295. To reduce a compound number to a common fraction, of a higher denomination. First reduce the given compound number to the lowest denomination mentioned for the numerator ; then reduce a unit of the denomination of the required fraction to the same denomination as the numerator, and the result will be the denominator. (Art. 281.) Obs. 1. The given number, and that of which it is said to be a part, must, in all cases, be reduced to the same denomination. (Art. 294.) 2. When the given number contains but one denomination, it of course requires no reduction. If the given number contains a fraction, the denominator of the fraction is the lowest denomination mentioned. Thus, in 6 s., the lowest denomination is fourths of a shilling; in a far., the lowest denomination is fifths of a farthing. 2. Reduce of a penny to the fraction of a pound. Solution. Since sevenths of a penny is the lowest and only denomination given, we simply reduce £1 to sevenths of a penny for the denominator. Now £1=240d., and 240d.x7=1680. Ans. 1o, or £t. Hence, 296. To reduce a fraction of a lower denomination to an equivalent fraction of a higher denomination. Reduce a unit of the denomination of the required fraction to the same denomination as the given fraction, and the result will be the denominator. Or, divide the given fraction by the same numbers as in reducing whole compound numbers to higher denominations. (Art. 281. II.) Thus in the last example, 4d.-12=., (Art. 227,) and 8s.; 20=£150,= £tu. Ans. Oes. When factors common to the numerator and denominator occur, the operation may be shortened by canceling those factors. (Art. 221.) 3. Reduce of a penny to the fraction of a pound. 4 the answer. 4 1 By Cancelation Ans. 7X12 x 205 x 12 x 20,5 £ UEST –295. How is a compound number reduced to a common fraction ? 296. How is a fraction of a lower denomination reduced to the fraction of a higher ? 4. Reduce 4žs. to the fraction of a pound. Ans. £ft, or £zt. 5. Reduce 4s. 7d. to the fraction of a pound. 6. Reduce 9d. 21 far. to the fraction of a pound. 7. What part of £1 is of 1 penny ? 8. What part of 1 lb. Troy is 7 ounces? 9. What part of 1 lb. Troy is 16 pwts. 3 grs ? 10. What part of 1 lb. avoirdupois is 8 oz. and 12 drams? 11. What part of 1 ton is 14 cwt. and 15 lbs ? 12. What part of 1 yd. is 2 ft. and 4 inches ? 13. What part of 1 mile is 82} rods ? 14. What part of 1 acre is 45. rods ? 15. What part of 1 square rod is 63 square feet? 16. Reduce of 1 qt. to the fraction of a gallon. 17. Reduce 7 gallons to the fraction of a hogshead. 18. Reduce { of 1 hour to the fraction of a day. 19. Reduce of 1 minute to the fraction of an hour. 20: Reduce of 1 second to the fraction of a week. • 21. What part of £3, 5s. 6d. Ifar. is £2, 1s. 3d. ? Solution.-Reducing both numbers to farthings, £3, 5s. 6d. 1far. =3145 far., and £2, 1s. 3d.=1980 far. (Art. 295. Obs.1.) Now 1980 is its of 3145, which is equal to 329. Ans. 22. What part of £2 is 7s. 6d. ? FRACTIONAL COMPOUND NUMBERS REDUCED TO WHOLE NUMBERS OF LOWER DENOMINATIONS. Ex. 1. Reduce of £1 to shillings and pence. Analysis.—5 of 1s.=ş of 5s. or is., consequently of 20s. (£1) is 20 times as much, and is. X20=100s. or 12s. and of a shilling. Reasoning as before, $ of id.=} of 4d., or d., and of 12d. (1s.) is 12 times as much; but d.X12=4&d., or 6d. Therefore £4=12s, 6d. Ans. Hence, 297. To reduce fractional compound numbers to whole numbers of lower denominations. First reduce the given numerator to the next lower denomination ; then divide the product by the denominator, and the quotient will be an integer of the next lower denomination. (Art. 281. I.) Proceed in like manner with the remainder, and the several quotients will be the whole numbers required, OBs. This operation is the same in principle as reducing higher denominations of whole numbers to lower. (Art. 281. I.) Whenever the fraction becomes improper, it is reduced to a whole or mixed number. (Art. 196.) 2. Reduce of £l to shillings. Ans. 16s. 3. Reduce of £1 to shillings and pence. 4. Reduce of 1s. to pence and farthings. 5. Reduce of 1 lb. Troy to ounces, &c. 6. Reduce of 1 ounce Troy to pennyweights. 