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 STUDENTS ! TAKE THIS AS A CHALLENGE !
SOLVE THESE 103 MATHEMATICAL PROBLEMS GIVEN TO INDIAN STUDENTS 1250 YEARS AGO
(MATHEMATICAL PROBLEMS FOR HIGH SCHOOL STUDENTS) Compiled from various ancient Indian mathematical text books By Dr. N. Gopalakrishnan , Ph. D., D.Litt, (Scientist & Hon. Director IISH) Bharatheeya Vidyaa Nikethan……. 1. What are the respective remainders obtained when the sums of 1 to 10 , each multiplied by 10, terms of the series whose first term and common difference are unity are severally subtracted from the sum of 100 terms of the same series ? 2. Multiply 1296 by 21, 896 by 37, 8065 by 60 3. Tell me the squares of 1 to 9 , 25, 36, 63, 432 and 7802 4. Quickly sat what are the cubes of 1 to 9, 15, 256 and 203 5. Say the sum of ½, 1/3, 1/6 and 1/12 and of 2 + ½, 3 – ¼, and 6 6. Friend, if you know the method of calculation , quickly say the sum of 1 ½ terms, ½ term, and of 1/3 term of the series whose first term ( aadi) and common difference (chaya) are each unity 7. Subtract ¼, 1/3 and 1/6 from 1 and say what remains. Also subtract 3 - ½ and 2 + 1/3 from 5 and say the remainder 8. Say what remains as the remainder when the sum of 2 plus ½ terms is subtracted from the sum of 5 + ½ terms of the series whose first term and common differnce are unity 9. 2 plus ½ is multiplied by 1 plus ½ and 60 plus 1/3 is multiplied by 5/2: what are the products say separately. 10. 6 + ¼ is divided by 2 + ½ and 60 + ¼ is divided by 3 + ½ ; say the quotients separately 11. Say, friend if you know , the square of 2 + ½ , of 15 + ¼ , of ½ and of 1/3. 12. Say, if you know, the cube of 7 + ½ of 17 + ¼ , of ¼, and of 1/3. 13. 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 fractions having the integers 3 to 9 for their denominators and the integers 2 etc., for their respective numerators. 14. Tell me the sum of ¼ of ½ of ½, 1/10 of 1/6 of 1/5 of 1/3 , and 1/7 of 1/6 of (2 +1/2) 15. It 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. 16. Say the amount, when 1 – ½, 5 – ¼ and 8 –1/3 are added together 17. What is obtained by adding (3-1/2)-1/4 of (3 –1/2) – 1/6 ((3 – ½) – 1/4of (3-1/2)) and ½ - ½ of ½-1/4 of (1/2 –1/3 of ½) ? 18. What amount is obtained by reducing 5 puranas, 3 panas, 1 kakini, - 1 varataka, - 1/5 of that of a varataka to puranas ? This is problem connected with ancient Indian coins/ currencies 19. What amount is obtained by adding together ½. ¼ of ¼, 1 divided by 1/3, ½ + ½ of ½, and 1/3 - ½ of 1/3 ? 21. If 1 1/3 palas of black pepper are obtained for 1 ¼ panas, then how much of that will be obtained for (10 – 1/3 ) panas ? 22. If one and a half dornas and three dukavas of grain is obtained for 8, say, if you know, for how much will one khari and one drona of that grain be obtained 23. If 60 +1/2 kharis of grain is obtained for 100 + 1/3 rupas, how much of that grain will be obtained for a quarter of a rupa ? 24. Where one suvarana gets 70 +1/3 rupas, say, friend, what will 1 masha as lessened by 1/10 of a masha get there. 25. A certain lame person goes to distance of 1/8 of a yojana in 1/3 of a day, say in how much time will he go to a distance of 100 yojanas 26. An insect goes to a distance of 1/6 of an angula in ¼ of a day, in how much time will it go to a distance of 10 and half a yojanas 27. The best amongst the elephants goes forward at the rate of ½ ( 1 +1/4)( 1 – 1/3) (1 +1/2) of a yojana in 6 x 1/5 x 1/9 x 1/3 (1 +1/4) of a day and comes back at the rate of 2(1 –1/3) yojanas in (1 +1/2) days. Say, friend, in how much time will he go to a distance of 100 yojanas 28. In how much time will a man , earning at the rate of ( 8 –1/2) rupas in ( 1 + 1/3) days and spending on his food at the rte of ½ per day, be a lord of 100 rupas ? 29. 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 pears a necklace. 30. Being measured by the masha of 5 raktikas, a quantity of gold amounts to 300 suvarnas, say how much would that quantity of gold, amount to, when measured by the masha of 6 raktikas 31. How much gold of 11 varnas can be had in exchange for 168 suvarnas of 16 varnas ? 32. Quickly say how many blankets of length 6 hastas and breadth 2 hastas can be made of of the yarn which yields 200 bl;ankets of length 9 cubits and breadth 3 cubits. 33. How much gold of 10 ¼ varnas will be obtained in exchange for 100 suvarnas and 8 mashas of gold of 12 ½ varnas ? 