52. Change to the decimal of an acre 135 sq. rd. 54 sq. ft. 53. Regarding a year as 12 months of 30 days each, what decimal of a year is 6 mo. 18 d.? 8 mo. 24 d.? 5 mo. 27 d. ? For other examples in reduction of denominate fractions, see page 171. 364. The operations upon compound numbers are similar to those upon simple numbers, the principal difference being that in operations upon compound numbers we use irregular scales, instead of the scale of tens. No special rules, therefore, are necessary for addition, subtraction, multiplication, and division. 365. ILLUSTRATIVE EXAMPLES. (I.) What is the sum of 11° 4′ 58′′, 37° 30′ 27′′, and 27° 24′ 54′′? (I.) WRITTEN WORK. 11° 4' 58" 37° 30′ 27′′ 27° 24' 54" Ans. 76° 0′19′′ Explanation. (I.) We write these numbers so that units of the same denomination shall be expressed in the same column. Adding the seconds, we have 139". Dividing 139′′ by 60 (60" 1'), we have 2′ 19′′. We write the 19" under the line in the seconds' place. Adding the 2′ with the minutes of the given numbers, and dividing the sum by 60 (60′ = 1°), we have 1° 0'. We write O' under the line in the minutes' place. the degrees of the given number, we have 76°. ADDITION. = Adding the 1° with Ans. 76° 0′ 19′′. (III.) 5 Ans. 9 m. 265 rd. 34 yd. 2 ft. or 9 m. 265 rd. 4 yd. Oft. 6 in. Ans. 196 bu. 2 pk. 3 qt. NOTE. Change any denominate fraction which occurs in an answer, or in an example, to units of the lower denominations given. (See examples 56, 57, and 58.) 366. Examples for the Slate. 54. What are the contents of three barrels which contain respectively, 45 gal. 2 qt., 42 gal. 3 qt., and 47 gal. 1 qt. ? 55. How much land in four lots which contain as follows: 7 A. 83 sq. rd. 31 sq. ft., 15 A. 146 sq. rd., 22 A. 52 sq. 13 sq. ft., and 5 A. 9 sq. rd. ? rd. 56. What is the length of three roads measuring respectively 15 m. 87 rd., 28 m. 40 rd., and 35 miles? 57. To 8° 17' 32′′ add 4.735° and §°. 58. Add together 7 d. 6 h., 63 d., and 0.375 of a week. 367. ILLUSTRATIVE EXAMPLE II. To of a gallon add of a quart. WRITTEN WORK. 3o1⁄2 gal. = 31⁄2 of 4 qt. = 1 qt. qt. Ans. 17 qt. WRITTEN WORK. 5 rd. 3 yd. 1 ft. 59. Add of a quart to 12 of a bushel. 61. Add 54 pounds to 4 of a ton. Perform such examples in exercises 206-208, page 171, as the teacher may indicate. Ans. 3 rd. 3 yd. 2 ft. Explanation. - That these fractions may be added they must first be expressed in the same denomination. They may be so expressed by changing 2 gal. to quarts, etc. SUBTRACTION. 368. ILLUSTRATIVE EXAMPLE. What is the difference between 5 rd. 3 yd. 1 ft. and 1 rd. 4 yd. 2 ft.? Explanation. We write these numbers as in simple subtraction, and subtract first the 2 feet of the subtrahend. As we have but 1 foot in the minuend, we cannot now take 2 feet away. So we change 1 of the 3 yards (leaving 2 yards) to feet. This 1 yard equals 3 feet. We add the 3 feet to the 1 foot, making 4 feet. Subtracting 2 feet from 4 feet we have 2 feet left, which we write as part of the remainder. As we have but 2 yards left in the minuend, we cannot now take 4 yards away, so we change 1 of the 5 rods to yards. This equals 5 yards, which, added to 2 yards, make 7 yards. Subtracting 4 yards from 7 yards, we have 3 yards left, etc. 65. 1 m. 80 rd. 2 yd. less 315 rd. 3 yd. equals what? 66. What is the difference between 5 ft. 6 in. and 3 rd. ? 67. A man who had of a square mile of woodland sold 51⁄2 square rods. How much had he left? 68. A man having of a pound of silver ore, gave away 31 pennyweights. How much had he left? 69. What is the difference between 0.378 of a day and 44.55 of a minute? 