9. A agreed to labor 21 days on condition that he should receive 1 cent for the first day, 2 cents for the second, 4 cents for the third, and so on; how much did he receive in all? 10. A gentleman, thinking $5000 too much for a farm containing 17 acres, agreed to pay 1 cent the first acre, 3 cents for the second, 9 cents for the third, and so on; which price was the greater, and how much? INFINITE SERIES. 273. An Infinite Series is a series which has an infinite number of terms. In a decreasing geometrical progression we subtract the last term multiplied by the ratio from the first term, and divide by 1 minus the ratio. In a decreasing geometrical progression of an infinite number of terms the last term becomes so small that it may be regarded as zero; hence from the above we have the following RULE. The sum of an infinite series equals the first term divided by 1 minus the ratio. WRITTEN EXERCISES. 1. Find the sum of the infinite series 1 + 1⁄2 +4, etc. 1 PROCESS. Sum = 1 1 = = 2, Ans. Find the sum— 2. Of the infinite series 1 + 1 + 1, etc. SECTION XIII. MISCELLANEOUS PROBLEMS. 274.-1. FIND the greatest common divisor of 1107, 1353, and 1599. 2. Find the least common multiple of 369, 697, and 861. 3. Reduce to its lowest terms. 567 693 4. Reduce 42+ 5. Reduce .6 of 43 of 9. What part of to an improper fraction. 6. Add + + 3 to 12 7. Find the value of of 12. 8. What part of 4 is 23? 10. Divide 163 by 233 31 .8 of 6 41 9 is .4? 26 61 11. Add 37 + 185 + 42o + 100%. 12. Reduce of 3 X 3 - 1 83 1 13. Divide .00084 by 57000. 14. Divide 7000 by .00002. 15. Reduce and 2 to decimals. 14 125 to a simple fraction. 130. of of of divided by § of of 151 12 16. Find the value of (.25 500 +7.5) X ÷ 110 .005 17. How many posts 6 ft. apart will be required to make 5 miles of fence? 18. How many dry gallons in 80 gallons liquid measure? 19. How many pounds Troy in 50 pounds avoirdupois ? 20. What will 5 bu. 3 pk. of berries cost at 8 cents a quart? 21. A railroad 60 ft. wide was run through a county 10 miles in length; how many acres did it cover? 22. If it requires 3 bu. of seed to sow an acre, how many pints will be required to sow a field 80 ft. square? 23. A cylindrical cistern is 6 ft. 6 in. deep, and 6 ft. in diameter; how many hogsheads of water will it hold? 24. The area of a square field is 4 A. 108 sq. rd.; find the perimeter in rods. 25. Simplify 83-23 — 95 — 11 12+57 + 41 26. A bin 8 ft. 6 in. long, 5 ft. 4 in. wide, and 6 ft. 8 in. high is full of wheat; if a bushel is equal to 14 cu. ft., what are the contents worth at 70 cents a bushel? 27. Find the exact number of days from May 5, 1890, to July 4, 1896. 28. When it is 3.15 A. M., Dec. 25th, at Vienna, longitude 16° 23′ east, what is the time at San Francisco, longitude 122° 23' west? 29. How deep must a circular cistern 5 ft. in diameter be to hold 50 barrels of 31 gallons each? 30. How many feet of railing will be required to enclose a square acre? How many to enclose a circular acre? 31. How often will a wheel 5 ft. in diameter revolve in going 10 miles? 32. Multiply 102117 by 45. 33. What is the difference in time between a city whose longitude is 76° 37' west and a city whose longitude is 13° 23′ east? 34. Divide 42682 by 62. 35. A rectangular field is 40 rd. wide and 15 ch. long; what is it worth, at $120 an acre? 36. What is 371% of 8%? 37. What per cent. of 15 is 51? 38. An agent sold goods for $1295, and lost 1 of the cost; how much did the goods cost him? 39. A man lost $475, which was 20% of what he had remaining; how much had he at first? 40. Find the simple interest of $420 for 3 yr. 4 mo. 12 da. at 51%. 41. Find the exact interest of $340 from May 8, 1894, to Jan. 11, 1895, at 6%. 42. Find the annual interest of $600 from June 10, 1890, to Aug. 15, 1894, at 6%, if the interest for the first two years is paid. 43. A sold a farm for $12460, and gained of the cost; how much did he gain? 44. An agent received $2000 to invest in goods after deducting 2% commission; what is the amount of his commission? 45. An agent sold goods on a commission of 3%; he remitted his employer $1500; what was the amount of goods sold? 46. Find the present worth and true discount of $800 for 1 yr. 6 mo. at 8%. 47. Find the contents of a pan whose altitude is 9 in., the radius of the lower base being 6 in. and of the upper base 8 in. 48. An agent's commission at 2% is $320; what sum did he remit to his employer? 49. What will it cost to paint the dome of a hall which is in the form of a hemisphere 40 ft. in diameter, at 40 cents a square yard? 50. The hypotenuse of a right triangle is 75 ft. and the base is 40 ft.; required the perpendicular. 51. In what time will $4080 at 5% give $668.10 interest? 52. Extract the square root of 430.064644. 53. What sum must be invested in 6% stock at 1171, brokerage 1%, to yield an annual income of $426? 54. What must I pay for 6% bonds to realize 51% on my investment, brokerage %? 55. Find the cube root of to three places of decimals. 56. M sold two houses for $4800 each: on the one he gained 25%, and on the other he lost 25%; did he gain or lose on the entire transaction, and how much? 57. If a bushel equals 11 cubic feet, how deep must a bin 8 ft. long and 4 ft. wide be to hold 120 bushels? 58. At what rate will $570.05 in 3 yr. 5 mo. 5 da. give $127.11+interest? 59. Find the 400th term of the series 3, 8, 13, 18, etc. 60. A rectangular box contains 30720 cu. in.; what are its dimensions, if they are to each other as 3, 4, and 5? 61. A 90-day note for $450 was discounted at a bank at 6%; required the proceeds. 62. I sold goods at 5% through an agent who charged me 3% my commission after paying the agent' was $240; required my entire commission. 63. Find the difference between the true discount and the bank discount of $1000 for 1 yr. 3 mo. 15 da. at 7%. can be bought for 64. How many shares of stock at 113 $7272, brokerage %? 65. Find the surface and contents of a sphere 12 in. in diameter. 66. Required the face of a 4-mo. note which when discounted at a bank at 6% yields $240. 1 67. Find the value of 1 4+圣 68. If a sphere 4 in. in diameter weigh 20 lb., what will a sphere 20 in. in diameter weigh? 4+ |