7. Reduce of 1 lb. avoirdupois to ounces, &c. 8. Reduce 4 of 1 cwt. to pounds, &c. 9. Reduce of 1 ton to pounds, &c. 10. Reduce $ of 1 yard to feet and inches. 11. Reduce of 1 rod to feet and inches. 12. Reduce of 1 mile to rods, feet, &c. 13. Reduce f of 1 gallon wine measure to quarts, &c. 14. Reduce } of i hogshead wine measure to gallons, &c. 15. Reduce of i peck to quarts, &c. Ans. 6 qts 1} pis. 16. Reduce ţof i bushel to quarts, &c. 17. Reduce of 1 hour to minutes and seconds. QUEST.–297. How are fractional compound numbers reduced to whole ones ? g* 18. Reduce % of 1 day to hours, &c. Solution. We reduce the numerator to pence, the denomination required, and divide it by the denominator, as in the last article. Thus, 2 X 20 X 12=480; and 480= 720=30. Therefore £=498d.=*=* or fd. Ans. Hence, 298. To reduce a fraction of a higher denomination to an equivalent fraction of a lower denomination. Reduce the given numerator to the denomination of the required fraction, and place the result over the given denominator. OBs. 1. This process is the same in principle as to reduce a whole compound number to a lower denomination. (Art. 281. I.) 2. When factors common to the numerator and denominator occur, the operation may be shortened by canceling those factors. (Art. 221.) 2 x 20 x12 Thus, in the last example, —the answer. 720 2 X 20 X 12 2X20 x 12 By Cancelation, =fd. Ans. 720 720,3 22. Reduce 17 of £1 to the fraction of a penny. 23. Reduce 277 of 1 lb. avoirdupois to the fraction of an ounce. 24. Reduce 7764 of 1 mile to the fraction of a rod. 25. Reduce 25 of a day to the fraction of an hour. 26. Reduce te of 1 week to the fraction of a minute. 27. Reduce of 1 yard to the fraction of a nail. 28. Reduce of 1 bushel to the fraction of a quart. 29. Reduce they of 1 hhd. wine measure to the fraction of a quart. 30. Reduce 75 of lb. Troy to the fraction of an ounce. 31. Reduce 776 of 1 pound Troy to the fraction of a pwt. 32. Reduce of an acre to the fraction of a rod. 33. Reduce to of a square yard to the fraction of a foot. 34. Reduce o of a degree to the fraction of a second. QUEST.-208. How is a fraction of a higher denomination reduced to the fraction of a lower denomination ? S. ADDITION OF COMPOUND NUMBERS. 299. The process of adding numbers of different denominations, is called COMPOUND ADDITION. 1. What is the sum of £6, 11s. 5d. 1 far. ; £4, 9s. 6d. 2 far.; £3, 12s. 8d. 3 far.; and £8, 6s. 9d. 1 far. ? Operation. Having placed the farthings under far£ d. far. things, the pence under pence, &c., we 6" 11 " 5 " 1 add the column of farthings together, as 411 9 6 2 in simple addition, and find the sum is 7, 311 12 8 3 which is equal to 1d. and 3 far. over. 8 " 6 9 1 Set the 3 far. under the column of far 0 5 " 3 Ans. things, and carry the ld. to the column of pence. The sum of the pence is 29, which is equal to 2s. and 5d. over. Place the 5d. under the column of pence, and carry the 2s. to the column of shillings. The sum of the shillings is 40, which is equal to £2, and nothing over. Write a cipher under the column of shillings, and carry the £2 to the column of pounds. The sum of the pounds is 23. Ans. £23, Os. 5d. 3 far. 300. Hence, we derive the following general 23 11 RULE FOR ADDING COMPOUND NUMBERS. I. Write the numbers so that the same denominations shall stand ander each other. II. Beginning with the lowest denomination, find the sum of each column separately, and divide it by that number which it requires of the column added, to make ONE of the next higher denomination. Set the remainder under the column added, and carry the quotient to the next column. III. Proceed in this manner with all the other denominations except the highest, whose entire sum is set down. PROOF.— The proof is the same as in Simple Addition. (Art. 55.) Obs. 1. Fractional compound numbers should be reduced to whole numbers of lower denominations, then added as above. (Art. 166.) QUEST.--299. What is Compound Addition ? 300. How do you write compound numbers for addition ? Which denomination do you add first? When the sum of any column is found, what is to be done with it? What is done with the last colurnn? |