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

OPERATIONS FOR FRACTIONS 15
(9-16) Operations for fractions
Rule for the addition of fractions :
32(i). Reduce the fractions to a common denominator
and then add the numerators. The denominator of a whole
number is unity.¹
Ex. 6. Say the sum of ½, ⅓, ⅙ and ¹/₁₂, and of 2 plus ½,
3 minus ¼, and 6.
Ex. 7. Friend, if you know the method of calculation
(gaṇita-vidhi), quickly say the sum of 1½ terms, ½ term, and of
⅓ term of the series whose first term (ādi) and common
difference (caya) are each unity.
Rule for the subtraction of fractions :
32(ii). Take the difference of the numerators of the
positive and negative fractions reduced to a common denomi-
nator.²
Ex. 8. Subtract ¼, ⅓ and ⅙ from 1 and say what remains.
Also, subtract 3 — ½ and 2+⅓ from 5 (and say the remainder).
Ex. 9. Say what remains as the remainder when the
sum of 2 plus ½ terms is subtracted from the sum of 5 plus ½
terms of the series whose first term and common difference are
(each) unity.
It may be remarked here that, according to the author of the
present work, rule 14(i) applies also to series of natural numbers
having a fractional number of terms. Series in A. P. having frac-
tional number of terms have generally no meaning but they may be
interpreted geometrically by means of figures called series-figures. See
infra, ‘Determinations pertaining to series’ (śreḍhī-vyavahāra).
¹ Cf. MSi, xv. 14(i) ; GT, p. 15, lines 20-21 ; SiŚe, xiii. 8 ; L
(ASS), p. 35, vs. 37 ; GK, I, p. 11, lines 6-7. Also see BrSpSi, xii. 2.
² Cf. BrSpSi, xii. 2(ii); MSi, xv. 14(ii) ; GT, p. 18, lines 3-4 ;
SiŚe, xiii. 8 ; L (ASS), p. 35, vs. 37; GK, I, p. 11, lines 6-7.

16 LOGISTICS
Rule for the multiplication of fractions :
33(i) The product (of the given fractions) is obtained on
dividing the product of the numerators by the product of the
denominators.¹
Ex. 10. 2 plus ½ is multiplied by 1 plus ½, and 60 plus ⅓
is multiplied by ⁵⁄₂ : what are the products, say separately.
Rule for the division of fractions :
33(ii). Having interchanged the denominator and the
numerator of the divisor, apply the previous rule (i.e., the
rule for multiplication).
Ex. 11. 6 plus ¼ is divided by 2+½, and 60 plus ¼ is
divided by 3+⅓ : say the quotients separately.
Rule for squaring a fraction :
34(i). The square of the numerator divided by the
square of the denominator is the square of the fraction.³
Ex. 12. Say, friend, if you know, the square of 2 plus ½,
of 15 plus ¼, of ½, and of ⅓.
Rule for finding the square root of a fraction :
34(ii). The square root of the numerator divided by
the square root of the denominator, gives the square root
(of the fraction).⁴
Rule for cubing a fraction :
35(i). The cube of the numerator divided by the cube
of the denominator gives the cube (of the fraction).⁵
¹ Cf. BrSpSi, xii. 3(ii) ; GSS, iii. 2; MSi, xv. 15(i) ; GT, p. 19,
line 26 ; SiŚe, xiii. 9(i) ; L (ASS), p. 36, vs. 39 ; GK, I, p. 12, lines 2-3.
² Cf BrSpSi, xii. 4 ; GSS, iii. 8(i) ; MSi, xv. 15(ii) ; GT, p. 21,
line 3-6 ; SiŚe, xiii. 10 ; L (ASS), p. 37, vs. 41 ; GK, I, p. 12, lines 11-12.
³ Cf. BrSpSi, xii. 5(i) ; GSS, iii. 13 ; MSi, xv. 16(i) ; GT p. 22,
line 24; SiŚe, xiii. 9(ii); L (ASS), p. 38, lines 2-3; GK, I, p. 12, lines 17-18.
⁴ Cf. BrSpSi, xii. 5(ii) ; GSS, iii. 13 ; MSi, xv 16(ii) ; GT, p. 23,
line 25 ; L (ASS), p. 38, lines 2-3 ; GK, I p. 12, lines 17-18.
⁵ Cf. GSS, iii. 13 ; MSi, xv. 17(i) ; GT, p. 25, line 13 ; L (ASS), I,
p. 38, lines 2-3 ; GK, I, p. 12, lines 17-18.

