a 3s. to 2s. on every game, and after a certain number of games found that he had lost 17 shillings. Now had A won 3 more from B, the number he would then have won, would be to the number B had won, as 5 to 4. How many games did they play? Aps. 9. Prob. 20. Two persons, A and B, can perform a piece of work in 16 days. They work together for 4 days, when A being called off, B is left to finish it, which he does in 36 days more. In what time would each do it separately? Ans. A in 24 days, and B in 48 days. Prob. 21. Some hours after a courier had been sent from A to B, which are 147 miles distant, a second was sent, who wished to overtake him just as he entered B; in order to which he found he must perform the journey in 28 hours less than the first did. Now the time in which the first travels 17 miles added to the time in which the second travels 56 miles is 13 hours and 40 minutes. How many miles does each go per hour! 9 Ans. the first goes 3, and the second 7 miles an hour. Prob. 22. Two loaded wagons were weighed, and their weights were found to be in the ratio of 4 to 5. Parts of their loads, which were in the proportion of 6 to 7, being taken out, their weights were then found to be in the ratio of 2 to 3; and the sum of their weights was then ten tons. What were the weights at first? Ans. 16, and 20 tons. Prob. 23. A and B severally cut packs of cards; so as to cut off less than they left. Now the number of eards left by A added to the number cut off by B, make 50; also the number of cards left by both exceed the number cut off, by 64. How many did each cut off ? Ans. A cut off 11, and B 9. Prob. 24. A and B speculate with different sums; A gains 1501, B loses 501, and now A's stock is to B's las 3 to 2. But bad A lost 501, and B gained 1001, then A's stock would have been to B's as 5 to 9. What was the stock of each ? Ans. A's was 3001, and B's 3501. Prob. 25. A Vintner bought a dozen of port wine and 3 dozen of white, for 121. 12 shillings; but the price of each afterwards falling a shilling per bottle, he had 20 bottles of port, and 3 dozen and 3 bottles of white more, for the same sum. What was the price of each at first ? Ans. the price of port was 3s. and of white 2s. per bottle. Prob. 26. Find two numbers in the proportion of 5 to 7, to which two other required numbers in the proportion of 3 to 5 being respectively added, the sums shall be in the proportion of 9 to 13; and the difference of those sums = 16. Ans. the two first numbers are 30 and 42; the two others, 6 and 10. Prob. 27. A Merchant finds that if he mixes sherry and brandy in quantities which are in the proportion of 2 to 1, he can sell the mixture at 783. per dozen; but if the proportion be as 7 to 2, be must sell it at 79 shillings a dozen. Required the price of each liquor. Ans. the price of sherry was 31s., and of brandy 72s. per dozen. Prob. 28. A number consisting of two digits when divided by 4, gives a certain quotient and a remainder of 3; when divided by I gives another quotient and a remainder of 8. Now the value of the digit on the left-hand is equal the quotient which was got when the number was divided by 9; and the other digit is ' equal 4th of the quotient got when the number was divided by 4. : Required the number. Ans. 71. Prob. 29. To find three numbers, such, that the . first with į the sum of the second and third shall be 120; the second with {th the difference of the third 1 and first shall be 70; and the sum of the three numbers shall be 95. Ans. 50, 65, and 75. Prob. 30. There are two numbers, such, that the greater added to } the lesser is 13; and if | the lesser be taken from the greater, the remainder is nothing. } What are the numbers ? Ans. 18, and 12. Prob. 31. There is a certain number, to the sum of whose digits if you add 7, the result will be three times the left-hand digit; and if from the number itself you subtract 18, the digits will be inverted. What is the number? Ans. 53. Prob. 32. A person has two horses, and a saddle worth 101.; if the saddle be put on the first horse, his value becomes double that of the second ; but if the saddle be put on the second horse, his value will not amount to that of the first horse by 131. What is the value of each horse ? Ans. 56 and 33. Prob. 33. A gentleman being asked the age of his two sons, answered, that if to the sum of their ages 18 be added, the result will be double the age of the elder; but if 6 be taken from the difference of their ages, the remainder will be equal to the age of the younger. What then were their ages ? Ans. 30 and 12. Prob. 34. To find four numbers, such, that the sum of the 1st, 2d, and 3d, shall be 13; the sum of the 1st, 2d, and 4th, 15; the sum of the 1st, 3d, and 4th, 18; and lastly the sum of the 2d, 3d, and 4th, 20. Ans. 2, 4, 7, 9. Prob. 35. A son asked his father how old he was. His father answered him thus. If you take away 5 . from my years, and divide the remainder by 8, the quotient will be of your age; but if you add 2 to f 2 your age, and multiply the whole by 3, and then subtract 7 from the product, you will have the number of the years of my age. What was the age of the father and son ? Ars. 53, and 18. Prob. 36. Two persons, A and B, had a mind to purchase a house rated at 1200 dollars; says A to B, if you give me of your money, I can purchase the house alone; but, says B to A, if you will give me 3th of yours, I shall be able to purchase the house. How much money had each of them? Ans. A had 800, and B 600 dollars. Prob. 37. There is a cistern into which water is admitted by three cocks, two of which are exactly of the same dimensions. When they are all open, fivetwelfths of the cistern is filled in 4 hours; and if one of the equal cocks be stopped, seven-ninths of the cistern is filled in 10 hours and 40 minutes. In how many hours would each cock fill the cistern? Ans. Each of the equal ones in 32 hours, and the other in 24. 38. Two shepherds, A and B, are intrusted with the charge of two flocks of sheep. A's consisting chiefly of ewes, many of which produced lambs, is at the end of the year increased by 80; but B finds his stock diminished by 20; when their numbers are in the proportion of 8:3. Now had A lost 20 of his sheep, and B had an increase of 90, the numbers would have been in the proportion of 7 to 10. What were the numbers ? Ans. A's 160, and B's 110. Prob. 39. At an election for two members of congress, three men offer themselves as candidates ; the number of voters for the two successful ones are in the ratio of 9 to 8; and if the first had had 7 more, his majority over the second would have been to the majority of the second over the third as 12: 7. Now if the first and third had formed a coalition, and had one more voter, they would each have succeeded by a majority of 7. How many voted for each? Ans. 369, 328, and 300, respectively. CHAPTER VI. ON THE INVOLUTION AND EVOLUTION OF NUMBERS, AND OF ALGEBRAIC QUANTITIES. 274. The powers of any quantity, are the successive products, arising from unity, continually multiplied by that quantity. Or, the power of the order om of a quantity, m being a whole positive number, is the product of that quantity continually multiplied m-1 times into itself, or till the number of factors amounts to the number of units in that given power. 975. INVOLUTION is the method of raising any quantity to a given power, EvoLUTION, or the extraction, of roots, being just the reverse of Involution, is the method of determining a quantity which, raised to a proposed power, will produce a given quantity. Note.--The terın root bas been already defined, (Art 15). § 1. INVOLUTION OF ALGEBRAIC QUANTITIES. 276. It has been observed, (Art. 13), that the powers of algebraic quantities, are expressed by placing the index or exponent of the power over the quantity. Hence, if a proposed root be a single letter and without a coefficient, any required power of it will be expresssed by the same leiter with the index of the power written over it, Thus, the nth power of a is =a", M being any positive number whatever. |