavas respectively.
Ex. 75. ½ pala of asafoetida (hiṅgu), 2 palas of long
pepper (pippalī), and 7 palas of dry ginger (śuṇṭhī), are each
obtained for one rūpa. Give me equal quantities (of each
of them) for one rūpa.¹
Rule for finding the rates of purchase and sale when mer-
chants investing unequal capitals become equally rich by
purchasing and selling articles at the same rates :
60. Reduce the capitals (vitta) to a common denomina-
tor, and then remove the denominators. This being done,
the greatest amount increased by an optional number (1, 2, 3,
etc.) gives the rate of sale (i.e., the rate at which the
articles are sold for one rūpa ; vikraya).
That (i.e., the rate of sale) multiplied by the price at
which each remnant article is sold (antyārgha), being dimini-
────────────────────────────────────────────────────────────
¹ Cf. GK, I, p. 57, lines 16-19.

MIXTURES, PURCHASE AND SALE 45
shed by 1 and then multiplied by the common denominator
(of the capitals), gives the rate of purchase (i.e., the rate at
which the articles are purchased for one rūpa ; kraya).¹
That is, if n persons, whose capitals reduced to a common deno-
minator are
C₁/D, C₂/D, C₃/D, ..., Cₙ/D rūpas respectively,
invest their capitals in purchasing certain articles at the rate of x articles
per rūpa, and if by selling those articles in multiples of y at the rate of y
articles per rūpa, and the remnant articles, at the rate of R rūpas per
article, they become equally rich, then
x = D(Ry – 1),
and y = C_λ + k,
where C_λ is the greatest of all C's, and k an arbitrary integer.
The rationale of this rule is as follows :
Suppose that by selling the articles in multiples of y, the n persons
earn P₁, P₂, P₃ ..., Pₙ rūpas respectively. Then the total amounts
earned by the n persons by selling all the articles (including the remnant
articles) are respectively
P₁ + R((C₁/D)x – P₁y), P₂ + R((C₂/D)x – P₂y), ... ,
and Pₙ + R((Cₙ/D)x – Pₙy) rūpas,
or (RC₁x/D) – P₁(Ry – 1), (RC₂x/D) – P₂(Ry – 1), ...,
and (RCₙx/D) – Pₙ(Ry – 1) rūpas.
Since these amounts are equal to one another, therefore we have
(RC₁x/D) – P₁(Ry – 1) = (RC₂x/D) – P₂(Ry – 1) = ...
= (RCₙx/D) – Pₙ(Ry – 1).
or RC₁x – P₁D(Ry – 1) = RC₂x – P₂D(Ry – 1) = ...
= RCₙx – PₙD(Ry – 1) = M say. (1)
To solve (1), let x = M = D(Ry – 1);
─────────────────────────────────────────────────────────────
¹ Also see GSS, vi. 102½ ; GK, I, pp. 94-97, vv. 36 (ii) – 38(i).

46 DETERMINATIONS
then
RCᵣ - Pᵣ = 1 , (2)
for r = 1, 2, ..., n .
But (Cᵣ/D)x / y = Pᵣ + Aᵣ / y ,
where Aᵣ is the remainder when (Cᵣ/D)x is divided by y , so that,
when x = D(Ry - 1),
Pᵣ = Cᵣ x / Dy - Aᵣ / y
= Cᵣ (Ry - 1) / y - Aᵣ / y
= RCᵣ - (Cᵣ + Aᵣ) / y (3)
Therefore, we must have
y > Cᵣ , for all r ,
or y > C_λ ,
where C_λ is the greatest of all C's.
Hence a solution of (1) is
x = D(Ry - 1),
y = C_λ + k ,
k being an arbitrary integer.
Sub-rule for finding another rate of purchase, when the capitals
are integral and the articles which are left after selling at
the general rate as also the amounts of money acquired by
selling the articles at the general rate have a common factor :
61. The articles which are severally left after selling
(at the general rate) (i.e., the remnant articles) as also the
amounts of money which are severally acquired by selling the
articles (at the general rate), being abraded by their common
factor (lit. divisor), the resulting equal amounts of money
(i.e., the equal sale proceeds) give another rate of purchase.
This rule relates to the case when D = 1 , and may be expressed
analytically as follows :
If d be the common factor of
P₁, P₂, ..., Pₙ
and ( C₁x - P₁y ), ( C₂x - P₂y ), ..., ( Cₙx - Pₙy ),
Indological Truths

