PROBLEM 10. 148. To reduce a fraction of a higher denomination to a fraction of a lower denomination. Ex. 1. Reduce of a penny, to the fraction of a farthing. As 1 penny is equal to 4 farthings, so any fraction of a penny will be 4 times as great a fraction of a farthing; .. d. = qr., timesqr. Ans. of a shilling to the fraction of a farthing. As 1 s. is equal to 12 d., so 4 s. and d. = 4 times qr. =4 12 times 4 d. = 4 d., RULE. Multiply the fraction by such numbers as are necessary to reduce the given to the required denomination. 3. Reduces. to the fraction of a farthing. 12)=} d. (=} qr. × 4)= 28 qr., Ans.; or NOTE 1. The sign of multiplication, in these examples, is written only between the numbers which are given before the canceling is begun; thus, in Ex. 3, no sign is written between 36 and 3, for they are not to be multiplied together, but the 3 is obtained by canceling 12 in 36. So in Ex. 4, the 12 comes from canceling 20 in 240, and the 3 from canceling 4 in 12. 4. Reduce of a ton to the fraction of a dram. 7 × 20 × 4 × 25 x 16 x 16 12 3 44800 dr., Ans. 3 of a rod to the fraction of a barleycorn. 240 5. Reduce NOTE 2. In the first statement of Ex. 5, the 161, in the numerator, is equal to 33, and, in the second statement, the 33 is retained in the numerator as a factor in the dividend, and the 2 is put in the denominator as a factor in the divisor. 148. Rule for reducing a fraction from a higher to a lower denomination? Explanation? How is Ex. 5 solved? grain. 6. Reduce of a pound, Troy Weight, to the fraction of a Ans. 108. 7. Reduce of a pound, Apothecaries' Weight, to the fraction of a grain. Ans. 108. 8. Reduce 4200 of a day to the fraction of a second. Ans. 144. 9. Reduce of a bushel to the fraction of a pint. Ans. 128. 10. Reduce of a gallon to the fraction of a gill. 11. Reduce 40 c. yd. to the fraction of a cubic inch. 12. Reduce 14. Reduce 15. Reduce 383 16. Reduce 17. Reduce 18. Reduce o fur. to the fraction of a link. Ans. 14. circ. to the fraction of a second. of a ton to the fraction of an ounce. 19. Reduce 3024 of a day to the fraction of a second. 21. Reduce of a bushel to the fraction of a pint. PROBLEM 11. 149. To reduce a fraction of a lower denomination to a fraction of a higher denomination. Ex. 1. Reduce of a barleycorn to the fraction of an inch. In 15 barleycorns there is only of 15 inches, so in of a barleycorn there is only of of an inch of an inch, Ans. of a gill to the fraction of a quart. 2. Reduce As 1 gill is a like reason. of a pint, so 3 gi. is of 3 pt. pt. and, for 42 RULE. Divide the given fraction by such numbers as are required to reduce the given to the required denomination. 149. Rule for reducing a fraction from a lower to a higher denomination? Explan 3. Reduce 28 qr. to the fraction of a shilling. 48 qr. (= 28 d. ÷ 4) = }d. (=}s.÷12)= 3 X 4 X 12 7 36 s., Ans.; or, s. Ans., as before. 4. Reduce 44800 dr. to the fraction of a ton. 7 3 × 16 × 16 × 25 × 4 × 20 240 5. Reduce 1989 b. c. to the fraction of a rod. tons, Ans. 6. Reduce 18 gr. to the fraction of a pound, Apothecaries' Weight. Ans. 1200 7. Reduce 18 gr. to the fraction of a pound, Troy Weight. 8. Reduce 25200 sec. to the fraction of a day. 9. Reduce in. to the fraction of a yard, Cloth Measure. 10. Reduce 168 sec. to the fraction of a week. 11. Reduce 432 sq. in. to the fraction of a yard. 12. Reduce 450 links to the fraction of a furlong. 13. Reduce 363 yd. to the fraction of an acre. 14. Reduce 720 seconds to the fraction of a sign. 15. Reduce gills to the fraction of a gallon. PROBLEM 12. Ans. 150. To reduce a fraction of a higher denomination to whole numbers of lower denominations. Ex. 1. Reduce £ to shillings and pence. £ (s. X 20)= Ans. 3s. 4d. s.