383. To Reduce a Denominate Fraction from a Lower to a Higher DenomiDation. EXAMPLE. — Reduce f of a grain to the fraction of a pound, Troy. OPERATION. EXPLANATION.- of a grain equals of at of a pen8 X X XXL X t = tatoo. nyweight; of 'of a pennyweight equals of of zo of an ounce; of 4 of fo of an ounce equals of the of sto of 1 of a pound, or tatoo of a pound. $gr. = tatoo lb., Troy. Rule.- Divide by the units in the scale, from the given to the required denomination. 3 384. To Reduce Denominate Fractions to Integers of Lower Denominations. EXAMPLE.-Reduce of a pound, Troy, to integers of lower denominations. OPERATION. EXPLANATION.—1 of a pound equals 1 of the 12 ounces X4=# oz. = 24 oz. in a pound, or if ounces, which reduced gives 24 ounces ; of an ounce equals 1 of the 20 pennyweights in an ounce, 1 x 0 = 5 pwt. or 5 pennyweights. Therefore, 1 of a pound, Troy, equals 2 ounces, 5 pennyweights. lb., Troy, = 2 oz. 5 pwt. Rule.-Multiply the denominate fraction by the unit next lower in the scale, and if the product be an improper fraction reduce it to a whole or mixed number. 385. To Reduce a Compound Denominate Number to a Fraction of a Higher Denomination, EXAMPLE.—Reduce 7 oz. 5 pwt. 9 gr. to the fraction of a pound, Troy. OPERATION. FIRST EXPLANATION.-Since 1 ounce equals 20 pennyweights, oz. 5 pwt. 9 gr. 7 ounces equal 140 pennyweights; 140 pennyweights plus 5 pen nyweights equals 145 pennyweights; since 1 pennyweight equals 20 24 grains 145 pennyweights equal 3480 grains; 3480 grains plus 9 140 pwt. grains equals 3489 grains; since 1 pound equals 5760 grains, 3489 5 pwt. grains equal $48% of a pound. 145 pwt. SECOND EXPLANATION.-Since 1 ounce equals 480 grains, 7 24 ounces equal 3360 grains; since 1 pennyweight equals 24 grains, 5 pennyweights equal 120 grains; 3360 grains, plus 120 grains, plus 9 grains equal 3489 grains; since 1 pound, Troy, equals 5760 grains, 3489 grains equal $48% of a pound. Therefore, 7 ounces, 5 pen3489 gr. = nyweights, 9 grains, equal $48% of a pound, Troy. 3480 gr. 9 gr. Rule.-Reduce the compound denominate number to its lowest denomination for a numerator, and a unit to the same denomination for a denominator; the fraction thus formed is the answer sought 7.60 pwt 14.4 gr. 386. To Reduce a Denominate Decimal to Units of Lower Denominations. EXAMPLE.—Reduce.865 of a pound, Troy, to integers of lower denominations. OPERATION. .865 lb. 12 EXPLANATION.—.865 of a pound equals .865 of the 12 ounces in 1 pound, or 10.38 ounces; 10.380 oz. .38 of an ounce equals.38 of the 20 pennyweights 20 in 1 ounce, or 7.6 pennyweights; .6 of a pennyweight equals .6 of the 24 grains in 1 penny weight, or 14.4 grains. Therefore, .865 of a 24 pound, Troy, equals 10 ounces, 7 pennyweights, 14.4 grains. .865 lb., Troy, = 10 oz. 7 pwt. 14.4 gr. Rule.-Multiply the decimal by that unit in the scale which will reduce it to units of the next lower denomination, and in the product point off as in decimals. Proceed in like manner with all decimal remainders. 387 To Reduce Denominate Numbers to Decimals of a Higher Denomination, EXAMPLE. — Reduce 8 oz. 3 pwt. 15 gr. to the decimal of a pound, Troy. OPERATION. 24 ) 15 gr. EXPLANATION.—Since 24 grains equal 1 pennyweight, 15 .625 grains equal or.625 of a pennyweight; 3 pennyweights plus 3. pwt. .625 pennyweights equal 3.625 pennyweights; since 20 penny. 20 ) 3.625 pwte weights equal 1 ounce, 3.625 pennyweights equal .18125 of an ounce, and 8 ounces plus .18125 of an ounce equal 8.18125 .18125 oz. ounces; since 12 ounces equal 1 pound, 8.18125 ounces equal 8. OZ. .681771, of a pound. Therefore, 8 oz. 3 pwt. 15 gr.=.681771 12 ) 8.18125 oz. lb., Troy. .681771 lb. Rule.