MIXTURES, PURCHASE AND SALE 47
then another rate of purchase is x' articles per rûpa, where
x' = P₁/d + R( C₁x - P₁y )/d
= P₂/d + R( C₂x - P₂y )/d
...
= Pₙ/d + R( Cₙx - Pₙy )/d ,
where x = Ry - 1, y = C_λ + k, C_λ being the greatest of all C's and
k an arbitrary integer; the rate of sale being the same as before
( i.e., C_λ+k articles per rûpa ).
The rationale of this rule is as follows :
When D = 1 , the equations (1) of the previous rule may be
written as
RC₁x - P₁(Ry - 1) = RC₂x - P₂(Ry - 1) = ...
= RCₙx - Pₙ(Ry - 1) = M. (1)
Dividing throughout by d , and writing x' for x/d , P_r' for P_r/d,
and M' for M/d, we have
RC₁x' - P₁'(Ry - 1) = RC₂x' - P₂'(Ry - 1) = ....
= RCₙx' - Pₙ'(Ry - 1) = M'. (2)
The values of x , y satisfying (1) give one set of rates of purchase
and sale, and the values of x' , y satisfying (2) give another set of rates
of purchase and sale.
When M = Ry - 1 , a solution of (1) is, as before,
x = Ry - 1 ⎫
⎬ (3)
y = C_λ+k ⎭
and the corresponding solution of (2) is
x' = (Ry - 1)/d , i.e., M' ⎫
⎬ (4)
y = C_λ+k . ⎭
The rates of purchase and sale given by (3) are those already given
in Rule 60 ; those given by (4) are those mentioned in the present rule.
Since, from (2),
M' = P₁/d + R(C₁x - P₁y)/d = P₂/d + R(C₂x - P₂y)/d
= ... = Pₙ/d + R(Cₙx - Pₙy) ,
it is clear that the value of x' given by (4) is the same as that stated
in the above rule. It can be easily seen that M' is an integer.
Note. When D ≠ 1, then another rate of purchase will be D(Ry - 1)/d
articles per rûpa, where y = C_λ + k.

48 DETERMINATIONS
Ex. 76. The capitals of (three) men are 1, 3, and 5
(rûpas) or ⅓, ¼ and ½ (rûpas) (respectively). By purchasing and
selling (certain articles) at the same rates and by selling the
remnant articles at the rate of 1 for 3 (rûpas), they become
possessed of equal riches. (Find the rates of purchase and
sale).¹
Sub-rule for finding the rates of purchase and sale when each
of the remnant articles is sold for a fractional price :
62. When the remnant-price (i.e., the price at which
each remnant article is sold) is fractional (lit. has a denomi-
nator), reduce it along with the capitals to a common
denominator (and remove the denominators as before), and
multiply the rates of purchase and sale resulting from them
(by applying Rule 60) by the common denominator.²
That is, if the capitals of the n persons and the remnant-price
reduced to a common denominator, be as follows
Capitals : C₁/D , C₂/D , C₃/D , ... , Cₙ/D rûpas
Remnant-price : (R/D) rûpas,
then x = D² ( (R/D)y - 1 ) ,
and y = D (C_λ + k) ,
C_λ being the greatest of all C's and k an arbitrary integer.
The rationale of this rule is as follows :
As before, equating the wealths of the n persons after all the articles
have been sold away, we get [see equations (1) of Rule 60 (notes)]
(R/D) · C₁x - P₁D ( (R/D)y - 1 ) = (R/D) · C₂x - P₂D ( (R/D)y - 1 )
= ... = (R/D) · Cₙx - PₙD ( (R/D)y - 1 ) ,
¹ Cf. GSS, vi. 103½, 104½; GK, I, p. 96, lines 2-6, and p. 97, lines 7-9.
² Cf. GSS, vi. 107½. For another rule, see GSS, vi. 109½; GK, I, p. 98,
lines 2-7.

