9. Required the 4th root of .967845, by logarithms. Ans. 9918624 10. Required the 7th root of .098674, by logarithms. 3. Required the .07 power of .00563, by logarithms. Ans. .6958821 4. Required the value of () 1 ×() 17 15 , by logarithms. Ans. .04279825 7 12 by loga 5. Required the value of x.012 226 MISCELLANEOUS QUESTIONS. 1. A person being asked what o'clock it was, replied ⚫ that it was between eight and nine, and that the hour and minute hands were exactly together; what was the time? Ans. 8h. 43min. 38 sec. 2. A certain number, consisting of two places of figures, is equal to the difference of the squares of its digits, and if 36 be added to it the digits will be inverted; what is the number? Ans. 48 3. What two numbers are those, whose difference, sum, and product, are to each other as the numbers 2, *3, and 5, respectively? Ans. 2 and 10 4. A person, in a party at cards, betted three shillings to two upon every deal, and after twenty deals found he had gained five shillings; how many deals did he win? Ans. 13. 5. A person wishing to enclose a piece of ground with palisades, found, if he set them a foot asunder, that he should have too few by 150, but if he set them a yard asunder he should have too many by 70; how many had he? Ans. 180 6. A cistern will be filled by two cocks, A and B, running together, in twelve hours, and by the cock a alone in twenty hours; in what time will it be filled by the cock B alone? Ans. 30 hours 7. If three agents, A, B, C, can produce the effects a, b, c, in the times e, f, g, respectively; in what time would they jointly produce the effect d. Ans. d÷(++) efg 8. What number is that, which being severally added to 3, 19, and 51, shall make the results in geometrical progression? Ans. 13 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 1151. 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 am-b Ans. x= and y = m-n m-n 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. 484 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 120l. less ; required the number of persons, and the sum divided. Ans. 9 persons, sum 2700l. 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 subic inch of copper to be 54 ounces, and that of a cubic inch of tin 44 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 xy2 +y6x2=7, to find the values of x and y. Ans. x=(√5+1), y=(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 x2+x2y2+y=11, to find the values of x and y by a quadratic. 31. Given the equation xin-2x3n+xn = a, to find the value of x by a quadratic. |