EXAMPLES. 1. Reduce, and to fractions having the least common de nominator possible. 4)3 4 8 3 1 2 4X3X2=24= least common denominator. 24-3×1-8 the first numerator; 24-4X3-18 the second numerator; 24-8x7=21 the third numerator. 13 Whence, the required fractions are, 4, 2 2. Reduce,, 3, and 4 to fractions having the least common denominator. 30 Ans. 38, 48, 4, and 48 CASE VII. 607 609 48. To reduce a fraction of one denomination to an equivalent fraction of a higher denomination. RULE.* Multiply the given denominator by the parts in the several denominations between it and that denomination to which it is to be reduced, for a new denominator, which is to be placed under the given numerator: Or, compare the given fraction with the several denominations between it and that denomination to which it is to be reduced, and then, by case 5th, reduce the compound fraction thus formed, to a single one, and the equivalent fraction of the required Idenomination will be obtained. Let this fraction be reduced to its lowest terms. As there are * The reason of the rule may be seen in the following manner. 12 pence in a fhilling, four-fifths of one penny can be only a twelfth part as much of 12 pence or a shilling, as it is of one penny. Hence, to reduce four fifths of a penny to the fraction of a fhilling, the given fraction must be diminished 12 times, or one twelfth of it will be the equivalent fraction of a fhilling. A fraction is diminished in value, according to the note to Cafe I. by multiplying the denominator by the whole number. Thus four fifths of a pen4 1 4 1 4 of of a fhilling. For the fame rea5 5 12 60 ny 4 fon, four fixtieths of a shilling can be only one twentieth as much of a pound, 4 of a fhilling, 4 1 4 1 1 of a 4 or 60 pound. Put these two operations together, and you have four-fifths of a penny, 20 300 The fame operation might have been performed thus. In a pound there are 240 pence. Then, four-fifths of a penny as before. And in general the fraction of one denomination must be as much diminished to be an equivalent fraction of a higher denomination, as is indicated by the number of parts of the given denomjuition to make one of the gher denomination. EXAMPLES. 1. Reduce of a cent to the fraction of a dollar. By comparing it, it becomes of of, which, reduced by case 5, will be 4X1 X1 = 4 D. Ans. and 7x10x10= 700 E. Ans. 5000 1560 2. Reduce of a mill to the fraction of an eagle. 5. Reduce 6. Reduce 7. Reduce Ans. of a shilling to the fraction of a moidore. guinea.. Ans.moidore. 8. Reduce of an ounce to the fraction of a . Avoirdupois. Aus.. *9. Reduce of a pound to the fraction of a guinea. Ans. 4 guin. 10. Reduce of a pwt. to the fraction of a pound Troy. Ans.. 11. Reduce of a lb. Avoirdupois to the fraction of 1 Cwt. 98 Ans. Cwt. 12. Express 51 furlongs in the fraction of a mile. Ans. mile. CASE VIII. To reduce a fraction of one denomination to an equivalent fraction of a lower denomination. RULE.† Multiply the given numerator by the parts in the denominations between it and that denomination you would reduce it to, for a This rule is the reverse of the preceding, and the propriety of it may be seen in a similar manner. The fraction of a higher denomination is obviously less than the equivalent fraction of a lower denomination; for instance, 4 of a pound is shillings or 5 shillings. Whence the value of the fraction must be increased, to render it an equivalent fraction of a lower denomination, so many times as there are parts of the less denomination in the higher. But, by the Note to Case I, the value of a fraction is increased by multiplying the numerator by a whole number. To reduce £ to the fraction of a shilling, I 400 new numerator, which place over the given denominator: Or, on◄ ly invert the parts contained in the integer, and make of them a compound fraction as before, then, reduce it to a simple one. EXAMPLES. 1. Reduce of a dollar to the fraction of a cent. By comparing the fraction it will be of 1 of 1o; then 1 X 10 X 10 100 4 =- c. Answer. 