23. Between Constantinople and Chicago? 24. Between Canton and San Francisco? For Dictation Exercises, see Key. 207. ADDITION AND SUBTRACTION OF DENOMINATE FRAC TIONS. ILL. EX., I. cwt. + qr. = what? These fractions, being of different denominations, must first be reduced to the same denomination. The fraction cwt. may be changed to quarters, or the qr. may be changed to parts of a cwt., and then addition can be performed. In the first case, the answer will be in qr.; in the second, in cwt. Another excellent method is first to reduce both fractions to integers of lower denominations, if necessary (Art. 198), and then add. Thus, qr. lb. Oz. Perform the following examples, and give the answers, as far as possible, in whole numbers of lower denominations. 208. MULTIPLICATION. ILL. Ex. 1 pt. 3 gi. × 7 = what? OPERATION. gi. pt. = 7 pt., which 7 X 3 gi. 21 gi. 5 pt.+1gi.; we write 1 gi. under gi. in the multiplicand, and reserve the 5 pt. to add with the pints in the product. 7x 1 pt. with the 5 pt. reserved 12 pt. 6 qt. 0 pt. Ans. 6 qt. 0 pt. 1 gi. 6 qt. O pt. 1 gi. Hence the 7 RULE FOR MULTIPLICATION OF COMPOUND NUMBERS. Multiply the number of the lowest denomination by the multiplier, and divide the product thus obtained by the number it takes of that denomination to make one of the next higher; write the remainder under the term multiplied, and add the quotient with the product of the next higher denomination; and thus continue till all the terms of the multiplicand are multiplied. EXAMPLES. 1. 7 bu. 4 pk. 3 qt. 1 pt. X 5 = ? Ans. 40 bu. 2 pk. 1 qt. 1 pt. 2. 28 gall. 2 qt. 1 pt. 3 gi. × 17 = ? Ans. 488 gal. O qt. 1 pt. 3 gi 3. 20 T. 16 cwt. 4 lb. 15 oz. X 25 =? 7. 18 y. 37 d. 23 h. 14 min. 7 sec. X 12 10. 7 £. 6 s. 3 d. 2 qr, X 73 = ? 11. 4 m. 7 f. 35 rd. 3 yd. 2 ft. 11 in. × 29 = ? 12. 118 cd. 4 cd. ft. 1 cu. ft, X 35 = ? 13. 76 cu. yd. 12 cu. ft. 184 cu. in. × 24 = ? 14. 4 sq. m. 320 A, 3 R. 25 sq. rd. 25 sq. yd. 7 sq. ft. 117 sq. in. X 9 = ? 15. How much land in 9 gardens each containing 34 sq. rd. 4 sq yd. 8 sq. ft. 67 sq. in. ? Art. 195, Note. 16. How many pickles in 17 jars, each jar holding 2 gall. 3 qt. 1 pt.? 17. If a car runs 18 m. 3 f. 29 r. 2 y. in of an hour, how far will it run in 7 h.? RULE FOR DIVISION OF COMPOUND NUMBERS. Divide the highest term of the dividend by the divisor; write down the quotient, and reduce the remainder to its value in the next lower denomination; add it to the number of that denomination, divide os before, and thus continue till every term is divided. PROOF. Compound Division may be proved by Compound Multiplication, and Compound Multiplication by Compound Division. EXAMPLES. 1. 14 £. 11 s. 3 d. 2 far. ÷ 8 =? Ans. 1 £. 16 s. 4 d. 32 far. 2. 56 cd. 5 cd. ft. 14 cu. ft. ÷ 5 = ? 3. 36° 18′ 36′′ Ans. 11 cd. 2 cd. ft. 12 cu. ft. 691 in. 40 =? 4. 74 ch. 3 rd. 22 l. 18 ? 5. 113 cu. yd. 22 cu. ft. 111 cu. in. 42 ? 6. 16 lb. 5 oz. 3 pwt. 21 gr. ÷ 13 =? ÷ 15 =? 12 oz. 3 dr. ÷ 6 = ? 19 =? 7. 38 ib, 8 3, 7 3, 2 9, 6 gr. =? 9 in. ÷ 12 = ? sq. in. 71=? 13. 30 y. 35 d. 7 h. 20 min. 35 sec. 29= ? 14. How far must a bird fly in one minute to fly 55 miles in an hour? 15. If 37 bu. 4 pk. of rye be divided between 7 men, what will each man receive? 16. If 65 A. 2 R. 18 sq. rd. 10 sq. yd. 14 sq. ft. be divided into 55 house-lots, what is the size of each? 17. How long will it take to travel 1 mile, at the rate of 75 miles in 10 h. 18 min. 12 s.? 18. Among how many men may 624 gall. 3 qt. be divided, that each man may receive 12 gall. 3 qt.? NOTE. Reduce each of the above to quarts before dividing. 19. How many bins, each containing 5 bu. 3 pk., will be required to hold 885 bu. 2 pk. of potatoes? 20. If a man walks 3 m. 6 fur. 26 rd. in one hour, how long will it take him to walk 23 m. 7 fur. 9 rd.? For Dictation Exercises, see Key. 210. LONGITUDE AND TIME. As the earth turns upon its axis once in 24 hours, it follows of 360°, or 15° of longitude, must pass under the sun in of 15°, or 15', must pass under the sun in 1 min. that 1 hour, and of time, and of 15′, or 15′′, must pass under the sun in 1 sec. of time; or, in a TABULAR FORM. 15° of longitude make a difference of 1 hour in time. Hence, to find the difference of longitude between any two places: Multiply the difference of time between the two places, expressed in hours, minutes and seconds, by 15. The product will express the number of degrees, minutes and seconds. NOTE. As the earth turns from west to east, sunrise occurs earlier in places east and later in places west of any given point. Hence the time is later in all places east, and earlier in all places west, of any given point than it is at that point. EXAMPLES. NOTE For table of latitude and longitude of places, see Art. 206, page 130. 1. The time in Pittsburg is 35 m. 54 s. earlier than in Boston; what is the difference of longitude between the two places? 2. What is the longitude of Pittsburg? 3. The time at St. Paul's is 1 h. 16 m. 192 s. New York; what is the longitude at St. Paul's? Ans. 8° 58' 30" Ans. 80° 2' W. earlier than in 4. The time in Copenhagen is 50 m. 19 s. later than in Greenwich; what is its longitude? 5. The time in Naples is 5 h. 41 m. 144 s. later than in Boston; what is its longitude? Ans. 14° 15′ 3′′ E. 211. From Art. 210 we also derive the following RULE. To find the difference of time between any two places: Divide the difference in longitude, expressed in degrees, minutes and seconds, by 15. The quotient will be the number of hours, minutes and seconds required. 1. What is the difference of time between Albany and Boston? Ans. 10 m. 443 s. 2. Between Paris and St. Petersburg? 3. Between Montreal and Mexico? 4. Between Cape Horn and Cape Good Hope? Ans. 5 h. 43 m. 0185 s. 5. Between Charleston, S. C., and Calcutta? 6. Between Canton and San Francisco? 7. When it is 8 o'clock P. M. in Washington, what is the time in London ? next day. Ans. 1 o'clock, 7 m. 374 s. A. M. of the 8. At 2 A. M., Jan. 1, 1864, at Paris, what was the time in New Orleans? 212. QUESTIONS FOR REVIEW. 1. When are denominate numbers simple? when compound? Give examples of each. May abstract numbers be compound? What is REDUCTION as applied to compound numbers? What is |