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 3far., 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. Hence, Ans. £T, or £. 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, d.÷12=s., (Art. 227,), and s.÷ 20=££. 16 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. Solution. By the last article, 4 = the answer. 7X12×20 7X12X207×12×20,5 QUEST-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 4s. to the fraction of a pound. Ans. £t, or £7. 5. Reduce 4s. 7d. to the fraction of a pound. 6. Reduce 9d. 24 far. to the fraction of a pound. 7. What part of £1 is 7 of 1 penny ? 8. What part of 1 9. What part of 1 10. What part of 1 11. What part of 1 lb. Troy is 7 ounces? lb. Troy is 16 pwts. 3 grs? lb. avoirdupois is 8 oz. and 12 drams? 12. What part of 1 yd. is 2 ft. and 4 inches? 14. What part of 1 acre is 45 rods? 15. What part of 1 square rod is 63 square feet? 21. What part of £3, 5s. 6d. 1far. 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 of 3145, which is equal to 329. 22. What part of £2 is 7s. 6d. ? 23. What part of £7, 3s. is £3? 24. What part of 2 bushels is 3 pecks? 25. What part of 10 bushels is 10 quarts? 26. What part of 16 rods is 40 feet? Ans. 27. What part of 3 weeks is 2 days and 7 hours? 28. What part of 2 hhds. 10 gals. is 45 gals.? 29. What part of 2 tons, 3 cwt. is 15 cwt. 65 lbs. ? 30. What part of 1 ton is 7 lbs. 10 ounces? 31. What part of 90° is 1° 15' 30"? 32. What part of 360° is 45° 15′ 10′′? 33. What part of 3 lbs. Troy is 1 lb. 3 oz.? 34. What part of 25 lbs. Troy is 10 lbs. 7 oz. 10 pwts.? 35. What part of 1 acre is 40 rods? 36. What part of 5 acres is 14 acres? FRACTIONAL COMPOUND NUMBERS REDUCED TO WHOLE NUMBERS OF LOWER DENOMINATIONS. Ex. 1. Reduce of £1 to shillings and pence. Analysis.― of 1s. of 5s. or s., consequently of 20s. (£1) s. X 20=100s. or 12s. and 4 of a shilling. of 1d. of 4d., or 4d., and is 20 times as much, and Reasoning as before, of 12d. (1s.) is 12 times as much; but d.X12=4&d., or 6d. Therefore £ 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 £1 to shillings. Ans. 16s. 3. Reduce of £1 to shillings and pence. of 1s. to pence and farthings. 5. Reduce of 1 lb. Troy to ounces, &c. 4. Reduce 6. Reduce of 1 ounce Troy to pennyweights. 7. Reduce of 1 lb. avoirdupois to ounces, &c. 8. Reduce of I 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 of 1 gallon wine measure to quarts, &c. 14. Reduce of 1 hogshead wine measure to gallons, &c. Ans. 6 qts 1 pis. 15. Reduce of 1 peck to quarts, &c. 16. Reduce of 1 bushel to quarts, &c. 17. Reduce of 1 hour to minutes and seconds. QUEST-297. How are fractional compound numbers reduced to whole ones? 18. Reduce 19. Reduce 20. Reduce 21. Reduce £ of 1 day to hours, &c. of 1 degree to minutes, &c. Solution. We reduce the numerator to pence, the denomination required, and divide it by the denominator, as in the last article. Thus, 2X20X12=480; and 480÷720-40. Therefore £48d.== or d. 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.) 22. Reduce of £1 to the fraction of a penny. 23. Reduce 17 of 1 lb. avoirdupois to the fraction of an ounce. 24. Reduce 25. Reduce 26. Reduce of 1 mile to the fraction of a rod. of a day to the fraction of an hour. 27. Reduce 45 of 1 yard to the fraction of a nail. of 1 bushel to the fraction of a quart. 29. Reduce of 1 hhd. wine measure to the fraction of a quart. 30. Reduce of 1 lb. Troy to the fraction of an ounce. 31. Reduce 25 of 1 pound Troy to the fraction of a pwt. of an acre to the fraction of a rod. 32. Reduce 33. Reduce T 34. Reduce of a square yard to the fraction of a foot. QUEST. 208. How is a fraction of a higher denomination reduced to the fraction of a lower denomination? ADDITION OF COMPOUND NUMBERS. 299. The process of adding numbers of different denomi nations, 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. £ S. d. far. 6" 11" 5" 1 9 4 " 3 " 12 8 "1 6 6 2 8 9 1 5 "3 Ans. Having placed the farthings under farthings, the pence under pence, &c., we add the column of farthings together, as in simple addition, and find the sum is 7, which is equal to 1d. and 3 far. over. Set the 3 far. under the column of farthings, and carry the 1d. to the column 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, 0s. 5d. 3 far. 23 0 of pence. 300. Hence, we derive the following general RULE FOR ADDING COMPOUND NUMBERS. I. Write the numbers so that the same denominations shall stand under 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 column? |