175 X 1 X 1 175 7 2. Reduce 3. Reduce 4. Reduce 5. Reduce 6. Reduce 7. Reduce 8. Keduce 9. Reduce 10. Reduce 11. Reduce 3 30000 11 400 37 28 Ans. 3m. Ans. 11m. Ans. 3d. Ans. qr. of an eagle to the fraction of a mill. of a dollar to the fraction of a mill. of a pound to the fraction of a penny. of a pound to the fraction of a farthing. of a guinea to the fraction of a penny. Ans. Ed. of a moidore to the fraction of a shilling. Ans. 1s. of a th Aroirdupois to the fraction of an ounce. of a guinea to the fraction of a pound. Ans. 4oz. Aus. £. of a Troy to the fraction of a pwt. Ans. pwt. of Cwt. to the fraction of a b Avoirdupois. CASE IX. Ans. b.. To find the value of a fraction in the known parts of the integer, as of coin, weight, measure, &c. KULE.* Multiply the numerator by the parts of the next inferior denomination, and divide the product by the denominator; and if any thing 4 * This rule follows from the preceding. Thus let£ be the fraction, whose value is to be found. By the preceding rule, £ of 5 of a shil 1335. And on the same principle, }s.— § of of a penny,=d. = 4d. Whence a farthing, gr.=3qr. qr. Therefore, £= 88. = Es. 6-d.=cs.€ d. qr. £6.67d.={s.€d. 27 i The same process is obviously applicable to every similar case. may be conducted thus; of remain, multiply it by the next inferior denomination, and divide by the denominator as before, and so on, as far as necessary; and the quotients placed after one another, in their order, will be the answer required; or, reduce the numerator, as if it were a whole number, to the lowest denomination, and divide the result by the denominator; the quotient will be the number of the lowest denomination, (which must be brought into higher denominations as far as it will go,) and the remainder will be a numerator to be placed overthe given denominator fora fraction of the lowest denomination. Note. From this rule, in connexion with what has been said of Reduction of Federal Money, it appears, that, annexing to the given numerator as many cyphers, as will fill all the places to the lowest denomination, and dividing the number so formed by the denominator, the quotient will be the answer in the several denominations, and the remainder a numerator to be placed over the given denominator, forming a fraction of the lowest denomination. $55000m. and 500°m.=625m.=62c, 5m. 2. What is the value of 17 of a dollar? $ d. c. m. 64)17 O 0 Q 128 420 384 360 320 40 Ans. as before. 3. What is the value of of an eagle? Ans. 2 qrs. 9b 10 oz. 7jdr. 10. What is the value of of a b Avoirdupois? 11. What is the value of 12. What is the value of 13. What is the value of Ans. 12oz. 124dr. 12. 14oz. 12,dr. Ans. 2qrs. 2n. Ans. 4qrs. 13n. Ans. 6fur. 26p. 11ft. Ans. 16h. 36m. 55,5. year is required? Ans. 257d. 19h. 45m. 5218. 13. The value of of a guinea is demanded? Ans. 188. Ans. 5s. 740. Ans. 21s. 74d. of an acre? Ans. 3r. 174p CASE X. To reduce any given quantity to the fraction of any greater denomiz nation of the same kind. RULE.* Reduce the given quantity to the lowest term mentioned, for a numerator; then reduce the integral part to the same term for a denominator; which will be the fraction required. Note. It appears from this rule and what has been said before, that, in Federal Money, where the given quantity contains no fraction of its lowest denomination, the annexing of as many cyphers to 1 of the required denomination, as will extend to the lowest denomination in the given quantity, will form a denominator, which placed under the given quantity used as one number for a numerator, will make the answer, which may be reduced to its lowest terms. Or, if there be a fraction of the lowest denomination, multiply the given whole numbers by its denominator, adding its numerator, for a numerator; and let the denominator itself at the left of as many cyphers as were mentioned above be a denominator; the fraction so formed will be the answer; which may be reduced to its lowest terms. This cafe is the reverse of the former, and the proof evident from that. NOTE. If there be a fraction given with the faid quantity, it must be farther reduced to the denominative parts thereof, adding thereto the numerator. |