MIXTURES. PURCHASE AND SALE 49
or RC₁x - P₁D² ( R/D y - 1 ) = RC₂x - P₂D² ( R/D y - 1 )
= ... = RCₙx - PₙD² ( R/D y - 1 ) = N, say. (1)
Let x = N = D²( R/D y - 1 ), then
RCᵣ - Pᵣ = 1, (2)
for r = 1, 2, ..., n.
But ((Cᵣ/D)x) / y = Pᵣ + Aᵣ/y,
where Aᵣ is the remainder when Cᵣx/D is divided by y, so that, when
x = D²( R/D y - 1 ),
Pᵣ = Cᵣx / Dy - Aᵣ / y
= [ Cᵣ D² ( R/D y - 1 ) ] / Dy - Aᵣ / y
= RCᵣ - (Cᵣ D + Aᵣ) / y. (3)
Therefore we must have
y > CᵣD, for all r,
or y > C_λD,
where C_λ is the greatest of all C's.
Hence a solution of (1) is
x = D²( R/D y - 1 ),
y = D(C_λ + k),
k being an arbitrary integer.
Ex. 77. The capitals of four men are 1½, 2, 3, and 5
rûpas. By purchasing and selling (certain articles) at the same
rates, and by selling the remnant articles at the rate of 1 for ½
(of a rûpa), they become possessed of equal riches. (Find the
rates of purchase and sale).¹

¹ Cf. GSS, vi. 108½, 110½; GK, I, p. 99, lines 2-5 and 15-18.

50 DETERMINATIONS
Rule for finding out how a specified number of creatures of
given rates can be bought for a specified price :
63-64. By the price of one creature of any variety
multiply the rate-creatures (of other varieties) in the order in
which they have been stated (in the problem) (and also the
number of creatures to be bought). From the products (corres-
ponding to the rate-creatures) severally subtract the respec-
tive rate-prices of the creatures (and from the product
corresponding to the number of creatures to be bought sub-
tract the specified price). Now multiply the various remain-
ders, excepting that obtained by subtracting the specified
price, by optional numbers (multipliers) which are to be
chosen in such a way that (i) the resulting products when
added together may yield (the remainder obtained by sub-
tracting) the specified price as sum, and (ii) on taking the
products of those multipliers and the respective rate-prices,
a negative number or zero may not be obtained for the multi-
plier of the creature which is without a multiplier. (The multi-
pliers for the various creatures, obtained in this way, when
multiplied by the respective rate-creatures, will give the
number of creatures of the different varieties that will be
bought for the specified price ; and the same multipliers
when multiplied by the rate-prices of the respective creatures
will give the prices that will be paid for the creatures of the
respective varieties).¹
For the explanation of this rule see under Ex. 78-79.
Ex. 78-79. Pigeons are sold at the rate of 5 for 3 (rûpas),
cranes at the rate of 7 for 5 (rûpas), swans at the rate of 9 for
7 (rûpas), and peacocks at the rate of 3 for 9 (rûpas). Knowing
the rates as stated above, bring 100 birds for 100 rûpas
for the amusement of the prince.²
¹ Cf. GK, I, p. 92, vv. 34(ii)-35. For other rules see GSS, vi. 146½, 151.
² The same example occurs in GSS, vi. 152-153; BBi (ASS), p. 163,
vs. 138; GK, I, p. 93, lines 2-5. A similar problem occurs in BM, III, E 3,
58 verso, which may be stated as : ‘A man earns 3 maṇḍas in a day, a
woman 1½ maṇḍas in a day, and a sûḍha ½ maṇḍa in a day. If 20 of
them earn 20 maṇḍas in a day, how many of each category are there?’

MIXTURES. PURCHASE AND SALE 51
Let the number of pigeons, cranes, swans, and peacocks bought for
100 rûpas, and the prices paid for the respective birds be as follows :
Pigeons Cranes Swans Peacocks
Number 5x 7y 9z 3u } (1)
Price 3x 5y 7z 9u
Then we have
5x+7y+9z+3u = 100 , (2)
3x+5y+7z+9u = 100 . (3)
Multiplying (2) by 3 (which is the price of one peacock) and
subtracting (3) therefrom, we have
12x+16y+20z = 200 . (4)
This is an indeterminate equation having an indefinitely large
number of solutions. But we are in the present case concerned only with
those positive and non-zero solutions of (4), which make
5x , 7y , 9z , and 3u
positive integers.
Proceeding by trial, we obtain the following sixteen such solutions
of (4) :
(1) x=3, y=4, z=5 . (9) x=4, y=7, z=2 .
(2) x=11, y=3, z=1 . (10) x=8, y=4, z=2 .
(3) x=1, y=8, z=3 . (11) x=12, y=1, z=2 .
(4) x=6, y=3, z=4 . (12) x=5, y=5, z=3 .
(5) x=2, y=6, z=4 . (13) x=9, y=2, z=3 .
(6) x=4, y=2, z=6 . (14) x=7, y=1, z=5 .
(7) x=3, y=9, z=1 . (15) x=1, y=3, z=7 .
(8) x=7, y=6, z=1 . (16) x=2, y=1, z=8 .
The corresponding values of u are obtained from (2).
Substituting these values of x, y, z, and u in (1), we obtain sixteen
valid solutions of the problem. (For actual solutions, see the answer).
The above analysis explains the basis of the foregoing rule (Rule
63-64). It may be observed that the multipliers mentioned in the rule
refer to the variables x, y, z, and u.
Indological Truths

