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Indian Mathematics c12.s2: Appendix II. Examples

Indian Mathematics c12.s2

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English

Appendix II. Examples

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1. One-half, one-sixth, and one-twelfth parts of a pole are immersed respectively under water, clay, and sand. Two *hastas* are visible. Find the height of the pole?

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*Answer*—8 hastas. *S.* 23.

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2. The quarter of a sixteenth of the fifth of three-quarters of two-thirds of half a *dramma* was given to a beggar by a person from whom he asked alms. Tell how many cowries the miser gave if thou be conversant in arithmetic with the reduction termed sub-division of fractions?

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*Answer*—1 cowrie. *L.* 32. (1,280 cowries=1 dramma).

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3. Out of a swarm of bees one-fifth settled on a blossom of *kadamba*, one-third on a flower of silīndhrī, three times the difference of those numbers flew to a bloom of kutaja. One bee, which remained, hovered and flew about in the air, allured at the same moment by the pleasing fragrance of a jasmine and pandanus. Tell me, charming woman, the number of bees?

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*Answer*—15. *L.* 54, *V.* 108.

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4. The third part of a necklace of pearls broken in an amorous struggle fell on the ground. Its fifth part was seen resting on the couch, the sixth part was saved by the lady and the tenth part was taken up by her lover. Six pearls remained on the string. Say, of how many pearls the necklace was composed?

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*Answer*—30. @H *S.* 26.

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5. A powerful, unvanquished, excellent black snake which is 32 *hastas* in length enters into a hole 7½ *aṅgulas* in 5/14 of a day, and in the course of a quarter of a day its tail grows by 2¾ *aṅgulas*. O ornament of arithmeticians, tell me by what time this same enters fully into the hole?

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*Answer*—76⅘ days. @H *M. v,* 31. (24 aṅgulas=1 hasta.)

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6. A certain person travels at the rate of 9 *yojanas* a day and 100 *yojanas* have already been traversed. Now a messenger sent after goes at the rate of 13 *yojanas* a day. In how many days will he overtake the first person?

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*Answer*—25. @H *M. vi,* 327.

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7. A white-ant advances 8 *yavas* less one-fifth in a day and returns the twentieth part of an *aṅgula* in 3 days. In what space of time will one, whose progress is governed by these rates of advancing and receding proceed 100 *yojanas*?

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*Answer*—98042553 days. @H *C.* 283. (8 yavas=1 aṅgula, 768000 aṅgulas=1 yojana).

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8. Twenty men have to carry a palanquin two *yojanas* and 720 *dīnāras* for their wages. Two men stop after going two *krośas*, after two more *krośas* three others give up, and after going half the remaining distance five men leave. What wages do they earn?

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*Answers*—18, 57, 155, 490. @H *M. vi,* 231. (4 krośas=1 yojana).

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9. It is well known that the horses belonging to the sun's chariot are seven. Four horses drag it along being harnessed to the yoke. They have to do a journey of 70 yojanas. How many times are they unyoked and how many times yoked in four ?

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*Answer*—Every 10 yojanas, and each horse travels 40 yojanas. *M. VI*, 158.

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10. If a female slave sixteen years of age brings thirty-two, what will one twenty cost ?

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*Answer*—25⅗. *L.* 76.

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11. Three hundred gold coins form the price of 9 damsels of 10 years. What is the price of 36 damsels of 16 years ?

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*Answer*—750. *M. V*, 40.

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12. The price of a hundred bricks, of which the length, thickness and breadth respectively are 16, 8 and 10, is settled at six *dīnāras*, we have received 100,000 of other bricks a quarter less in every dimension. Say, what we ought to pay ?

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*Answer*—2531¼. *C.* 285.

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13. Two elephants, which are ten in length, nine in breadth, thirty-six in girth and seven in height, consume one *droṇa* of grain. How much will be the rations of ten other elephants which are a quarter more in height and other dimensions ?

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*Answer*—12 droṇas, 3 prasthas, 1½ kuḍavas. *C.* 285. (64 kuḍavas=16 prasthas=1 droṇa).