REDUCTION OF FRACTIONS 17
Ex. 13. Say, if you know, the cube of 7 plus ½ ,
of 17 plus ¼ , of ¼ , and of ⅕ .
Rule for extracting the cube root of a fraction :
35(ii). The cube root of the numerator divided by the
cube root of the denominator gives the cube root (of the
fraction).¹
(17-22) Reduction of fractions (kalâ-savarṇa)
Rule for reducing fractions of the bhâga class (i.e., fractions
connected by + and – signs):
36. In the bhâga class, in order to reduce the (two given)
fractions to a common denominator, remove the common
factor, if any, from the denominators, and then by each of
them (i.e., the reduced denominators) multiply the denomina-
tor and numerator of the other fraction.²
Ex. 14. What sum is obtained by adding together the
fractions having (the integers) 2 to 6 for their denominators,
and 1 for their numerators, and by adding together the frac-
tions having (the integers) 3 to 9 for their denominators and
(the integers) 2 etc. for their respective numerators.
Alternative rule for reducing fractions of the bhâga class :
37. By the lower denominator multiply the upper
numerator, (then) by the upper denominator multiply the
lower denominator, and (then) add the product of the
numerator and the denominator in the middle to the upper
numerator.³
Suppose, for example, that we want to reduce the fraction
2/3 + 4/5
───────────────────────────────────────────────────────────
¹ Cf. GSS, iii. 13 ; MSi, xv. 17(ii) ; GT, p. 26, line 27 ; L (ASS),
p. 38, lines 2-3 ; GK, I, p. 12, lines 17-18.
² Cf. BrSpSi, xii. 2(i) ; MSi, xv. 13(ii) ; GT, p. 30, line 16 ;
SiŚe. xiii. 11(i) ; L (ASS), p. 28, line 9 ; GK, I, p. 9, vs. 26(i). Also see
GSS, iii. 55 (i);
³ Cf. SiŚe, xiii. 12.

18 LOGISTICS
Writing these fractions one below the other without the lines of
separation as the Hindus used to do, we get
2
3
4
5
[In this scheme, 2 is the upper numerator and 3 the upper denomi-
nator ; similarly 4 is the lower numerator and 5 the lower denominator.]
Now according to the rule, multiplying the upper numerator by the
lower denominator, and multiplying the lower denominator by the upper
denominator, we get
2 × 5      10
  3        3
     i.e.,
  4        4
5 × 3      15
Next, adding the product of the numerator and the denominator in
the middle (viz. 3 and 4) to the upper numerator, we get
(10 + 12) / 15   i.e.,   22 / 15   or   22 / 15
rubbing out the figures in the middle, which are not required.
Rule for reducing fractions of the prabhāga class (i.e., fractions
connected by 'of') :
38(i). In the prabhāga class, one should multiply the
numerators together, and the denominators together.¹
That is,
(a / b) of (c / d) of (e / f) = ace / bdf .
Ex. 15. Tell (me) the sum of
1/4 of 1/2 of 1/2 , 1/10 of 1/6 of 1/5 of 1/3 , and 1/7 of 1/6 of ( 2 + 1/2 ).
Rule for reducing a fraction of the bhāga-bhāga class (i.e., a
whole number divided by a fraction) :
38(ii). Having multiplied the whole number (rūpa) by
the denominator of the fraction, remove the denominator :
¹ Cf. BrSpSi, xii. 8(ii) ; GSS, iii. 99 ; MSi, xv. 13(i) ; SiŚe, xiii.
11(i) ; L (ASS), p. 31, vs. 32 ; GK, I, p. 9, vs. 26(ii).
Indological Truths