- Divide the lowest denomination given by the number in the scale next higher, and to the quotient add the integers of the next higher denomination. So continue to divide by all the successive orders of units in the scale. ADDITION OF DENOMINATE NUMBERS. 02. 388. EXAMPLE.—Find the sum of 2 lb. 5 oz. 13 pwt. 4 gr., 17 lb. 11 oz. 18 pwt. 20 gr., and 9 lb. 9 oz. 6 pwt. 15 gr. OPERATION. EXPLANATION.—Since each of the given expressions is a ib. pwt. gr. compound number of the same class, and they all have the 2 5 13 4 same varying scale, their addition may be performed the same as in simple numbers; in reducing the sum of each column 17 11 18 20 from a lower to a higher order, observe the units in the 9 9 6 15 ascending scale. 30 lb. 2 oz. 18 pwt. 15 gr. Rule.-I. Write the numbers of the same unit value in the same column. II. Beginning with the lowest denomination, add as in simple numbers, and reduce to higher denominations according to the scale. SUBTRACTION OF DENOMINATE NUMBERS. 389. EXAMPLE.–Subtract 11 lb. 7 oz. 13 pwt. 9 gr. from 23 lb. 4 oz. 17 pwt. Rule.- Write the numbers as for simple subtraction; take each subtrahend term from its corresponding minuend term for a remainder. In case any subtrahend term be greater than the minucnd term, borrow 1 as in simple subtraction, and reduce it to the denomination required. MULTIPLICATION OF DENOMINATE NUMBERS. 390. EXAMPLE.Each of five bars of silver weighed 16 lb. 3 oz. 10 pwt. 21 gr. What was the total weight? EXPLANATION.-Multiply 21 grains by 5 and obtain 105 OPERATION. grains, which reduce to pennyweigbts by dividing by 24, 10. oz. pwt. and obtain 4 pennyweights, with a remainder of 9 grains; gr. 16 3 10 multiply 10 pennyweights by 5, add the 4 pennyweights, 21 and reduce to ounces by dividing by 20, obtaining 2 ounces, 5 14 pennyweights; multiply 3 ounces by 5, add the 2 ounces 81 lb. 5 oz. 14 pwt. 9 gr. and divide by 12, obtaining 1 pound, 5 ounces; multiply 16 pounds by 5, add the 1 pound and obtain 81 pounds. Rule.- Beginning with the lowest denomination, multiply each in succession, and reduce the product to higher denominations by the scale. REMARKS.–1. In order that the pupil may have all problems under each denominate subject given together, and so make an exhaustive study separately of each, it has seemed proper to include all of the reductions under a typical subject, that of TROY WEIGHT, and hereafter, as may be needed, reference will be made to such reductions. 2. The teacher will appreciate the above change, as each subject will thus be made to include enough work for a lesson, and the confusion often arising from giving in the same lesson several tables, with varying scales, may be avoided. DIVISION OF DENOMINATE NUMBERS. 391. EXAMPLE.—If 7 lb. 1 oz. 12 pwt. 18 gr. of silver be made into 6 plates of equal weight, what will be the weight of each? OPERATION. EXPLANATION.-One plate will weigh as much as 16. oz. pwt. 6 plates. Write the dividend and divisor as in short divi. gr. sion. Divide 7 pounds by 6, obtaining a quotient of 1 6 ) 7 n 12 18 pound and an undivided remainder of 1 pound; reduce 1 lb. 3 oz. 5pwt. 11 gr. this remainder to ounces (12) and add to the 7 ounces of the dividend, obtaining 19 ounces, which divide by 6, obtaining 3 ounces and an undivided remainder of 1 ounce; reduce this remainder to pennyweights (20) and add to the 12 penny. weights of the dividend, obtaining 32 pennyweights, which divide by 6, obtaining 5 pennyweights and an undivided remainder of 2 pennyweights; reduce this remainder to grains (48) and add to the 18 grains of the dividend, obtaining 66 grains, which divide by 6, obtaining 11 grains, and thus completing the division. Therefore, the weight of each plate will be 1 pound, 3 ounces, 5 pennyweights, 11 grains. Rule.