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14. One bestows alms on holy men in the third part of a day, another gives the same in half a day and a third distributes three in five days. In what time, keeping to these rates, will they have given a hundred ?

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*Answer*—17 6/7. *C.* 282.

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¹ \* *L*=the Līlāvati, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra *S*=Srīdhara, *C*=Chaturveda. \* *L*=the Līlāvatī, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra, *S*=Srīdhara, *C*=Chaturveda. \* *L*=the Līlāvatī, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra, *S*=Srīdhara, *C*=Chaturveda. \* *L*=the Līlāvatī, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra, *S*=Srīdhara, *C*=Chaturveda. \* *L*=the Līlāvatī, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra, *S*=Srīdhara, *C*=Chaturveda.
* L= the Lilāvatī, V= Vīja Gaṇita, both by Bhāskara, M= Mahāvīra, S= Srīdhara, C= Chaturveda. \* L=the Līlāvatī, V=Vīja Gaṇita, both by Bhāskara, M=Mahāvīra, S=Srīdhara, C=Chaturveda. \* L=the Līlāvatī, V=Vīja Gaṇita, both by Bhāskara, M=Mahāvīra, S=Srīdhara, C=Chaturveda. \* *L* = the Līlāvatī, *V* = Vija Gaṇita, both by Bhāskara, *M* = Mahāvīra, *S* = Sridhara, *C* = Chaturveda.
* L=the Līlāvati, V=Vīja Gaṇita, both by Bhāskara, M=Mahāvīra, S=Srīdhara, C=Chaturveda.

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15. Say, mathematician, what are the apportioned shares of three traders whose original capitals were respectively 51, 68 and 85, which have been raised by commerce conducted by them in joint stock to the aggregate amount of 300?

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16. One purchases seven for two and sells six for three. Eighteen is the profit. What is the capital?

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*Answer*—18÷((3/6)/(2/7) — 1)=24. *Bakhshālī Ms.* 54.

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17. If a *pala* of best camphor may be had for two *nishkas*, and a *pala* of sandal wood for the eighth part of a *dramma* and half a *pala* of aloe wood also for the eighth of a *dramma*, good merchant, give me the value of one *nishka* in the proportions of 1, 16 and 8; for I wish to prepare a perfume?

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*Answer*—Prices—Drammas 14 2/9, 8/9, 8/9+4/9, 64/9, 32/9 *L.* 98. (16 drammas=1 nishka).

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18. If three and a half *mānas* of rice may be had for one *dramma* and eight of beans for the same price, take these thirteen *kākinīs*, merchant, and give me quickly two parts of rice and one of beans: for we must make a hasty meal and depart, since my companion will proceed onwards?

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*Answer*—7/12 and 7/24. *L.* 97, *V.* 115. (64 kākinīs=1 dramma).

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19. If the interest on 200 for a month be 6 *drammas*, in what time will the same sum lent be tripled?

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*Answer*—66⅔ months. *C.* 287.

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20. If the principal sum with interest at the rate of five on the hundred by the month amount in a year to one thousand, tell the principal and interest respectively?

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*Answer*—625, 375. *L.* 89.

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21. In accordance with the rate of five per cent. (per mensem) two months is the time for each instalment ; and paying the instalments of 8 (on each occasion) a man became free in 60 months. What is the capital ?

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*M. vi*, 64.

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22. Five hundred *drammas* were a loan at a rate of interest not known. The interest of that money for four months was lent to another person at the same rate and it accumulated in ten months to 78. Tell the rate of interest on the principal ?

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*Answer*—60. *C.* 288.

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23. Subtracting from a sum lent at five in the hundred the square of the interest, the remainder was lent at ten in the hundred. The time of both loans was alike and the amount of interest equal ?

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*Answer*—Principal 8. *V.* 109.

