9. It is required to find two geometrical mean proportionals between 3 and 24; and four geometrical means between 3 and 96. Ans. 6 and 12; and 6, 12, 24, and 48 10. It is required to find six numbers in geometrical progression such, that their sum shall be 315, and the sum of the two extremes 165. Ans. 5, 10, 20, 40, 80, and 160 11. The sum of two numbers is a, and the sum of their reciprocals is b; required the numbers. 12. After a certain number of men had been employed on a piece of work for 24 days, and had half finished it, 16 men more were set on, by which the remaining half was completed in 16 days: how many men were employed at first; and what was the whole expence, at 1s. 6d. a day per man? Ans. 32 the number of men ; and the whole expence 115l. 4s. 13. It is required to find two numbers such, that if the square of the first be added to the second, the sum shall be 62, and if the square of the second be added to the first, it shall be 176. Ans. 7 and 13 14. 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 15. 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. ba -N Ans. x and y= am-f 16. 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; and so on for a third and fourth time; which being done, 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 17. 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, and 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 27001. 18. 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. 19. 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 51 ounces, and that of a cubic inch of tin 41 ounces. Ans. 420 oz. of copper, and 85 oz. of tin 20. 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 overtaken. Ans. 72 miles 21. In how many different ways is it possible to pay 100l. with seven shilling pieces and dollars of 4s. 6d. each ? Ans. 6 different ways 22. 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) 23. 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) 24. 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 25. Given x3y+y3x=3, and x®y2+y®x2=7, to find the values of x and y. Ans. x=(√5+1), g=√(√/5—1) 26. Given x+y+z=23, xy+xz+yz=167, and xyz =385, to find x, y, and z. Ans. x=5, y=7, z=11 27. 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, z=11, and w=20 28. Given x+yz=384, y+xz=237, and z+xy=192, to find the values of x, y, and z. Ans. x=10, y=17, and z=22 29. Given x2+xy=108, y2+yz=69, and z2+xz=580, to find the values of x, y, and z. Ans. x=9, y=3, and z=20 30. Given x2+xy+y2=5 and x+x2y2+y=11, to find the values of x and y by a quadratic. 2 1 2 Ans. x=√10+ √5, y=√/10-√5 31. Given the equation x4"-2x3n+x2 = a, to find the value of x by a quadratic. Ans. x=1+~{ ±√(a+; })} 32. 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. Ans. By 144, nearly 33. Required the least number of weights, and the weight of each, that will weigh any number of pounds from one pound to a hundred weight. Ans. 1, 3, 9, 27, 81 34. 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 35. It is required to find the least number, which being divided by 6, 5, 4, 3, and 2, shall leave the remainders 5, 4, 8, 2, and 1, respectively. Ans. 59 36. Given the cycle of the sun 18, the golden number 8, and the Roman indiction 10, to find the year. Ans. 1717 37. Given 256x - 87y=1, to find the least possible values of x and y in whole numbers. Ans. x=52, and y=153 38. It is required to find two different isosceles triangles such, that their perimeters and areas shall be both expressed by the same numbers. Ans. Sides of the one 29, 29, 40; and of the other 37, 37, 24 39. It is required to find the sides of three right angled triangles, in whole numbers, such, that their areas shall be all equal to each other. Ans, 58, 40, 42; 74, 24, 70; 113, 15, 112 1 40. Given x = 1.2655, to find a near approximate value of x. Ans. 3.82013 41. Given xy of x and y. = 5000, and y 3000, to find the values Ans. a 4.691445, and y=5.510132 42. Given xx+y=285, and y* xy= 14, to find the values of x and y. Ans. x 4.016698, and y = 2.825716 43. To find two whole numbers such, that if unity be added to each of them, and also to their halves, the sums, in both cases, shall be squares. Ans. 48 and 1680 44 Required the two least nonquadrate numbers x and y such, that x2+y2 and x3+y3 shall be both squares. Ans. x=364 and y=273 45. It is required to find two whole numbers such, that their sum shall be a cube, and their product and quotient squares. Ans. 25 and 100, or 100 and 900, &c. 46. It is required to find three biquadrate numbers such, that their sum shall be a square. Ans. 124, 154, and 204 47. It is required to find three numbers in continued geometrical progression such, that their three differences shall be all squares. Ans. 567, 1008, and 1792 48. It is required to find three whole numbers such, that the sum or difference of any two of them shall be square numbers. Ans. 434657, 420968, and 150568 49. It is required to find two whole numbers such, that their sum shall be a square, and the sum of their squares a biquadrate. Ans. 4565486027761 and 1061652293520 50. It is required to find four whole numbers such, that the difference of every two of them shall be a square number. Ans. 1873432, 9288168, 2399057, and 6560657 |