34. If the interest on 100 for a month be 5, what is the interest on 60 for a year ? from the interest say the time, and from them both the unknown principal 35. If 1 ½ be the interest on 100 ½ for one third of a month, what will be the interest on 60 ¼ for ( 8 – ½) months 36. When the price of a suvarna of gold of 16 varnas is 60 , then say the price of 63 suvarnas of gold of 10 varnas 37. If 8 drobnas of rice are carried to a distance of one yojana for 6 panas, say for how much will a khari together with a drona of rice be carried to a distance of 3 yojanas 38. If 3 laborers earn 5 rupas in 2 days , say what will 8 laborers earn in 9 days ? 39. 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 ? 40. If a rectangular piece of stone with length, breadth and thickness equal to 9, 5 and 1 hastas respectively costs 8, what will two other rectangular pieces of stone of dimensions 10, 7 and 2 hastas cost ? 41. If the diet of an elephant of diamter 2 hastas , height 6 hastas and length 7 hastas is one drona, what should be diet of an elephant of diameter 3 hastas , height 9 hastas and length 10 hastas 42. If 2 palas of dry ginger re obtained for 6 and one pala of long pepper for 9, hoqw much of long pepper will then be obtained for 6 palas of dry ginger ? 43. If 16 mangoes are obtained for 2 panas and 100 wood apples for 3 panas, say then how many wood apples will be obtained for 6 mangoes. 44. If 16 workers of age 16 get 200, say then, o mathematician , how much will 2 workers of 20 years of age get ? 45. If 3 camels of 10 years of age get 108 puranas, say then what will 5 camels of 9 years of age get. 46. The rate of interest being 5 % per month, a certain sum is seen to amount to 96 in a year, Say, friend what is the capital and what the interest ? 47. The interest on 100 ½ for one month and a quarter being 1 ½, a certain sum amounts to 36 ½ in a period of 7 ½ months. Find the sum and the interest accrued thereon. 48. The rae of interest being 5% per month, the commission of the surety (bhavyaka) 1 % per month, the fee of the calculator (vrutti) ½ % per month and the charges for the scribe ¼ % per month, a certain 50. There are four bonds capitals amounting to 100, 200, 300, and 400 are given to someone on intrest at the raes of 2,3,4 and 5 % per month in the respective order; and months amounts to 2,3,5 and 4 multiplied by 2, have passed. Say, how would a single bond ( eka patra) be now made out of these. 51. O’ Learned man how a single bond be made of our the above 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 . 52. 57. ½ pala of asafetida , 2 palas of long pepper, and 7 palas of dry ginger are each obtained for one rupa. Give me equal quantities of each of them for one rupa. 58. The ca[pitals of three men are 1, 3 and 5 rupas or 1/3, ¼ and ½ rupas respectively. By purchasing an selling certain articles at the same rates and by selling the remnant articles at the rate of 1 for 3 rupas, they become possessed of equal riches . find the rates of purchase and sale. 59. The capitals of four men are 1 ½, 2, 3 and 5 rupas. 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 rupa , they become possessed of equal riches. Find the rates of purchase and sale. 60. Pigeons are sold at the rate of 5 for 3 rupas, cranes at the rate of 7 for 5 rupas, swans at the rate of 9 for 7 rupas and the peacocks at the rate of 3 for 9 rupas. Knowing the rates as stated above bring 100 birds for 100 rupas for the amusement of the princess. 61. The rate of sales of pomegranates, mangoes and wood apples are respectively 1 for 2 rupas, 5 for 3 rupas and 2 for 1 rupas. Bring 100 fruits for 80 rupas. 62. When a person , traveling at the speed of 8 yojanas per 5 minus ½ days, has alreddy traveled for 6 minus ¼ days, another person, who travels at the speed of 3 yojanas a day starts traveling from the same place along the same track. Say after calculating, when the latter traveler would overtake the former 63. One man travels at the speed of 8 yojanas a day and another at 2 yojanas a day from the same place and after reach the destination come back by the same track. The length of the track is 100 yojana. Say where is the meeting of the two. One going ahead and the other coming back 64. 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 panas as wages, a hole happned 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 hould be paid to the porter ? 