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24. There is 1 part of 1 *varṇa*, 1 part of 2 *varṇas*, 1 part of 3 *varṇas*, 2 parts of 4 *varṇas*, 4 parts of 5 *varṇas*, 7 parts of 14 *varṇas*, and 8 parts of 15 *varṇas*. Throwing these into the fire make them all into one and then what is the *varṇa* of the mixed gold ?

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*Answer*—10½. *M. vi*, 170.

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[The term *varṇa* corresponds to 'carat' or measure of 'purity of gold.']

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25. Gold 1, 2, 3, 4 *suvarṇas*, and losses 1, 2, 3, 4 *māshakas*.

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The average loss is (1.1+2.2+3.3+4.4)/(1+2+3+4)=3.

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*Bakhshālī Ms.* 27.

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26. Of two arithmetical progressions with equal sums and the same number of terms the first terms are 2 and 3, the increments 3 and 2 respectively and the sum 15. Find the number of terms?

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*Bakhshālī Ms.* 18.

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27. A merchant pays octroi on certain goods at three different places. At the first he gives ⅓ of the goods, at the second ¼, and at the third ⅕. The total duty paid is 24. What was the original amount of the goods?

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*Answer*—40. *Bakhshālī Ms.* 25.

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28. One says: "Give me a hundred and I shall be twice as rich as you, friend!" The other replies: "If you deliver ten to me I shall be six times as rich as you. Tell me what was the amount of their respective capitals?

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*Answer*—40 and 170. *V.* 106, 156.

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29. *A* gives a certain amount, *B* gives twice as much as *A*, *C* gives 3 times as much as *B*, *D* gives 4 times as much as *C* and the total is 132.

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*Answer*—*A* gives 4, etc. *Bakhshālī Ms.* 54.

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30. Four jewellers possessing respectively 8 rubies, 10 sapphires, 100 pearls and 5 diamonds, presented each from his own stock one apiece to the rest in token of regard and gratification at meeting; and thus they became owners of stock of precisely the same value. Tell me, friend, what were the prices of their gems respectively?

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*Answer*—24, 16, 1, 96 [These are relative values only]. *L.* 100.

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31. The quantity of rubies without flaw, sapphires, and pearls belonging to one person, is five, eight and seven respectively. The number of like gems belonging to another is seven, nine and six. One has ninety, the other sixty-two rupees. They are equally rich. Tell me quickly, then, intelligent friend, who art conversant with algebra, the prices of each sort of gem ?

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*Answer*—14, 1, 1, etc. *V.* 105 & 156.

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[Bhāskara ‘assumes’ relative values.]

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32. The horses belonging to these four persons respectively are five, three, six and eight; the camels belonging to them are two, seven, four and one; their mules are eight, two, one and three; and the oxen owned by them are seven, one, two and one. All are equally rich. Tell me severally, friend, the rates of the prices of horses and the rest ?

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*Answer*—85, 76, 31, 4, etc. *V.* 157.

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33. Say quickly, friend, in what portion of a day will four fountains, being let loose together fill a cistern, which, if opened one by one, would fill it in one day, half a day, the third and the sixth parts respectively ?

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*Answer*—1/12. *L.* 95.

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34. The son of Pṛithā, angered in combat, shot a quiver of arrows to slay Karṇa. With half his arrows he parried those of his antagonist; with four times the square-root of the quiverful he killed his horse; with six arrows he slew Salya; with three he demolished the umbrella, standard and bow; and with one he cut off the head of the foe. How many were the arrows which Arjuna let fly ?

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*Answer*—100. *L.* 67, *V.* 133.

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35. For 3 *paṇas* 5 *palas* of ginger are obtained, for 4 *paṇas* 11 *palas* of long pepper and for 8 *paṇas* 1 *pala* of pepper. By means of the purchase money of 60 *paṇas* quickly obtain 68 *palas* ? *M. vi*, 150.

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*Answer*—Ginger 20, long pepper 44, pepper 4.

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36. Five doves are to be had for three *drammas*; seven cranes for five ; nine geese for seven and three peacocks for nine. Bring a hundred of these birds for a hundred *drammas* for the prince's gratification ? *V.* 158-9 ; *M. vi*, 152.