REDUCTION OF FRACTIONS 19
this is the process (of reduction) for the bhāga-bhāga class.¹
That is,
a ÷ b/c = ac / b .
Ex. 16. Friend, if you know (the method), say after
thinking, what sum will be obtained by adding together (the
fractions) :
1 ÷ 1/6 , 1 ÷ 1/10 , 1 ÷ 1/3 , 1 ÷ 1/9 , and 1 ÷ 1/2 .
Ex. 17. 1 has been severally divided by fractions
having (the integers) 3 to 6 for their denominators and (the
integers) 2 etc., for their respective numerators. Say what
sum will be obtained when they are added together.
Rule for reducing a fraction of the bhāgānubandha class (i.e., a
whole number increased by a fraction, or a fraction increa-
sed by a fraction of itself) :
39. In the bhāgānubandha class, the whole number
(rūpagaṇa) as multiplied by the denominator (of the fraction)
should be increased by the numerator (of the fraction) ; or,
the upper denominator having been multiplied by the lower
denominator, the initial numerator (i.e., the upper numerator)
should be multiplied by the sum of the lower numerator and
denominator.²
That is,
(i) a + b/c = (ac + b) / c ,
(ii) a/b + c/d of a/b ( which was written in the Hindu
way as³ [ (a/b) / (c/d) ] ) = a (d + c) / bd .
─────────────────────────────────────────────────────────────
¹ Cf. BrSpSi, xii. 9(i) ; GSS, iii. 99(ii) ; MSi, xv. 19(i).
² Cf. BrSpSi, xii 9(ii) ; GSS, iii. 113 ; MSi, xv. 11(ii)-12 ; GT,
p. 34, lines 15-16 ; L (ASS), p. 32, vs. 34 ; GK, I, p. 9, vs. 27.
³ Actually it was written as a
b
c
d

20 LOGISTICS
Ex. 18. What sum is obtained by adding together
1+½, 5+¼, and 8+⅓ ?
Ex. 19. What is the sum of
(3 + ½) + ¼ of (3 + ½) + ⅙ of {(3 + ½) + ¼ of (3 + ½)}
and ½ + ⅓ of ½ + ¼ of {½ + ⅓ of ½} ?
Rule for reducing a fraction of the bhāgāpavāha class (i.e., a
whole number minus a fraction, or a fraction minus a
fraction thereof) :
40. In the bhāgāpavāha class, the numerator of the frac-
tion should be subtracted from the whole number multiplied
by the denominator (of the fraction) ; or having multiplied
the upper denominator by the lower denominator, the upper
numerator should be multiplied by the lower denominator as
diminished by the lower numerator.¹
That is to say,
(i) a - b / c = (ac - b) / c ,
(ii) a / b - c / d of a / b (which was written in the Hindu way
a
b
as² ) = a (d - c) / bd .

• c
  d
  Ex. 20. Say the amount, when 1 - ½ , 5 - ¼ , and 8 - ⅓
  are added together.
  Ex. 21. What is obtained by adding together :
  (3 - ½) - ¼ of (3 - ½) - ⅙ {(3 - ½) - ¼ of (3 - ½)}
  and ½ - ⅓ of ½ - ¼ of (½ - ⅓ of ½) ?
  ──────────────────────────────────────────────────────────
  ¹ Cf. BrSpSi, xii. 9(ii) ; GSS, iii. 126 ; MSi, xv. 11(ii)-12; GT, p. 37,
  lines 2-5 ; L (ASS), p. 32, vs. 34 ; GK, I, p. 9, vs. 27.
  ² Actually, it was writtten as
  a a
  b or b
  c+ ˙c
  d d
  the fractions being written without the separating line, and + or (˙) being
  written to denote subtraction.

REDUCTION OF FRACTIONS 21
The commentator interprets ‘rûpatrayamardhonam’ as meaning
3 - ½ of 3 and not (3 - ½), so the first expression according to him is
(3 - ½ of 3) - ½ of (3 - ½ of 3) - ⅓ of [(3 - ½ of 3) - ½ of (3 - ½ of 3)].
The interpretation given by us is more natural and plausible,
especially when we see that a similar interpretation has been given to
the verse of Ex. 19, which is analogous to this one.
Rule for reducing a chain (vallî) of measures :
41. To reduce a chain (vallî), multiply the foremost
denominator and numerator by the lower denominator, and
(then) from or to the foremost numerator subtract or add
(as the case may be) the lower numerator.¹
Ex. 22. What amount is obtained by reducing
5 purâṇas , 3 paṇas , 1 kâkiṇî , - 1 varâṭaka , - ⅕ of that (i.e., of
a varâṭaka) (to purâṇas) ?
Converting each measure to the previous one, and writing them one
below the other in the Hindu way, we have
5
1
3
16
1
4
1+
20
1+
5
where + indicates subtraction.
Now applying the rule to the uppermost two fractions of this chain,
we get
83
16
1
4
1+
20
1+
5
¹ Cf. MSi, xv. 18 ; GT, p. 39, lines 7-10.