65. One man saw a dance for one quarter of the day, another for two quarters of the day, another for three quarters of the day and yet another till the end of the day. The dancing party has to be paid by them a sum of ninety six rupas in all. If payment is to be made in proportion to the time of seeing the dance, how much of that sum should be paid by each of them separately ? 66. A palanquin is to be carried to a distance of 3 krosas by 10 men for 100 rupas. Of those men, 2,3 and 5 stop away after going over 1,2, and 3 krosas respectively. Calculate the wages of each of them separately. 67. Five scholars, enchanters of Vedas were invited by a person to take part in the worship of the five faces of the five faced god Siva on a dakshina of 300 rupas. And they,. On the completion of worship of, two, three, four and five faces respectively, went away one by one. Say what are their dakshinas. 68. In what time will the four drains, which severally fill up a cistern in ½, 1/4., 1/5 and 1/6 of a day , fill up that cistern if they are opened simultaneously , to flow into it ? 69. If for carrying 24 jack fruits over a distance of 5 krosas a porter is to get 9 of those jack fruits, what will he get if he carried them over a distance of 2 krosas ? 70. Twenty four jackfruits were carried to a distance by a man for 4 out of those 24 jack fruits as wages; the remaining jack fruits were carried over the remaining distance by another person for 5 of them as wages . the load was thus carried by the two persons over a distance of 5 krosas. Say how much of that distance was gone over by each of them ? 71. O’ friend, a cook prepares varieties of food with the six savours , pungent, bitter, astringent, acid, saline and sweet. Say what is the possible number of varieties ? 72. One fourth, one third, and one sixth of a pillar are respectively buried under the water, mud and sand of a river and three cubits are visible. Give out the measure of the length of that pillar 73. After gioving away one half of a quantity, then 2/3 of what remains, then ¾ of what remains there after and then 4/5 of what remains there after, the residue left is 3 . what was the quantity ? 74. Of a her of cows, one half went away towards the east and one fourth towards the west, the difference of the two as multiplied by 2 and divided by 5 went away towards the north and three cows are left. What is the numerical strength of the herd ? 75. A number is diminished by its square root, what remains is diminished by its one- sixth, what remains after that is diminished by its square root, what remains after that is diminished by is one fifth and what remains after that is diminished by twice the square root of itself; the residue now left is 8 . Find out the number . 76. One third of a troop of monkeys together with one third of itself has gone to the tank; the square root of the whole troop is afflicted with thirst; and the remaining two monkeys are sitting under the mango tree . what is the number of the monkeys in the troop ? 77. After giving away one , then one sixth of what remains, then one fourth of what remains after that, then one third of what remains after that, and then the square root of the original number, the residue left is 5. what is the original number ? 78. Say what is that number which being multiplied by 5/2 , then divided by 3, then squared, then increased by 9, then reduced to its square root, and then diminished by 1, becomes 4. 79. What is the sum of 5 terms of the series whose first term is 2 and common difference 3 ? and what of one half of a term ? Say the sum of one fifth of a term of the series whose common difference is 5 and first term 2 ? 80. In a leathern oil bottle full of oil there occurs a minute hole, and the oils leaks through it. The bottle has to be carried to a distance of 3 yojanas. If the wages for the first yojana be 10 panas and those for the subsequent yojanas successively less by 2 panas, what are the wages for a krosa ? 81. One man gets 3 rupas and the other men get 2 rupas more in succession; say , what do the first 4 ½ men get . 82. If a laborer gets 1 ½ in the first month and 1/3 more in succession in the following months, what will he get in the first 3 ½ months. 83. A man taking 3 rupas with him, went out to make profit. If his capital becomes double after every month, what will it become after 3 years ? 