52 DETERMINATIONS
Ex. 80. (The rates of sale) of pomegranates, mangoes,
and wood-apples are respectively 1 for 2 (rûpas), 5 for 3 (rûpas),
and 2 fruits for 1 (rûpa). Bring 100 (fruits) for 80 (rûpas).¹
(v) MEETING OF TWO TRAVELLERS
Rule for finding the time in which the fast traveller, who starts
travelling on the same track after the slow traveller has
already covered a specified distance, would overtake the slow
traveller :
65. By the difference between the speeds per day of the
fast and slow travellers divide the distance already travelled
by the slow traveller. This gives the time (in days in which
the fast traveller would overtake the slow one).²
Ex. 81-82. When a person, travelling ( at the speed of )
8 yojanas per 5 minus ½ days, has already travelled for 6 minus
¼ days, another person, who travels ( at the speed of) 3 yojanas
a day, starts travelling (from the same place) along the same
track. Say, after calculating, when the latter traveller would
overtake the former.³
Rule for finding the time when two travellers, one fast and
the other slow, who, starting simultaneously from the
same place, are destined to go to a specified distance
and then to come back by the same track, will meet each other
on the way, one going ahead and the other coming back :
66(i). The (length of the) track divided by half the
sum of the speeds per day (of the two travellers) gives the
time (in days) at meeting.⁴
The time at meeting is evidently measured since the start of travel.
Ex. 83. One man travels ( at the speed of) 8 yojanas
(a day) and another (at the speed of) 2 yojanas (a day).
(They start simultaneously from the same place, and after
reaching the destination come back by the same track).
¹ Cf. GSS, vi. 147½-149, 150.
² Cf. BM, III, A 13, 3 recto; GSS, vi. 326½.
³ Cf. BM, III, B4, 4 recto; E 2, 53 recto; E 2, 53 verso; GSS, vi. 327½.
⁴ Cf. GSS, vi. 319 (ii).

MIXTURES. WAGES AND PAYMENTS 53
The (length of the) track is 100 yojanas. Say where is the
meeting (of the two), one going ahead and the other coming
back.¹
Sub-rule for finding the length of the track :
66(ii). The time of meeting multiplied by half the sum
of the speeds per day (of the two travellers) gives the length of
the track.
Sub-rule for finding the speed of one of the two travellers :
67(i). The (length of the) track as divided by the time
of meeting, being doubled and then diminished by the speed
(of one of the two travellers) gives the other speed (i.e., the
speed of the other traveller).
(vi) WAGES AND PAYMENTS
Rule for finding the wages for carrying a bottle of oil when
some oil falls down on the way due to a hole in the bottle :
67(ii). Give half the wages for the quantity fallen down
(on the way) and full for the rest.²
It can be easily seen that the wages prescribed in this rule corres-
pond to the average load carried by the porter.
Ex. 84-85. While a leathern oil-bottle (kutapa), filled
with 200 palas of oil, was being carried (by a porter) to a
distance of 8 yojanas for 5 paṇas (as wages), a hole hap-
pened to occur in the bottom of it through which the oil
leaked out (on the way) continuously. If 20 palas of oil be left
(in the bottle), what wages should be paid (to the porter)?³
Rule for finding the amounts to be paid to a dancing party
by the spectators, who see the dance for different parts of the
day, the amount to be paid to the dancing party for the whole
day being given:
68. The succeeding parts of the day, each diminished
by the preceding one, should be multiplied by the ‘fruit’
¹ Cf. BM, III, 9 verso ; GSS, vi. 321-321½
² Cf. GK, I, p. 102, lines 6-7.
³ Read bhṛte for dhṛte in the text. For a similar example, see GK, I,
p. 103, lines 2-5.