22 LOGISTICS
Now applying the rule to the uppermost two fractions of this reduced
chain, we get
333
64
1 +
20
1 +
5
Applying the same rule to this chain, it reduces to
6659
1280
1 +
5
and then to
33294 / 6400 or 16647 / 3200
Hence the required result is
16647 / 3200 or 5 647/3200 purâṇas.
Rule for reducing fractions of the bhâgamâtâ class :
42. That (class of fractions) in which fractions of the
'bhâga and other classes occur in combination (of two or
more) is called bhâgamâtâ. In that (class), the result is obtained
by applying the aforesaid rules separately.¹
Ex. 23. What amount is obtained by adding together
½, ¼ of ¼, 1 ÷ ⅓, ½+½ of ½, and ⅓-½ of ⅙ ?
Ex. 24. If you know the method of calculation, say
what amount will be obtained by the addition of ¼, ½ of ⅓,
1 ÷ ½, and 8+½ ?
(23) Rule of three (trairâśika)
Rule of three :
43. In (solving problems on) the rule of three, the
argument (pramâṇa) and the requisition (icchâ), which are
of the same denomination, should be set down in the first and
last places ; the fruit (phala), which is of a different deno-
mination, should be set down in the middle. (This having
¹ Cf. GSS, iii. 138.

RULE OF THREE 23
been done) that (middle quantity) multiplied by the last
quantity should be divided by the first quantity.¹
Ex. 25. If 1 pala and 1 karṣa of sandal wood are ob-
tained for ten and a half paṇas, then for how much will
9 palas and 1 karṣa (of sandal wood of the same quality) be
obtained ?
Here
argument = 1 pala and 1 karṣa
= 1¼ or 5/4 palas ;
fruit = 10½ or 21/2 paṇas ;
and requisition = 9 palas and 1 karṣa
= 9¼ or 37/4 palas
Writing these quantities as directed in the rule, we have
┌───┬────┬────┐
│ 5 │ 21 │ 37 │
│ 4 │ 2 │ 4 │
└───┴────┴────┘
Then applying the rule, the required result
= [(21/2) × (37/4)] / [5/4] = (21×37) / (2×5) paṇas
or 4 purāṇas, 13 paṇas, 2 kākiṇīs and 16 varāṭakas.
Ex. 26. If 1⅓ palas of black pepper are obtained for
1¼ paṇas, then how much of that (black pepper) will be
obtained for (10 - ⅓) paṇas? Say quickly.
Ex. 27. If one and a half droṇas and three kuḍavas of
grain is obtained for 8, say, if you know, for how much will
one khārī and one droṇa (of that grain) be obtained.
Ex. 28. If 60+½ khārīs of grain is obtained for 100+⅓
rūpas, how much of that (grain) will be obtained for a quarter
of a rūpa ? ²
──────────────────────────────────────────────
¹ Cf. Ā, ii. 26; BrSpSi, xii. 10 ; GSS, v. 2(i) ; MSi, xv. 24-25(i) ; GT,
p. 68, vs. 86 ; SiŚe, xiii. 14 ; L (ASS), p. 71, vs. 73 ; GK, I, p. 47, vs. 60.
² Examples similar to Exs. 25-28 occur in GSS, iii. 3-6 ; v. 8, 9, 13,
14 ; GT, p. 68, vs. 87, p. 69, vs. 88 ; p. 70, vs. 90 ; p. 71, vv. 91, 92;
L (ASS), pp. 72-73, vv. 74, 76 ; GK, I, p. 47, lines 13-14, 17-18, p. 48,
lines 8-11 ; and also in Bhāskara I's comm. (629 A.D.) on Ā, ii. 26-27.