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*Answer*—Prices 3, 40, 21, 36.

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Birds 5, 56, 27, 12.

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(This class of problem was treated fully by Abú Kāmil-el-Misri (*c.* 900 *A.D.*). See H. Suter : *Das Buch der Seltenheit, etc.* Bibliotheca Mathematica 11 (1910-11), pp. 100-120.

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37. In a certain lake swarming with red geese the tip of a bud of a lotus was seen half a *hasta* above the surface of the water. Forced by the wind it gradually advanced and was submerged at a distance of two *hastas*. Calculate quickly, O mathematician, the depth of the water ?

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*Answer*—1 5/4. *L.* 153 ; *V.* 125.

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38. If a bamboo measuring thirty-two *hastas* and standing upon level ground be broken in one place by the force of the wind and the tip of it meet the ground at sixteen *hastas*, say, mathematician, at how many *hastas* from the root it is broken ?

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*Answer*—12. *L.* 148.

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39. A snake’s hole is at the foot of a pillar 9 *hastas* high and a peacock is perched on the summit. Seeing a snake, at a distance of thrice the pillar, gliding towards his hole, he pounces obliquely on him. Say quickly at how many *hastas* from the snake’s hole do they meet, both proceeding an equal distance ?

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40. From a tree a hundred *hastas* high, a monkey descended and went to a pond two hundred *hastas* distant, while another monkey, jumping a certain height off the tree, proceeded quickly diagonally to the same spot. If the space travelled by them be equal, tell me quickly, learned man, the height of the leap, if thou hast diligently studied calculation ?

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*Answer*—50. *L.* 155 ; *V.* 126.

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41. The man who travels to the east moves at the rate of 2 *yojanas*, and the other man who travels northward moves at the rate of 3 *yojanas*. The latter having journeyed for 5 days turns to move along the hypotenuse. In how many days will he meet the other man ?

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*Answer*—13. *M. vii*, 211.

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42. The shadow of a gnomon 12 *aṅgulas* high is in one place 15 *aṅgulas*. The gnomon being moved 22 *aṅgulas* further its shadow is 18. The difference between the tips of the shadows is 25 and the difference between the lengths of the shadows is 3. Find the height of the light ?

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*Answer*—100. *C.* 318 ; *Ar.* 16 ; *L.* 245.

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43. The shadow of a gnomon 12 *aṅgulas* high being lessened by a third part of the hypotenuse became 14 *aṅgulas*. Tell, quickly, mathematician, that shadow ?

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*Answer*—22½. *V.* 141.

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44. Tell the perpendicular drawn from the intersection of strings mutually stretched from the roots to summits of two bamboos fifteen and ten *hastas* high standing upon ground of unknown extent?

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*L.* 160.

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45. Of a quadrilateral figure whose base is the square of four and the face two *hastas* and altitude twelve, the flanks thirteen and fifteen, what is the area?

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*Answer*—108. *S.* 77.

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46. In the figure of the form of a young moon the middle length is sixteen and the middle breadth is three *hastas*. By splitting it up into two triangles tell me, quickly, its area?

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*Answer*—24. *S.* 83.

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47. The sides of a quadrilateral with unequal sides are 13×15, 13×20 and the top side is the cube of 5 and the bottom side is 300. What are all the values here beginning with that of the diagonals?

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*Answers*—315, 280, 48, 252, 132, 168, 224, 189, 44100. *M. vii*, 59.

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If A² + B² = C², and a² + b² = c² then the quadrilateral Ac, Bc, aC, bC is cyclic and the diagonals are Ab + aB and Aa + Bb, the area is ½ (ABc² + abC²), &c. In the present case A = 15, B = 20, C = 25; a = 5, b = 12, c = 13. The diagonals are 315, 280; the area 44100. For full details see the *Līlāvatī*, § 193.

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48. O friend, who knowest the secret of calculation, construct a derived figure with the aid of 3 and 5 as elements, and then think out and mention quickly the numbers measuring the perpendicular side, the other side and the hypotenuse?

