sum shall be 62, and if the square of the second be added to the first, it shall be 176. Ans. 7 and 13 19. The fore wheel of a carriage makes six revolutions more than the hind wheel, in going 120 yards; but if the circumference of each wheel was increased by three feet, it would make only four revolutions more than the hind wheel in the same space; what is the circumference of each wheel? Ans. 12 and 15 feet 20. A number consisting of three digits in arithmetical progression being divided by the sum of its digits gives 26, and if 198 be added to it the digits will be inverted; what is the number? Ans. 234 21. If 6000 soldiers be placed in the form of an oblong, so that the number of men in rank shall be to those in file as 3 to 2, how much ground will they stand on, supposing the distance between man and man to be 23 yards? 96 Ans. 92ac. 3ro. 35p. 7 yards 22. It is required to divide a given number a into two such parts, x and y, that the sum of mx and ny shall be equal to some other given number b. Ans. x= and y b. an = am-b 23. Out of a pipe of wine, containing 84 gallons, 10 gallons were drawn off, and the vessel replenished with 10 gallons of water; after which, 10 gallons of the mixture were again drawn off, and then 10 gallons more of water poured in: now the like process having been repeated four times, it is required to find how much pure wine remained in the vessel, supposing the two fluids to have been thoroughly mixed each time? Ans. 48 gallons 24. It is required to find the least whole number such, that the sum of its aliquot parts shall exceed itself by 7. Ans. 196 25. It is required to find two such numbers, x and ولا that the sum of the aliquot parts of x shall be equal to y, and the sum of the aliquot parts of 2 26. A sum of money is to be divided equally among a certain number of persons; now if there had been 3 claimants less, each would have had 150l. more, but if there had been 6 more, each would have had 1207. less; required the number of persons, and the sum divided. Ans. 9 persons, sum 2700l. 27. From each of sixteen pieces of gold, a person filed the worth of half a crown, and then offered them in payment for their original value, but the fraud being detected, and the pieces weighed, they were found to be worth, in the whole, no more than eight guineas; what was the original value of each piece? Ans. 13s. 28. A composition of tin and copper, containing 100 cubic inches, was found to weigh 505 ounces; how many ounces of each did it contain, supposing the weight of a cubic inch of copper to be 5 ounces, and that of a cubic inch of tin 44 ounces. Ans. 420 oz. of copper, and 85 oz. of tin 29. A privateer, running at the rate of 10 miles an hour, discovers a vessel 18 miles a head of her, making way at the rate of 8 miles an hour; how many miles will the latter run before she is overAns. 72 miles taken. 30. At what time, next after 12 o'clock, will the hour, minute, and second hand of a watch, be all at the greatest distance from each other? Ans. 12 hours, 21 minutes 31. The sum of three numbers in harmonical proportion is 39, and the sum of their squares 549; what are the numbers? Ans. 9, 12, and 18 32. Given the sum of two numbers = 2, and the sum of their ninth powers =32, to find the numbers by a quadratic equation. Ans. 1±√(2√34–11) 33. It is required to find two numbers such, that their product shall be equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes. Ans. 5 and (5+√5) 34. The arithmetical mean of two numbers exceeds the geometrical mean by 13, and the geometrical mean exceeds the harmonical mean by 12; what are the numbers? Ans. 234 and 104 35. Given x'y+y'x=3, and x'y' + y2x2=7, to find the values of r. and y. Ans. x=(√5+1), y=4(√5— ́1) 36. Given a + xy=1932 and y-xy=4709 to find the values of x and y. Ans. 12 y = 17 37. Given x'y+yx=512500 and x'y-y'x= 5500 to find the values of x and y. Ans r=25 and y = 20 38. Given x+y+2=23, xy + xz+yz=167, and xyz=385, to find x, y, and z. Ans. x=5, y=7, x=11 39. To find four numbers, x, y, z, and w, having the product of every three of them given; viz. xyz=231, xyw=420, yzw=1540, and xzw=660. Ans. x=3, y=7, x=11, and w=20 40. Given x+yz=384, y+xz=237, and x+ xy=192, to find the values of x, y, and z. Ans. 10, y=17, and x=22 41. Given x2+xy=108, y2+yz=69, and 22+ xz=580, to find the values of x, y, and z. Ans. x=9, y=3, and z=20 42. Given x2 + xy + y2 = 5 and x* + x2y2+y* = 11 to find the values of x and y by a quadratic. 2 2 Ans. x= √10+ √5, y=√10−√5 43. The equation x-2x+x"= a being given, it is required to find the value of x. 44. It is required to find the number by which a people must increase annually, so that they may be doubled at the end of every century. be before there are 10 times as many as there were Ans. 231 years nearly at first? 46. Supposing there are 100000 inhabitants in VOL. I. 2 D a province, and that its population increases every 1 year by a part; what will be the number of its 30 inhabitants at the end of a century? Ans. 2654874 47. The earth having been repeopled, after the deluge, by the three sons of Noah and their wives, it is required to find what number of inhabitants there would be at the time of the building of the tower of Babel, 210 years afterwards, supposing their numbers to have been doubled every 14 years? Ans. 196608 ཎྜཎྜ 48. Required the least number of weights, and the weight of each, that will weigh any number of pounds from one pound to twenty-nine hundred weight. Ans. 1, 3, 9, 27, 81, 243, 729, and 2187 49. It is required to find two square numbers such, that either of them, when added to its aliquot parts, shall make the same sum. Ans. 106276 and 165649 50. It is required to find four whole numbers such, that the square of the greatest may be equal to the sum of the squares of the other three. Ans., 3, 4, 12, and 13 51. It is required to find the least number, which being divided by 6, 5, 4, 3, and 2, shall leave the remainders 5, 4, 3, 2, and 1, respectively. Ans. 59 52. Given the cycle of the sun 18, the golden number 8, and the Roman indiction 10, to find the year. Ans. 1717 |