84. The first bangle is obtained for 8 panas, and the last bangle for 13 panas. If the total number of the bangles be 24 , say what is the price of all of them. 85. One man goes with initial speed 3 yojanas per day and acceleration 1 yojana per day per day and another man goes with the constant speed of 10 yojanas per day. In what time will they cover the same distance ? 86. After one man had traveled for 6 days with some initial speed (adi) and acceleration (uttara) another man went by the same track with an unknown initial speed and acceleration of 2 units per day per day. Say how will they meet each other two times . 87. In a gamble two persons alternately won 30, 10, 100 and 8 casts of dice beginning with 9 and increasing successively by 6. say who is the winner. 88. In the casts of dice ( alternatively won by the two persons) be 7, 3, 9, and 12 and the first term and common difference of the series formed by the take money, as stated before, then say after calculation who wins. 89. Say what is the sum of i. The sum of the first five natural numbers, ii. The square of 5 and iii. The cube of 5. 90. Friend, quickly say what is the sum of the cubes of 10 terms of the series whose first term and common difference are each unity; and also the sum of the successive sums of those terms. 91. O’ Friend, if you know then say after calculation the sum of i. The sum of successive sums of the first 6 natural numbers, ii. The sum of the squares of the first 6 natural numbers and iii. The sum of the cubes of the first 6 natural numbers. 92. Tell me the sum of the squares of the first six terms of the arithmetic series who first term is two and the common difference three. 93. Say after adding together the cubes of the four terms which begin with 5 and increase successively by 2. 94. In an equilateral quadrilateral, the face, the base and the altitude are all equal to the flank sides, each being 1 ½ hastas in length. Say, friend what is the area of the quadrilateral. 95. Give out the area of that rectangular quadrilateral in which the base and face are each 5 ½ hastas, and the flank sides and altitude each 3 hastas. 96. In a triangle the flank sides are 4 – ¼ and 3 ¼ hastas, the base is 3 ½ hastas and the altitude is 3 hastas, what is the area of that triangle ? 97. In an equilateral triangle the base is 8 ½ hastas and the altitude is 7 hastas and 8 2/3 angulas. What is the area thereof ? 98. If you know the method of finding the area of place figures, say the area of the isosceles triangle , whose flank sides are each 5 hastas, altitude 3 hastas and the base 8 hastas 99. In a quadrilateral the face is 1 1/3 hastas , the base is 9 1/3 hastas, the flank sides are each 5 hastas and the altitude is 3 hastas what is the area ? 100. In an in-equilateral quadrilateral with equal altitudes, the base is 10 hastas, the face is 4 1/6, the flank sides are 9 – 1/3 and 6 + ½ hastas and the altitude is 6 ½ hastas minus 1/60 of an angula . what is the area of the figure ? 101. What is the area of an elephant’s tusk whose base is 2 cubits and altitude cubits; and also of the figure of the shape of a felloe whose base and face are 3 cubits and altitude is 10 cubits ? 102. The central length of a crescent moon is 8 cubits and the central width 3 cubits. Treating it as made up of a paid of triangle, quickly say what its are is ? 103. In a thunderbolt, the central length is 10 hastas, the faces are each 5 hastas, and the central width is 2 hastas. What is its area, if it be regarded as made up of two quadrilaterals ? Sanskrit terms and the measurements used in mathematical books Sankalitha - addition, vyavakalith - subtraction, pratyutpanna - multiplication, bhagahara - division, varga – square, varga mula – square root, ghana – cube, Ghana mula – cube root Table of money measurements: one purana = 16 panas; one pana = four kakinis, one kakini = 20 varatakas/ coudies, 12 panas = one drama, 36 dramma = nishka. Table of weights: one masha = 5 gunjas, 16 masha = one karsha, one karsha of gold = suvarna , 4 karshas = one pala. Table of the measurements of capacity: One khari = 16 dronas, one drona = 4 adhakas, one adhaka = 4 prasthas, one prastha = 4 kudavas Table of linear measurements 24 angula = one hasta, 4 hasta – one danda, 2000 danda = one krosa 4 krosa = one yojana. Table of time measurements |
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