64. A ship sailed from Boston to Liverpool; on the fourth day the master took an observation of the sun at noon, and found by his chronometer that it was 1 hr. 5 min. and 40 sec. earlier than the Boston time: how many degrees east of Boston was the ship? Solution.-1 hr. 5 m. 40 sec.=3940 sec., (Art. 281,) and 3940 sec. 4985'. The ship had therefore sailed 985' east, which is equal to 16° 25'. Ans. 65. The difference of time between Albany and Buffalo is 19 minutes what is the difference of their longitude? 66. The difference of time between Richmond and New Orleans is 51 min. 4 sec.: what is the difference of their longitude? 67. The difference of time between Boston and Cincinnati is 53 min. 32 sec.: what is the difference of their longitude? COMPOUND NUMBERS REDUCED TO FRACTIONS. 294. That one concrete number may properly be said to be a part of another, the two numbers must necessarily express objects of the same kind, or objects which can be reduced to the same kind or denomination. Thus, 1 penny is of a pound, but 1 penny cannot properly be said to be a part of a foot, or of a year ; for, feet and years cannot be reduced to pence. So, 1 orange is of 5 oranges; but 1 orange cannot be said to be 5 pumpkins; for apples and pumpkins cannot oranges. Ex. 1. Reduce 2s. 7d. to the fraction of a pound. of 5 apples, or be reduced to Analysis.—The object in this example is to find what part of 1 pound, 2s. 7d. is equal to. To ascertain this, we must reduce both the given numbers to the same denomination, viz: pence. Now 2s. 7d.=31d., and £1=240d. (Art. 281. I.) The question, therefore, resolves itself into this: what part of 240 is 31? The answer is; consequently 2s. 7d. (31d.) is of a pound. Hence, QUEST.-294. When can one concrete number be said to be a part of another? 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. £10, or £2. 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 £160 £21. Ans. = 809 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. 7X12X20 1 -£ 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 ! 178 REDUCTION. [SECT 4. Reduce 4s. to the fraction of a pound. Ars. £14, 5. Reduce 4s. 7d. to the fraction of a pound. 6. Reduce 9d. 2 far. to the fraction of a pound. 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 drains 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 824 rods? 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. 1 =3145 far., and £2, 1s. 3d.=1980 far. (Art. 295. Obs..) N 1980 is of 3145, which is equal to 3. Ans. 22. What part of £2 is 7s. 6d. ? 23. What part of £7, 3s. is £3? 629. 24. What part of 2 bushels is 3 pecks? 26. What part of 16 rods is 40 feet? 27. What part of 3 weeks is 2 days and 7 hours ? 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 1 acres? FRACTIONAL COMPOUND NUMBERS REDUCED TO WHOLE NUMBERS OF LOWER DENOMINATIONS. Ex. 1. Reduce of £1 to shillings and pence. of 20s. (£1) Analysis.— of 1s.= of 5s. or 1⁄2s., consequently is 20 times as much, and §s. ×20=10os. or 12s. and of a shilling. Reasoning as before, of 1d.= of 4d., or 4d., and 4 of 12d. (1s.) is 12 times as much; but d.×12=48d., 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 1 lb. Troy to ounces, &c. 8. Reduce 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 of 1 gallon wine measure to quarts, &c. 14. Reduce of 1 hogshead wine measure to gallons, &c. 15. Reduce of 1 peck to quarts, &c. Ans. 6 qts. 1 pts. 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. Rduce 19. Reduce 20. Reduce 21. Reduce £ of 1 day to hours, &c. of 1 minute to seconds. of 1 degree to minutes, &c. to the fraction of a penny. Solution. We reduce the numerator to pence, the denomination required, and divide it by the denominator, as in the last article. Thus, 2×20×12=480; and 480-720-480. Therefore £40d.=4%=4 or 3d. 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.) the op 2. When factors common to the numerator and denominator occur, eration 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. of 1 week to the fraction of 1 minute. 27. Reduce 45 of 1 yard to the fraction of a nail. 28. Reduce 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. 32. Reduce 33. Reduce 34. Reduce of an acre to the fraction of a rod. of a square yard to the fraction of a foot. QUEST.-298. How is a fraction of a higher denomination reduced to the fraction of a lower denomination? |