24 LOGISTICS
The term rūpa stands for any coin whatever. “By the word rūpaka
is meant a coin of gold, etc., bearing the name paṇa, etc.”¹
Ex. 29. Where one suvarṇa gets 70+⅓ rūpas, say,
friend, what will 1 māṣa as lessened by ⅒ (of a māṣa) get
there.²
Ex. 30. A certain lame person goes to a distance of ⅛
of a yojana in ⅓ of a day, say in how much time will he go
to a distance of 100 yojanas.³
Ex. 31. An insect goes to a distance of ⅙ of an aṅgula in
¼ of a day, in how much time will it go to a distance of 10
and a half yojanas ?⁴
Forward and backward motion:
44(i). On subtracting the backward motion per day
from the forward motion per day, the remainder is the (true)
motion per day.⁵
Ex. 32. The best amongst the elephants goes forward
at the rate of ½(1 + ¼) (1 − ⅓) (1 + ½) of a yojana in
6 × ⅕ × ⅑ × ⅓ (1 + ¼) of a day, and comes back at the rate of
2 (1 − ⅓) yojanas in (1 + ½) days. Say, friend, in how much
time will he go to a distance of 100 yojanas.⁶
Ex. 33. In how much time will a man, earning at the
rate of (8 − ½) rūpas in (1+⅓) days and spending on his food
at the rate of ½ per day, be a lord of 100 (rūpas) ?⁷
¹ रूपकशब्देन पणादिसंज्ञितं स्वर्णादिद्रव्यम् See Parameśvara's commentary
on Ā, ii 30.
² Cf. GT, p. 73, vs. 95.
³ Cf GSS, v. 3, 4 ; GT, p. 72, vs. 93.
⁴ Cf. GSS, v. 5 ; GT, p. 73, vs. 94 ; GK, I, p. 48, lines 16-19 ff.
⁵ Cf. BM. III, L 1, 60 recto ; GSS, v. 23 ; MS, xv. 30.
⁶ Cf. GSS, v. 27. Similar examples occur also in BM. See, for
instance, BM, III, M1, 21 verso ; M4, 36 verso. Also see Pṛthûdaka’s
example quoted by Sudhakara Dvivedi in his comm. on BrSpSi, xii.
10-12, p. 179, lines 14-17; and Bhâskara I's example in his comm. on
Ā, ii. 26-27(i).
⁷ Cf. BM, III, L 1, 60 recto ; M 11, 44 recto and verso ; M 12,
43 recto ; GSS, v. 26.

INVERSE RULE OF THREE 25
(24) Inverse rule of three (vyasta-trairāśika)
Inverse rule of three :
44(ii). When there is change in the unit of measure-
ment, the middle quantity multiplied by the first quantity and
divided by the last quantity gives the result.¹
Ex. 34. When (a given quantity of pearls is) measured
at 8 pearls a necklace, the number of necklaces is twenty ;
say, mathematician, what the number of necklaces would be
(when the same quantity of pearls is measured) at 6 pearls a
necklace.²
Ex. 35. Being measured by the māṣa of 5 raktikās, a
quantity of gold amounts to 300 suvarṇas. Say how much
would that (quantity of gold) amount to, when measured by
the māṣa of 6 raktikās.³
Raktikā is the same as guñjā.
Ex. 36. How much gold of 11 varṇas can be had in
exchange for 168 suvarṇas of 16 varṇas ?⁴
Ex. 37. Quickly say how many blankets of length 6
(cubits) and breadth 2 (cubits) can be made out of (the yarn
which yields) 200 blankets of length 9 (cubits) and breadth
3 (cubits).⁵
Ex. 38. How much gold of 10½ varṇas will be obtained
in exchange for 100 suvarṇas and 8 māṣas (of gold) of
12½ varṇas ?⁶
(25-27) Rules of five, seven and nine (pañca-sapta-nava-rāśika)
Rule of five, seven, and nine :
45. After transposing the fruit from one side to the
other, and then having transposed the denominators (in like
¹ Cf. BrSpSi, xii. 11(i) ; MSi, xv, 25(ii). Also see L (ASS), p. 74,
vs. 77-78 ; GK, I, p. 49, vs. 61.
² The same example occurs in GT, p. 74, lines 8-9.
³ Cf. L (ASS), p. 76, vs. 81 ; GK, I, p. 50, lines 3-4, 7-8. A similar
example occurs in Bhāskara I's comm. on Ā, ii. 26-27(i).
⁴ Cf. GT, p. 74, vs. 96. Also Cf. GSS, v. 18 (i) ; vi. 173½, 174½;
L (ASS), p. 75, vs. 80. For the meaning of varṇa see infra p. 36.
⁵ Cf. GSS, v. 19. ⁶ Cf. GSS, v. 18(ii).

