17. 1 cub. in., and 1 cub. ft. each to the decimal of 1 cub. yd. 19. 1 gill; 1 pt.; 1 qt.; and 1 pk., each to the decimal of 1 gal. 21. 1 dr. and 1 oz., each to the decimal of 1 lb. av. 22. 1 and 1, each to the decimal of 1 day. 3. 3 fur. 25 po. 3 yd. 2 ft. 7 in. to the fraction of 1 mile. 2 ft. 7 in. 9 li. to the decimal of 1 yd. 2 rd. 35 po. 15 yd. 7 ft. to the fraction of 1 acre. 4. 5. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 20 ft. 354 in. (cub. meas.) to the fraction of 1 yd. 5 bush. 3 gal. 1 qt. to the fraction of 1 quarter. 25 gal. 3 qt. 1 pt. to the fraction of 1 hhd. (wine meas.) 8 cwt. 2 qr. 20 lb. 12 oz. to the fraction of 1 ton. 5 oz. 14 dr. to the decimal of 1 lb. av. 7 oz. 5 dr. 2 sc. 16 gr. to the fraction of 1 lb. (apoth. wt.) " 6 oz. 14 dwt. 20 gr. to the fraction of 1 lb. (troy wt.) 4. 1hhd. (wine meas.) to cubic feet. 1 lb. troy to the decimal of 1 lb. av. 1 lb. av. to pounds troy. 1gr.; 1 dwt.; 1 oz. (troy); each to the decimal of 1 lb. av. 5. 6. 7. 8. 9. 58 gal. 3 qt. 1 pt. to cubic feet. 10. 5 cub. yd. 14 ft. 1532 in. to gallons. 15 lb. 8 oz. 14 dwt. 22 gr. to lb. av. 11. 24 lb. 5 dr. 2 sc. 16 gr. (apoth. wt.) to lb. av. 12. 1 qr. 16 lb. 12 oz. 8 dr. to pounds troy. 147. A compound number may be considered either, 1st, as a sum of numbers altogether or partly fractional, or, 2nd, as a number belonging to an irregular system of numeration, in which the different orders of units are expressed and individually combined by means of the decimal system; but the number of ones or parts of one of any order, which make one of the next order is not 10 or, but a number fixed by the table of some arbitrary weight or measure. In the first case, the sums, differences, products, and quotients of compound numbers may be found by the rules for the addition, subtraction, multiplication, and division of fractional expressions. In the second, the difference between these operations with decimal and with compound numbers consists in denoting every order of units, not necessarily by one, but sometimes by two or more figures; and in reducing from any order of units or denomination to the preceding or following by dividing or multiply. ing, not by 10, but by a number fixed by the table of an arbitrary weight or measure. Some of the inconveniences of the existing system of compound numbers have been pointed out in Art. 142. To familiarise the reader with a better system, it is recommended that, in the examples which follow, every compound number be reduced to a decimal fraction, either by means of the tables in Art. 143. a, or by the rules of reduction; and that the calculation required be then made, independently, by both methods, that is, by compound rules and by decimals. As a test of accuracy, either result may be reduced to an equivalent number expressed in the form of the other result. ADDITION OF COMPOUND NUMBERS. 148. In Addition of Compound, as in addition of whole numbers, the thing required is to find a single number which shall be equal to certain proposed numbers taken all together. The sole difference between the operations consists in the different way of carrying from the lower denominations to the higher. (Art. 147.) As an example, let it be required to find the sum of £35 17s. 73d., £86 15s. 94d., and £74 18s. 101⁄2d. The units of the same order, or the same denominations, of the compound number, being ranged under each other as in whole numbers (Art. 16. and 147.) thus the farthings are added together, and their sum found to be 6=1}d. is written under the column of farthings, and 1d. is carried to the column of pence the sum of the column of pence is 26, and 26+ 1 carried=27d.= 2s. 3d.; 3 is written under the column of pence, and 2s. are carried to the column of shillings: the sum of the column of shillings is 50, and 50+2 carried = 52s. = £2 12s.; 12 is written under the column of shillings, and £2 carried to the pounds, which are added as whole numbers. The sum of the proposed compound numbers is thus found to be £197 12s. 3 d. The rule for reducing numbers from a lower to a higher denomination is required in addition of compound numbers. That for reducing a fractional part of any denomination to a compound number expressed in terms of the lower denominations of the weight or measure to which the proposed fraction belongs, may sometimes be necessary; and also the rule for reducing fractions to a common denominator. As an example requiring these reductions, let it be proposed to find the sum of £; s.; and 4d. £, s., and 4d. may be reduced to the same denomination, and added together as fractional parts of pounds, or of shillings, or of pence, or of farthings. But if it is required to express the result in the form of a compound number, then Following with the two preceding examples, the course recommended in Art. 147., it is found that and £90738095188. 