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*Answer*—16, 30, 34. *M. vii*, 94.

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That is construct a triangle of the form 2mn, m²-n², m²+n², where m = 5, n = 3.

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49. In the case of a longish quadrilateral figure the perpendicular side is 55, the base is 48 and then the diagonal is 73. What are the elements here?

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50. Intelligent friend, if thou knowest well the spotless Līlāvatī, say what is the area of a circle the diameter of which is measured by seven, and the surface of a globe or area like a net upon a ball, the diameter being seven, and the solid content within the same sphere?

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*Answer*—Area 38 2423/5000; surface 153 1173/1250; volume 179 1487/2500 *L.* 204.

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51. In a circle whose diameter is ten, what is the circumference? If thou knowest, calculate, and tell me also the area?

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*Answer*—√(1,000), √6250. *S.* 85.

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52. The measure of Rāhu is 52, that of the moon 25, the part devoured 7.

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*Answer*—The arrow of Rāhu is 2, that of the moon 5. *C.* 311.

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This is an eclipse problem and means that circles of diameters 52 and 25 intersect so that the portion of the line joining the two centres common to the two circles is 7. The common chord cuts this into segments of 5 and 2.

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53. The combined sum of the measure of the circumference, the diameter and the area is 1116. Tell me what the circumference is, what the calculated area, and what the diameter?

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*Answer*—108, 972, 36. *M. vii*, 32.

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The rule given is circumference = √(12 (combined sum + 64)) - √64 which assumes that π=3.

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54. The circumferential arrows are 18 in number. How many are the arrows in the quiver?

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The rule given is n=((c+3)²+3)/12 where c is the number of arrows in the outside layer.

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55. Tell me, if thou knowest, the content of a spherical piece of stone whose diameter is a *hasta* and a half? *Answer*—1⅔5/2. *S.* 93.

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The rule given is v=d³ (½+1/2.18).

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56. A sacrificial altar is built of bricks 6 *aṅgulas* high, half a *hasta* broad and one *hasta* long. It is 6 *hastas* long, 3 *hastas* broad and half a *hasta* high. Tell me rightly, wise man, what its volume is and how many bricks it contains. *Answer*—9, 72. *S.* 96. 24 aṅgulas=1 hasta.

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57. If thou knowest, tell me quickly the measure of a mound of grain whose circumference is 36 and height 4 *hastas*? *Answer*—144. *S.* 102.

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The rule used assumes that π=3.

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58. In the case of a figure having the outline of a bow, the string measure is 12, and the arrow measure is 6. The measure of the bow is not known. Find it, O friend. *Answer*—√360. *M. vii*, 75.

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59. In the case of a figure having the outline of a bow the string is 24 in measure, and its arrow is taken to be 4 in measure. What is the minutely accurate value of the area? *Answer*—√5760. *M. vii*, 72.

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The rule used is S=(a c√(10.))/4 60. Multiplier consisting of surds two, three and eight : multiplicand the surd three with the rational number five. Tell quickly the product ?

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61. What is the number which multiplied by five and having the third part of the product subtracted, and the remainder divided by ten ; and one-third, one half and a quarter of the original quantity added gives two less than seventy ?

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*Answer*—48. *L.* 51.

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The solution may be summarised this : f(x)=68, f(3)=17/4 therefore x=(3×68)/(17/4)=48.

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62. The eighth part of a troop of monkeys squared was skipping in a grove and delighted with their sport. Twelve remaining were seen on the hill amused with chattering to each other. How many were there in all ?

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*Answer*—48 or 16. *V.* 139.

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63. The fifth part of the troop less three, squared, had gone to a cave and one monkey was in sight, having climbed on a branch. Say how many there were ?

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*Answer*—50 or 5. *V.* 140.

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"But," Bhāskara says, "the second is not to be taken for it is incongruous. People do not approve a negative absolute number."

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64. Say quickly what the number is which added to five times itself divided by thirteen becomes thirty ?

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*Answer*—65/3. *V.* 168.