26 LOGISTICS
manner) and multiplied the numbers (so obtained on either
side), divide the side with larger number of quantities (nume-
rators) by the other.¹
The two sides referred to in this rule are known as (i) the argument
side (pramāṇarāśi-pakṣa) and (ii) the requisition side (icchārāśi-pakṣa). In
Ex. 40, for instance, these sides are as follows :
(i) argument side ⎧ 100½ or 201/2
⎨ ⅓
⎩ 1½ or 3/2
(ii) requisition side ⎧ 60¼ or 241/4
⎩ 8 - ½ or 15/2
These are written as
(i) (ii)
┌────────────┬────────────┐
│ 201 │ 241 │
│ 2 │ 4 │
│ │ │
│ 1 │ 15 │
│ 3 │ 2 │
│ │ │
│ 3 │ 0 │
│ 2 │ │
└────────────┴────────────┘
0 being written in place of the desired quantity (unknown).
Then the rule is applied as follows :
Transposing the fruit, we get
(i) (ii)
┌────────────┬────────────┐
│ 201 │ 241 │
│ 2 │ 4 │
│ │ │
│ 1 │ 15 │
│ 3 │ 2 │
│ │ │
│ 0 │ 3 │
│ │ 2 │
└────────────┴────────────┘
─────────────────────────────────────────────────────────────────────────────
¹ Cf. BrSpSi, xii. 11(ii)-12 ; GSS, v. 32 ; MSi, xv. 26, 27 ; GT, p. 74,
vs. 97 ; SiŚe, xiii, 15; L (ASS), pp. 76-77, vs. 82 ; GK, I, p. 50, vs. 62.

RULE OF FIVE, SEVEN, AND NINE 27
Transposing the denominators, we get

¹ Cf. GSS, v. 33, 41 ; GT, p. 75, vs. 98 ; L (ASS), p. 77, vs. 83;
GK, I, p. 50, lines 18-21. Similar examples occur in Bhâskara I's comm.
on Â, ii. 26-27(i).
² Cf. GSS, v. 34 ; GT, p. 76, vs. 99 ; GK, I, p. 51, lines 6-9. Similar
examples occur in Bhâskara I's comm. on Â, ii. 26-27(i).
³ Cf. GSS, v. 35.
⁴ Cf. GSS, v. 36 ; GT, p. 78, vs. 101.

28 LOGISTICS
Ex. 44. If 3 labourers earn 5 rūpas in 2 days, say what
will 8 labourers earn in 9 days ?¹
Ex. 45. If a blanket, whose breadth is 2 (cubits) and
length 8 (cubits), gets 10, what will 2 other (similar) blankets
of breadth 3 (cubits) and length 9 (cubits) get ?²
Ex. 46. If a ( rectangular piece of ) stone with length,
breadth, and thickness equal to 9, 5, and 1 cubits respectively
costs 8, what will two other (rectangular pieces of stone) of
dimensions 10, 7, and 2 cubits cost ?³
Ex. 47. If the diet of an elephant of diameter 2 (cubits),
height 6 (cubits) and length 7 (cubits) is one droṇa, what
should be the diet of an elephant of diameter 3 (cubits), height
9 (cubits) and length 10 (cubits) ?⁴
(28) Barter of commodities (bhāṇḍa-pratibhāṇḍa)
Rule for problems on the barter of commodities :
46(i). In (problems on) the barter of commodities,
having transposed the prices (of the commodities), apply the
previous rule (i.e., rule of five, etc.).⁵
Ex. 48. If 2 palas of dry ginger are obtained for 6
and one pala of long pepper for 9, how much of long
pepper will then be obtained for 6 palas of dry ginger ?⁶

¹ Cf. GT, p. 77, vs. 100.
² This example is the same as that given in GT, p. 78, vs. 102.
³ This example is the same as that given in GT, p. 79, vs. 103.
Also cf. L (ASS), p. 79-80, vs. 85; GK, I, p. 52, lines 2-5.
⁴ Similar examples occur in Bhāskara I's comm. on Ā, ii. 26-27(i)
and in Pṛthûdaka's comm. on BrSpSi, xii. 10-12.
⁵ Cf. BrSpSi, xii. 13(i); MSi, xv. 28(i); GT, p.80, lines 16-17; SiŚe, xiii,
16(i); L (ASS), p. 83 vs. 88; GK, I, p. 53, lines 2-3. Also see GSS, vi. 18.
⁶ Cf. GSS, v. 37, 38 and vi. 19-20; GT, p. 81, lines 6-9.

