The U. S. Post-Offices receive 15 grams (though a little over weight) as half an ounce avoirdupois. The unit of weight in most commercial dealings is the kilogram; for very large quantities, the tonneau. CHAPTER XIII. REDUCTION OF DENOMINATE NUMBERS. 364. Reduction is the process of changing the de nomination of a quantity without changing its value. 365. There are two kinds of Reduction : 1. Reduction Descending, in which a higher denomination is changed to a lower; as, pounds to ounces, or pounds and ounces to drams. 2. Reduction Ascending, in which a lower denomination is changed to a higher; as, drams to ounces, or to ounces and pounds. 366. Reduction Descending. EXAMPLE 1.-Reduce £23 4d. 3 far. to farthings. £1 equals 20s. Therefore, £23 equal 23 times 20s.; but, as either factor may be made the multiplier, for convenience we say 20 times 23, or 460, shillings. We have now reduced the given number to 460s. 4d. 3 far. 1s. equals 12d. Therefore, 460s. equal 12 times 460, or 5520, pence-which, with the 4d., make 5524d. We have now reduced the number to 5524d. 3 far. 1d. equals 4 far. Therefore, 5524d. equal 4 times 5524, or 22096, farthings—which, with the 3 far., make 22099 far. Ans. £23 4d. 3 far. 20 460s. 12 5524d. 4 22099 far. Ans. 367. RULE.-Multiply the number of the highest denomination given, by the number required of the next lower denomination to make 1 of this higher; and to the product add the given number, if any, of such lower denomination. Treat this result, and those successively obtained, in the same way, till the required denomination is reached. Ex. 2.-How many drams in 343 tons? 1 ton, as was learned from the Table of equivalents memorized under Avoirdupois Weight, equals 512000 drams. Therefore, 343 tons equal 343 times 512000 drams. To multiply 343 and 512000 together, is shorter than to multiply successively by 20, 4, 25, 16, and 16, according to the rule. When no intermediate denominations are given, one multiplication, by the 343 512000 686000 343 1715 number of equivalent units, may, as in this Ans. 175616000 dr. case, most easily effect the reduction. 368. Reduction Ascending. Ex. 3.—Reduce 22099 far. to higher denominations. 4 farthings make 1 penny. Therefore, in 22099 far. are as many pence as 4 is contained times in 22099, or 5524d. 3 far. 12 pence make 1s. Therefore, in 5524d. are as many shillings as 12 is contained times in 5524, or 460s. 4d. We have now reduced the given number to 460s. 4d. 3 far. 20s. make £1. Therefore, in 460s. are as many pounds as 20 is contained times in 460, or £23; and we have thus reduced the given number to £23 4d. 3 far. 4) 22099 far. 12) 5524d. 3 far. Ans. £23 4id. 369. RULE.-Divide the given number by the number required of its denomination to make one of the next higher. Divide the quotient in the same way, and thus proceed till the required denomination is reached. The last quotient and the several remainders form the answer. 370. Proof.-In Ex. 1, Art. 366, we reduced £23 4d. 3 far. to 22099 farthings. In Ex. 3, Art. 368, we reduced 22099 farthings to £23 4d. 3 far. It will thus be seen that Reduction Descending and Reduction Ascending may be used to prove each other. Reduction of Federal Money. 371. Applying to Federal Money the rules just given, and using the short methods of multiplying and dividing by 10 and 100, we obtain the following rules, the reasons for which may be explained by the student. RULES.-I. To reduce dollars to mills, annex three naughts; to reduce dollars to cents, two; to reduce cents to mills, one. II. To reduce dollars and cents to cents,—or dollars, cents, and mills, to mills,-remove the dollarmark and the decimal point. III. To reduce mills to dollars, point off three figures from the right; to reduce cents to dollars, two; to reduce mills to cents, one. Construct two rules similar to the above for the reduction of francs to centimes, and centimes to francs (Art. 315). 372.-EXAMPLES FOR PRACTICE IN REDUCTION. Reduce the following: 1. $247 to cents. 2. $2.47 to mills. 3. 1563 m. to dollars. 4. $0.09 to mills. 6. 1516 cen. to francs. 7. 45830 c. to dollars. 8. $6245 to mills. 9. $413.65 to cents. 5. $1983.415 to mills. 10. 347 fr. to centimes. 11. Reduce 519841 cubic inches to cubic yards, etc. Ans. 11 cu. yd. 3 cu. ft. 1441 cu. in. 12. How many tons, etc., in 2548938 dr.? Ans. 4 T. 19 cwt. 2 qr. 6 lb. 12 oz. 10 dr. 13. Reduce to higher denominations 40,000,000 sec.; 164745". First ans. 1 yr. 97 d. 23 h. 6 min. 40 sec. Reduce the following to higher denominations, proving each result (Art. 370):— 14. 516423 far. 15. 827591 min. 16. m14627. 17. 201971 rd. 18. 23750 gr. Troy. 19. 156824 gr. Apoth. 20. 18003 links. 21. 4562 pt. (dry meas.). 22. 846 pt. (liq. meas.). 24. Reduce the following to cents, and add the results: 1459 mills; ten thousand and eleven dollars; 4 E.; 23 dimes; half-dollars; 14 half-eagles. Ans. 1012825.9 ct. Ans. 48142.5 ft. 25. Reduce 9 mi. 37 rd. 4 yd. to feet. 26. How many seconds in 49° 38′ 27′′? 27. How many more seconds are there in a leap-year than in a common year? 28. Find how many pence are equal to each of the following, and add the results: 17 guineas; 8216 far.; £4 11s. 9d.; 3 crowns; 10; 1 half-crown; 480 far.; 7 florins. Ans. 8007d. 29. What is the difference, in grains, between 1 dram Apothecaries' Weight and 1 dram Avoirdupois? Between 1 sc. and 1 dwt.? 30. If in a certain steel-pen factorý 28800 pens are made every working-day, how many gross will be made in 14 weeks and 3 working-days? 31. How many seconds are there in a lunar month? Ans. 2551442.84 sec. 32. Reduce each of the following to dollars, and add the results: 15 E.; 24414 c.; 15 double eagles; 1415 mills; 64 quarter-eagles; 195 dimes. Ans. $875.055. 33. In 5838 inches how many rods, etc.? Dividing successively by 12 and 3, we reduce the given number to 162 yd. foot, we 12) 5838 in. 3) 486 ft. 6 in. 162 yd. 6 in. 11) 324 half-yd. 29 rd. 5 half-yd. 29 rd. 2 yd. 14 ft. ft. Ans. 29 rd. 2 yd. 2 ft. In reducing yards to rods, therefore, after dividing by, if there is a remainder, divide it by 2 to bring it to yards. So, in reducing square yards to square rods, we have to divide by 301, or 121. This is equivalent to multiplying by what, and dividing by what? Multiplying by 4 reduces the square yards to what? On dividing by 121, therefore, what will be the denomination of the remainder, and how can we reduce it to square yards? 34. Reduce 14720196 square feet to acres, etc. Ans. 337 A. 3 R. 28 sq. rd. 20 sq. yd. 3 sq. ft. 35. If a vessel sailing due west on the equator changes |