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65. A certain unknown quantity is divided by another. The quotient added to the divisor and the dividend is fifty-three. What is the divisor ?

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*Answer*—5, 8. *M. vi*, 274.

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¹ \* L=the Līlāvatī, V=Vīja Gaṇita, both by Bhāskara, M=Mahāvīra, S=Srīdhara, C=Chaturveda.

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66. What number is that which multiplied by nought and added to half itself and multiplied by three and divided by nought amounts to the given number sixty-three ? *Answer*—14. This assumes that 0/0=1. *L.* 46.

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67. What four numbers are such that their product is equal to twenty times their sum, say, learned mathematician who art conversant with the topic of the product of unknown quantities ? *Answer*—5, 4, 2, 11. *V.* 210.

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Bhāskara puts arbitrary values for three of the quantities and gets 11 for the fourth.

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68. If you are conversant with operations of algebra tell the number of which the fourth power less double the sum of the square and of two hundred times the simple number is ten thousand less one ? *Answer*—11. *V.* 138.

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This may be expressed by x⁴-2 (x²+200 x)=9999. It is the only case in which the fourth power occurs.

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69. The square of the sum of two numbers added to the cube of their sum is equal to twice the sum of their cubes ? *Answer*—1, 20 ; 5, 76, *etc.* *V.* 178.

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70. Tell me, if you know, two numbers such that the sum of them multiplied respectively by four and three may when added to two be equal to the product of the same numbers ? *Answer*—5, 10 and 11, 6. *V.* 209, 212.

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71. Say quickly, mathematician, what is the multiplier by which two hundred and twenty-one being multiplied and sixty-five added to the product, the sum divided by a hundred and ninety-five becomes cleared ? *Answer*—5, 20, 35 &c. *L.* 253 ; *V.* 65.

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¹ \* *L*=the Līlāvatī, *V* =Vīja Gaṇita, both by Bhāskara, *M*=Mahāvīra, *S*=Srīdhara, *C*=Chaturveda.
* L=the Līlāvatī, V=Vīja Gaṇita, both by Bhāskara, M=Mahāvīra, =Srīdhara, C=Chaturveda.

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72. What number divided by six has a remainder of five, divided by five has a remainder of four, by four a remainder of three and by three one of two?

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*Answer*—59. *Br. xviii*, 7; *V.* 160.

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73. What square multiplied by eight and having one added to the product will be a square?

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*V.* 82.

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Here 8u² +1=t² and u=6, 35, etc. t=17, 99, etc.

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74. Making the square of the residue of signs and minutes on Wednesday multiplied by ninety-two and eighty-three respectively with one added to the product an exact square; who does this in a year is a mathematician.

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*Br. xviii*, 67.

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(1) 92 u² +1=t² (2) 83 u² +1=t².

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*Answer*—(1) u=120, t=1151. (2) u=9, t=82.

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75. What is the square which multiplied by sixty-seven and one being added to the product will yield a square-root; and what is that which multiplied by sixty-one with one added to the product will do so likewise? Declare it, friend, if the method of the 'rule of the square' be thoroughly spread, like a creeper, over thy mind?

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*V.* 87.

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(1) 67 u² +1=t². (2) 61 u +1=t².

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*Answers*—(1) u=5967, t=48842. (2) u=226,153,980, t=1,766,319,049.

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76. Tell me quickly, mathematician, two numbers such that the cube-root of half the sum of their product and the smaller number, and the square-root of the sum of their squares, and those extracted from the sum and difference increased by two, and that extracted from the difference of their squares added to eight, being all five added together may yield a square-root—excepting, however, six and eight?

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*V.* 190.

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[3](xy+y)/2+√(x²+y²)+√(x+y+2)+√(x-y+2)+√(x²-y²+8)=t²

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*Answers*—x=8 ; 1677/4 ; 15128, etc. ; y=6, 41 ; 246, etc.

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∗ *L*=the Līlāvati, *V*=Vīja Gaṇita, both by Bhāskara, *M*=Mahāvira, *S*=Śrīdhara, *C*=Chaturveda.

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