SALE OF LIVING BEINGS 29
Setting down the argument and requisition sides, we have
┌───┬───┐
│ 6 │ 9 │
│ 2 │ 1 │
│ 6 │ │
└───┴───┘
Transposing the prices of the commodities, we get
┌───┬───┐
│ 6 │ 9 │
│ 1 │ 2 │
│ 6 │ │
└───┴───┘
Now we have to apply the rule of five, etc. But the fruits (i.e. prices)
have been already transposed, and there are no denominators on any
side. Hence the required answer
6 × 1 × 6
= —————————
9 × 2
= 2 palas of long pepper.
Ex. 49. If 16 mangoes are obtained for 2 paṇas and
100 wood-apples for 3 (paṇas), say then how many wood-
apples will be obtained for 6 mangoes.¹
(29) Sale of living beings (jīva-vikraya)
Rule for problems on the sale of living beings :
46(ii). In (problems on) the sale of living beings, the
same rule (of five, etc.) is applied after transposing the ages
(of the living beings).²
Ex. 50. If 5 women of 16 years of age get 200, say
then, O mathematician, how much will 2 women of 20 years
of age get ?³
────────────────────────────────────────────────────────
¹ Cf. GSS, v. 38; GT, p. 80, lines 23-26 ; L (ASS), p. 84, vs. 89;
GK, I, p. 53, lines 5-8. A similar example occurs in Pṛthûdaka’s comm.
on BrSpSi, xii. 13.
² Cf. GT, p. 81, vs. 108; SiŚe, xiii. 16 (ii); GK, I, 53, vs. 63.
³ Cf GSS, v. 40; GT, p. 81, vs. 109; L (ASS), p. 74, vs. 79; GK,
p. 53, lines 15-18.

30 LOGISTICS
Setting down the argument and requisition sides, we have
┌──────────┬──────────┐
│ 5 │ 2 │
│ 16 │ 20 │
│ 200 │ │
└──────────┴──────────┘
Transposing the ages (of the women), we get
┌──────────┬──────────┐
│ 5 │ 2 │
│ 20 │ 16 │
│ 200 │ │
└──────────┴──────────┘
Now we have to apply the rule of five, etc. Therefore, transposing
the fruit, we have
┌──────────┬──────────┐
│ 5 │ 2 │
│ 20 │ 16 │
│ │ 200 │
└──────────┴──────────┘
There being no denominators on any side, the required amount
= (2 × 16 × 200) / (5 × 20) = 64.
Ex. 51. If 3 camels of 10 years of age get 108 purāṇas,
say then what will 5 camels of 9 years of age get.¹
──────────────────────────────────────────────────────────
¹ A closely similar example occurs in GT, p. 82, vs. 110. Also
Cf. GSS, v. 39.

DETERMINATIONS 31
DETERMINATIONS (Vyavahāra)
(1) Determinations pertaining to mixtures of things
(miśraka-vyavahāra)
(i) SIMPLE INTEREST
Rule for finding the capital and the interest, when the
amount and the rate of interest are given :
47. Multiply the argument (pramāṇa-rāśi) by its time,
and the other time by the fruit (phala) ; divide each of those
(products) by their sum, and multiply by the amount (i.e.,
capital plus interest). The results (thus obtained) give the
capital (mūla-dhana) and the interest (vṛddhi-dhana) respectively.¹
That is,
argument × time × amount
Capital = ──────────────────────────────────────────────
argument × time + fruit × other time
fruit × other time × amount
Interest = ────────────────────────────────────────────── ·
argument × time + fruit × other time
Ex. 52. The rate of interest being 5 per cent per
month, a certain sum is seen to amount to 96 in a year. Say,
friend, what is the capital and what the interest ?²
Here argument = 100, time = 1 month, fruit = 5, amount = 96,
and other time = 1 year, i.e., 12 months. Therefore
100 × 1 × 96
Capital = ──────────────────── = 60,
100 × 1 + 5 × 12
5 × 12 × 96
Interest = ──────────────────── = 36.
100 × 1 + 5 × 12
Ex. 53. The interest on 100½ for one month and a
quarter being 1½, a certain sum amounts to 36½ in a period
of 7½ months. (Find the sum and the interest accrued
thereon).
Rule for separating the capital, interest, and the commission
of the surety etc., from the given amount :
48. Divide the product of the argument and its time
──────────────────────────────────────────────────────────────────────────
¹ Cf MSi, xv. 31; GT, p. 82, vs. 111 ; SiŚe, xiii. 17; L (ASS), p. 85,
vs. 90. Also see BrSpSi, xii. 14 (ii); GSS, vi. 21, 23.
² The same example occurs in GT, p. 83, vs. 112. Also see GSS,
vi. 22, 24; L (ASS), p. 86, vs. 91; GK, I, p. 60, lines 10-11. Similar
examples occur also in Pṛthūdaka's comm. on BrSpSi, xii. 14.
Indological Truths