70. Cape Horn is in 55° 58′ 4′′ south latitude, and the Cape of Good Hope is in 34° 22' south latitude. Which is farther south, and how much? The difference of latitude between places on opposite sides of the equator is found by adding the latitudes. The difference of longitude between places on opposite sides of the first meridian is found by adding the longitudes. If their sum exceeds 180°, the difference of longitude equals 360° minus that sum. For a table of longitudes, see page 159. What is the difference of longitude between 71. Albany and Chicago? 73. Rome and New York? 72. Berlin and Paris? 74. San Francisco and Calcutta? 75. What is the difference in latitude between Philadelphia 39° 57' north latitude, and Buenos Ayres 34° 3′ south latitude? To find the Number of Years, Months, and Days from one Date to another. NOTE. The following method of finding the time is generally used in computing interest. 370. ILLUSTRATIVE EXAMPLE I. What is the time in years, months, and days from Jan. 11, 1877, to May 5, 1881 ? Explanation. From Jan. 11, 1877, to Jan. 11, 1881, is 4 years; from Jan. 11, 1881, to April 11, 1881, is 3 months; from April 11 to April 30 is 19 days, and from April 30 to May 5 is 5 days more. Ans. 4 y. 3 m. 24 d. Rule. 371. To find the difference in time between two dates: First find the number of entire years between the two dates, then the number of calendar months remaining, and lastly, the remaining days. 372. Oral Exercises. a. How many years, months, and days are there from Feb. 3, 1875, to Oct. 17, 1878? b. How many years, months, and days are there from Sept. 25, 1874, to Jan. 4, 1882 ? c. Mozart was born Jan. 27, 1756, and died Dec. 5, 1791; at what age did he die? d. Goethe died March 22, 1832, and Bryant was born Nov. 3, 1794; what was Bryant's age when Goethe died? 373. ILLUSTRATIVE EXAMPLE II. How many days are there from Nov. 12, 1875, to March 10, 1876 ? Explanation. There are 18 days remaining in November, 31 days in December, 31 in January, 29 in February, and 10 in March. 18 +31 +31 + 29 + 10 = 119. Ans. 119 days. e. How many days from March 7 to July 1, 1878 ? f. How many days from Oct. 9, 1876, to Feb. 11, 1877? g. How many days from January 15 to August 7, 1875? 374. A Table showing the Number of Days From any To the corresponding Day of the following 31 92 122 153 184 214 245 275 30 91 122 153 183 214 244 123 153 184 214 61 92 Jan. Feb. Mar. Apr. May. June. July. Aug. Sept. Oct. Nov. Dec. January.. 365 31 59 90 120 151 181 212 February. 334 365 28 59 89 120 150 181 March.... 306 337 365 61 April 275 306 334 365 May..... 245 276 304 335 365 June 214 245 273 304 334 365 30 61 92 July....... 184 215 243 274 304 335 365 31 62 92 August... 153 184 212 243 273 304 334 365 31 September 122 153 181 212 242 273 303 334 365 October .. 92 123 151 182 212 243 273 304 335 365 31 61 November 61 92 120 151 181 212 242 273 304 334 365 30 December 31 62 90 121 151 182 212 243 274 304 335 365 122 153 183 123 153 61 92 122 30 61 91 ..... 61 31 NOTE. In leap years, if the last day of February is included in the time, a day must be added to the number obtained from the table. Find from the table above the number of days h. From April 19 to June 19. i. From Jan. 1 to March 4. WRITTEN WORK. 243 273 304 334 212 242 273 303 13 sq. rd. 72 sq. ft. 4 Perform such examples of exercises 209 and 210, page 171, as the teacher may indicate. j. From Dec. 5 to Feb. 5. k. From Oct. 12 to Feb. 15. MULTIPLICATION. 375. ILLUSTRATIVE EXAMPLE. How much land is there in 4 gardens, each containing 13 sq. rd. 72 sq. ft.? Explanation. - Multiplying 72 sq. ft. by 4, we have 288 sq. ft. for a product, which equals 1 sq. rd. and 15 sq. ft. We write the 15% sq. ft. and carry the 1 sq. rd. to the square 53 sq. rd. 153 sq. ft. Ans. rods in the product. 13 sq. rd. multiplied by 4 are 52 sq. rd., which, with the 1 sq. rd. carried, are 53 sq. rd. Ans. 53 sq. rd. 154 sq. ft. |