-Write the terms as in short division; divide as in integers, and reduce remainders, if any, to next lower orders by the scale. REMARKS.-1. Should the highest dividend order not contain the divisor, reduce its units to the order next lower, and so proceed to the end. 2. The above and like divisions may be accomplished by the reduction of the denominate expressions to the lowest order in its scale, then effecting the division and afterwards reducing the quotient to higher denominations. COMPOUND DENOMINATE DIVISION. 392. EXAMPLE.—How many plates, each weighing 1.1b. 3 oz. 5 pwt. 11 gr., can be made from 7 lb. 7 oz. 12 pwt. 18 gr. of silver? EXPLANATION.-Reduce each of the given OPERATION. expressions to its equivalent in grains. Since 1 lb. 3 oz. 5 pwt. 11 gr. = 7331 gr. one plate weighs 7331 grains, and the weight of the silver to be used is 43986 grains, as many 7 lb. 7 oz. 12 pwt. 18 gr. = 43986 gr. plates can be made as the weight of one plate, 43986 gr. • 7331 gr. = 6 7331 grains, is contained times in the 43986 grains to be so used, or 6 plates. Rule.-Reduce the dividend and divisor to the same denomination, and divide as in simple numbers. EXAMPLES FOR PRACTICE. 393. '1. Reduce 31 lb. 10 oz. 13 pwt. to pennyweights. Reduce 'g of a pennyweight to the fraction of a pound. 17. Add 236 lb. 4 oz. 15 pwt., 83 lb. 11 oz. 21 gr., 46 lb. 16 pwt., 105 lb. 9 oz. 11 gr. 18. What is the sum of 16 lb. 16 pwt. 16 gr., 100 lb. 1 oz. 5 pwt. 20 gr., 76 lb. 7 oz. 6 pwt. 13 gr., 19 lb. 2 oz. 10 pwt. 20 gr.? 19. Find the equivalents in lower denominations of .1425 oz. 20. 1 pwt. 15 gr. is what decimal of a pound ? 21. Subtract 41 lb. 11 oz. 6 pwt. 18 gr. from 50 lb. 2 oz. 22. What is the difference between 19 lb. 9 oz. 11 pwt. and 11 oz. 16 pwt. 22 gr.? 23. What will be the cost of 15 gold chains, each weighing 1 lb. 3 oz. 18 pwt. 18 gr., at 74 per grain ? 24. I bought 7 lb. 7 oz. 12 pwt. 18 gr. of old gold, at $1.05 per pwt. What was the sum paid ? 25. A manufacturer made 18 vases from 7 lb. 8 oz. 8 pwt. 18 gr. of silver. What was their average weight ? If 12 rings be made from 1 lb. 8 oz. of gold, what will be the weight of each ? 27. A miner having 77 lb. 10 oz. 5 pwt. of gold dust, divided of it among nis laborers, and had the remainder made into chains averaging 3 oz. 3 pwt. 3 gr. of pure gold each. If he sold the chains for $52.50 each, how much did he receive for them ? 28. What is the aggregate weight of five purchases of old silver, weighing respectively 4 lb. 9 oz. 20 gr., 13 lb. 17 pwt. 22 gr., 20 lb. 1 oz. 17 pwt. 4 gr., 8 lb. 2 oz., and 27 lb. 12 pwt. 21 gr.? 29. I bought 27 lb. 11 oz. 1 gr. of old silver, and after having used 15 lb. 15 pwt. 15 gr., sold the remainder at 5¢ per pwt. What quantity was sold, and how much was received for it? 30. A goldsmith bought 3 lbs. 9 oz. 1 pwt. 16 gr. of old gold, at 80¢ per pwt., and made it into pins of 40 grains weight each, which he sold at $2 apiece. How much did he gain or lose ? AVOIRDUPOIS WEIGHT. 394. Avoirdupois Weight is used for all ordinary purposes of weighing. Table. 16 ounces = 1 pound .. lb. 100 pounds. = 1 hundred-weight.. cwt. 20 hundred-weight., or 2000 pounds = 1 ton... T. Scale, descending, 20, 100, 16; ascending, 16, 100, 20. REMARK.-At the United States Custom Houses, in weighing goods on which duties are paid and to a limited extent in coal and iron mines, the long ton of 2240 pounds is still used. Long Ton Table, 16 ounces = 1 pound.... lb. 28 pounds... = 1 quarter. qr. 4 quarters, or 112 pounds.. = 1 hundred-weight... cwt. 20 hundred-weight, or 2240 pounds = 1 ton T. |