32 DETERMINATIONS
as also the fruit (i.e., interest), etc., as multiplied by the other
time by their sum, and then multiply them by the mixed
amount : then are obtained the capital etc. in their respective
order.¹
This rule is similar to the previous one.
Ex. 54. The rate of interest being 5 per cent per
month, the commission of the surety (bhāvyaka) 1 per cent per
month, the fee of the calculator (vṛtti) ½ per cent per month,
and the charges of the scribe ¼ per cent per month, a certain
sum amounts to 905 in a year. (Find the capital, the interest,
and the shares of the surety, calculator, and the scribe).²
Here argument = 100, time = 1 month, first fruit = 5, second
fruit = 1, third fruit = ½, fourth fruit = ¼ ; amount = 905, and other
time = 1 year, i.e., 12 months.
∴ Capital = (argument × time × amount) / [argument × time + (sum of fruits) × other time]
= (100 × 1 × 905) / [100 × 1 + (5 + 1 + ½ + ¼) × 12]
= 500.
Interest = (first fruit × other time × amount) / [argument × time + (sum of fruits) × other time]
= (5 × 12 × 905) / [100 × 1 + (5 + 1 + ½ + ¼) × 12]
= 300.
Commission of surety = (second fruit × other time × amount) / [argument × time + (sum of fruits) × other time]
= (1 × 12 × 905) / [100 × 1 + (5 + 1 + ½ + ¼) × 12]
= 60.
Similarly, fee of the calculator = 30, and charges of the scribe = 15.
Rule for finding the time in which a sum lent out at simple
interest will be paid back by equal monthly instalments :
49-50. To get the desired time (īpsitakālopanaye) sub-
tract from the capital (mūlāt) the present worths of the first
monthly instalment, the second monthly instalment, etc.,

¹ Cf. GT, p. 83, vs. 113; SiŚe; xiii. 18;
² The same example occurs in GT, p. 83, vs. 114.

MIXTURES. SIMPLE INTEREST 33
(mûla), one after another (pṛthak pṛthak). [This will give
the number of complete months elapsed (gata-mâsa) and a
residue of the capital (mûla-śeṣa), if any]. Now calculate
interest for a month on the residue (śeṣasya-mâsika-phalaṁ),
and subtract that from the amount of monthly instalment
(mâsikopanayât) : by the remainder divide the interest for
a month on the residue (mâsikaphala) as multiplied by the
number of complete months elapsed (mâsa) and increased by
the residue of the capital (mûlaśeṣa).¹ The quotient added
to the number of complete months elapsed (gatamâsa) gives
the time (in months) of recovery of the capital (together
with interest).²
Let C be the capital lent out, i the rate of interest per cent per
month, I the amount of monthly instalment of payment, and T + 1/t,
t > 1, the time in months in which the money is recovered with interest.
Now suppose that the P.W.s of the 1st, 2nd,....... Tth instalments
having been subtracted one after another from C, the residue left is R.
Then evidently, R is the P.W. of I/t for (T + 1/t) months, so that
100 × (I / t)
R = ------------------------ ,
100 + (T + 1/t) i
(Ri / 100) · T + R
whence 1/t = ------------------- ·
I - (Ri / 100)
Hence the required time T + 1/t
(Ri / 100) · T + R
= T + ---------------------- months.
I - (Ri / 100)
Ex. 55-56. A rich man lent to somebody a sum of 100
rûpas at 5 per cent (per month simple interest) and from him
¹ The reading mâsaśeṣayug occurring in the Sanskrit text should be
mûlaśeṣayug.
² Cf. GK, I, p. 72, lines 14-15 ; and p. 73, lines 1-6.

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