137d., it is concluded that these results are correct. With money expressed in pounds, florins, cents, and mils, the reductions are unnecessary; the number of decimal figures is fewer (only three); and the whole calculation is shortened to that of finding the sum of, for example, 35-882, 86.788, and 74.943. a. The rule for the addition of whole numbers may, with some slight changes, be also made the rule for the addition of compound numbers, thus Having found the values of fractional parts of any other denomination than the lowest, in terms of the lower denominations, and reduced fractional parts of this denomination with different denominators to equivalent fractions having a common denominator; and having written the numbers under each other in horizontal lines so arranged that the same denomination in all the compound numbers falls in the same vertical column; add together the numbers in the right vertical column; reduce the sum to a compound number expressed in terms of the last and the preceding denomination; write the units of the last denomination under the last column and carry the number expressing the units of the next higher denomination to the second vertical column. Add together the number carried and the numbers in the second column. Reduce the sum to a compound number expressed in terms of this and the next higher denomination. Write the number of units of the second denomination under the second vertical column; carry the units of the third denomination to the third vertical column: and in this manner proceed till the sum of all the columns has been found. 149. Examples. Add together, 1. £82 7s. 5d.; £59 13s. 8d.; and £196 8s. 6d. 2. £54 17s. 6d., £39 8s. 4d., £27 18s. 9d., and £9 5s. 11d. 3. 4. 5. 6. 15 mi. 7 fur. 25 po. 4 yd., 32 mi. 5 fur. 38 po. 3 yd., and 146 mi. 3 fur. 18 po. lyd. 42 ac. 3 ro. 17 po. 25 yd., 18 ac. I ro. 35 po. 16 yd., and 27 ac. 2 ro. 29 po. 8 yd. 5 hhd. 28 gal. 3 qt., 9 hhd. 57 gal. 1 qt., and 17 hhd. 38 gal. 2 qt. 1 pt. 8 t. 17 cwt. 1 qr. 18 lb., 4 t. 9 cwt. 3 qr. 24 lb., and 12 t. 8 cwt. 2 qr. 27 lb. 7. 23 lb. 7 oz. 18 dwt. 21 gr., 52 lb. 9 oz. 12 dwt. 18 gr., and 35 lb. 12 dwt. 8. 9. 10. 23 gr. 39 d. 14 h. 25 m. 39 s., 28 d. 23 h. 48 m. 50 s., and 19 d. 17 h. 37 m. 54s. £372 8s. 4 d., £568 17s. 111⁄2d., £796 9s. 8d., and £937 13s. 93d. 15 cwt. 3 qr. 17 lb. 8 oz., 9 cwt. 27 lb. 13 oz. 15 dr., 6 cwt. 1 qr. 19 lb. 12 dr., and 2 qr. 24 lb. 14 oz. 11. 25 mi. 3 fur. 25 po. 4 yd. 1 ft. 9 in., 37 mi. 7 fur. 2 yd., 51 mi. 34 po. 2 ft. 7 in., and 9 mi. 5 fur. 28 po. 4 yd. 21 ft. 12. 8 qr. 6 bu. 3 pk. 1 gal. 2 qt., 15 qr. 7 bu. 2 pk. 3 qt., 9 qr. 4 bu. 1 pk. 1 qt., and 6 bu. 3 pk. 2 qt. 13. 53 ac. 1 ro. 35 po. 26 yd. 8 ft., 79 ac. 29 po. 18 yd. 5 ft., and 154 ac, 3 ro. 17 po. 25 yd. 7 ft. 14. £20 78. 41d., 53 guineas, and 85 half-crowns. 15. 25 cwt. 2 qr. 17 lb., 6 cwt. 86 lb., and 15 cwt. 3 qr. 12-7 lb. 16. 32 ac., 5067 sq. yd., and 50000 square links. 17. 57.5£, 29°34£, 87·176£, and 153-942£. 18. 137-05 yd., 496.346 yd, 874-849 yd., and 682.574 yd. 19. 314-74 lb., 592-819 lb., and 476.59 lb. (av. wt.) 20. 53.25 lb., 29-36 lb., 19.542 lb., and 8-425 lb. (troy: the sum to be given in lb. av.) 21. £, s., and 3d. 22. mi., fur., and po. 23. ton., cwt., qr., and lb. 24. ac., ro., † po., and yd. SUBTRACTION OF COMPOUND NUMBERS. 150. The difference of two compound numbers is found in the same manner as the difference of two whole numbers; excepting that, when the number expressing any denomination of the subtrahend is greater than the number expressing the same denomination of the minuend, the latter is not augmented by 10 but by as many of this denomination as make 1 of the next higher denomination. As an illustration let it be required from 181b. 7 oz. 18 dwt. 15 gr. to subtract 9 lb. 8 oz. 14dwt. 20gr. troy weight. Writing the compound numbers so that, in each, the same denominations may fall under each other, thus lb. oz. dwt. gr. 18 7 18 15 9 8 14 20 and subtracting the numbers in the lower line from the corresponding numbers in the upper line, (15-20) grs. To render the subtraction possible, 1 dwt. (=24 gr.) is borrowed. 15+24=39 gr., and 39-2019 gr.=the gr. of the remainder. The remainder, 19, is written under the column of grains, and 1 dwt. for that borrowed is carried to the dwt. of the subtrahend. == 14+1 carried 15 dwt., and 18-15-3 dwt. the dwt. of the remainder. The remainder, 3, is written under the column of dwt. (7-8) oz, To render the subtraction possible 1 lb. (= 12 oz.) is borrowed. 7+12=19 oz. and 19-8-11 oz. the oz, of the remainder. 11 is written under the column of oz., and 1 lb. for that borrowed is carried to the lb. of the subtrahend. 9 + 1 carried = 10 lb., and 18-10-8 lb. the lb. of the remainder. = 8 is written under the column of lb. and the difference of the proposed numbers is found to be 8 lb. 11 oz. 3 dwt. 19 gr. |