=33s.; again s. (=d. × 12) = 4d.;.. £3 s. 4 d., Ans. Hence, RULE. Reduce the given fraction to a fraction of the next lower denomination (Art. 148); then, if the fraction is improper, reduce it to a whole or mixed number (Art. 140). If the result is 150. Rule for reducing a fraction of a higher denomination to integers of lower denominations? Explanation? a mixed number, reduce the fractional part of it to the next lower denomination, as before, and so proceed as far as desirable. NOTE. If, at any time, the reduced fraction is proper, there will be no whole number of that denomination. 2. Reduce £ to whole numbers of lower denominations. JJ £ (=12 s. X 20) = s. 41 s.; 1 s. (=red.X12)= 3 ad. d., a proper fraction; d. (4 qr. X 4)=3 qr. ; .. 13 £= 4 s. 0 d. 3 qr., Ans. 3. Reduce of an acre to lower denominations. Ans. 1 r. 17 rd. 18 yd. 1 ft. 50g in. 4. Reduce of a furlong to rods, yards, etc. 5. Reduce of a week to days, etc. 6. Reduce Ans. 18 rd. 3 yd. 2 ft. of a rod, Long Measure, to yards, etc. 7. Reduce 1887 of a circumference to signs, etc. 216 8. Reduce of a ton to hundred weights, etc. 9. Reduce 17. to ounces, drams, scruples, etc. 10. Reduce 230 circ. to signs, degrees, etc. 10368 11. Reduce of a civil year (365 days) to days, etc. 12. What is the value of 76% of a pound Troy? 13. What is the value of 12 of a bushel? 14. What is the value of 13 of a gallon? 24 15. What is the value of of a pound, Apothecaries' Weight? 16. Reduce of a mile to furlongs, chains, etc. 17. Reduce of a cord to cord feet, cubic feet, etc. 18. Reduce of a yard to quarters, nails, etc. PROBLEM 13. 151. To reduce whole numbers of lower denominations to the fraction of a higher denomination. Ex. 1. One farthing is what part of a penny? Ans. Since 4 farthings make a penny, 1 farthing is of a penny. 2. Six pence and 1 farthing are what part of a shilling? 6d.+1qr.25qr.; and 1s. 48qr.; .. 6d. and 1qr.s., Ans. To determine what part one thing is of another, considered as a unit or whole thing, the part is always made the numerator of a fraction, and the unit or whole thing is put for the denominator; thus, the fraction expresses the part that 3 miles is of 5 miles. Before the comparison can be made, the part and the whole must be of the same kind or denomination; thus, 3 pecks is not of 5 bushels, but, reducing the 5 bushels to 20 pecks, we have 3 pecks equal to 23 of 20 pecks, i. e. 3 of 5 bushels. Hence, RULE 1. Reduce the given quantity to the lowest denomination it contains, for a numerator; and reduce a unit of the higher denomination to the same denomination as the numerator, for a denominator. 3. Reduce 6 rd. 5 ft. 9 in. to the fraction of a furlong. 6 rd. 5 ft. 9 in.1257 in. and 1 fur.=7920 in. .. 6 rd. 5 ft. 9 in.=337 fur. fur., Ans. = 2640 4. Reduce 7.oz. 4 dwt. to the fraction of a pound. Ans. 3. 5. Reduce 9 rods, 1 foot, and 6 inches to the fraction of a furlong. 9 rd. 1 ft. 6 in. 1800 in. and 1 fur.=7920 in.; .. 9 rd. 1 ft. 6 in.: (a) In Ex. 5, 6 in. ft.; 1 ft. 4828 fur.= 80 7920 fur., Ans. yd. rd. and 9 rd. = YP rd.=51⁄2 fur., Ans., as by Rule 1. Hence, RULE 2. Divide the number of the lowest denomination given by the number required to reduce it to the next higher denomination, and annex the fractional quotient so obtained to the given number of that higher denomination; divide the mixed number so formed by the number required to reduce it to the NEXT higher denomination, annex the quotient to the given number of that denomination, and so proceed as far as necessary. NOTE 1. This rule is frequently preferable to the 1st, because it enables us to use smaller numbers and gives the result in lower terms. 151. Rule for reducing the lower denominations of a compound number to a fraction of a higher denomination? Explanation? Principle? Second rule for reducing integers of lower denominations to the fraction of a higher denomination? Explanation? Why referable to Rule 1? |