2x8x20 Or, the same as before, by câncelling 3X 6 X 11 11 the 3's, and dividing by 5's. 3. Reduce of to a simple fraction. Ans. 4. Reduce 1 of 1 off to a simple fraction. 23 5. Reduce of of 34 to a simple fraction. 3 6. Reduce of į of 1 of 4 to a simple fraction. Ans. 7. Reduce 2 and off to a fraction. } Ans. 13 Ans. $. CASE VI. To Reduce Fractions of Different Denominators, to Equivalent Fractions having a Common Denominator. * Multiply each numerator by all the denominators except its own, for the new numerators : and multiply all the denominators together for a common denominator. Note, It is evident tbat in this and several other operations, when any of the proposed quantities are integers, or mixed numbers, or compound fractions, they must first be reduced, by their proper Rules, to the form of simple fractions. EXAMPLES. 2 1. Reduce }, }, and , to a common denominator. 1 X3 X 4 =12 the new numerator for ditto 3 3 X 2 X 3 =18 ditto 2 X 3 X 4 =24 the common denominator. Therefore the equivalent fractions are 4*, **, and . Or the whole operation of multiplying may be best performed mentally, only setting down the results and given fractions thus : , , , = }, 11, 1 = 1, is, is by abbreviation. 2. Reduce and to fractions of a common denominator. 2 Ans. 3$. 13 3. Reduce i, j, and }, to a common denominator. Ans. 49. 30 4. Reduce , % and 4, to a common denominator. Ans. 25, 19. 130 Note I. When the denominators of two given fractions have a common measure, let them be divided by it; then 63' 3 This is evidently no more than multiplying each numerator and its denomi. nator by the same quantity, and consequently the value of the fraction is not al. oprio 3. Reduce multiply the terms of each given fraction by the quotient arising from the other's denominator. Ex. is, and is = 17' and , by multiplying the former 5 7 by 7, and the latter by 5. 2. When the less denominator of two fractions exactly divides the greater, maltiply the terms of that wbich has the less denominator by the quotient. Er. 4 and = and , by mult. the fortner by 2. 2 3. When more than two fractions are proposed, it is sometimes convenient, first to reduce two of them to a common denominator ; then these and a third ; and so on till they he all reduced to their least common denominator. Erand and = and fand 1=1 and 1 and 2: CASE VIT. To find the value of a Fraction in parts of the Integer. Multiply the integer by the numerator, and divide the product by the denominator, by Compound Multiplication and Division, if the integer be a compound quantity. Or, if it be a single integer, multiply the numerator by the parts in the next inferior denomination, and divide the product by the denominator. Then, if any thing remains, multiply it by the parts in the next inferior denomination, and divide by the denominator as before ; and so on as far as necessary; so shall the quotients, placed in order, be the value of the fraction required. * The numerator of a fraction being considered as a reinainder, in Division, and the denominator as the divisor, this rule is of the same nature as Compound Division, or the valuation of remainders in the Rule of Three, betore explained. Note, by the Editor.- Fractions may be reduced to their least common deno: mrinator as follows. Let 24, 27, 30, 32, 36, 40, 45, 48 be the denominators: reduce cach denomina. tor into the product of the powers of its prime factors, and the given numbers become 23 x3, 3), 2X3 X5, 25, 2* X36, 23 X5, 39 X5, 2* *3: now take the highest power of each prime factor and we have 26, 33,5; the product of which 24 x33 X5=32 X 27X 5=4320, is the least common denominator required. Again, let 2, 3, 4, 5, 6, 7, 8, 9, 10 be the denominators. In this case the powers of the primes in each number are 2, 3, 22, 5, 2X3,7, 23, 39, 2X5; and the highest powers of the primes are 23, 33, 5, 7, of which the product is 23 X32 X5X7= 8X9X5X7=63X40=2520, which is the least common denominator. This method is advantageous when the prime factors are easily discovered, in other cases we may proceed in the following manner. Find the greatest common divisor of the first and second given numbers; divide the product of the first and second given numbers by this greatest common divisor, and call the quotient c: in like manner divide the product of c and the third given number by their greatest common divisor, and call the quotient :: proceed in like man. ter with o and the fourth given number, and the last number thus found will be the EXAMPLES. 1. What is the of 21 68 ? 12. What is the value off of il? By the former part of the Rule By the 2d part of the Rule, 21 6s 2 4 20 5) 94 Ans. 3) 40 (13s 4d Ans. 11 16s 9d 239. 1 12 3) 12 (4d 3. Find the value of of a pound sterling. Ans. 7s 6d. 4. What is the value off of a guinea ? Ans. 4s 8d 5. What is the value of of a half crown ? Ans. 1s 10.1d. 6. What is the value of of 4s 10d ? Ans. 1s 11 d. 7. What is the value of Ib troy ? Ans. 9 oz 12 dwts. 8. What is the value of of a cwt ? Ans. 1 qr 7 lb. the least common multiple of the given numbers; that is, the least common denoininator of the given fractions. Ex. 1. Let 24, 27, 30, 32, 36, 40, 45, 48 be the given numbers. The greatest common divisor of 24 and 27 is found by the common rule to be 3, then 24x27 216=c. Again, the greatest common divisor of 216 and 30 is found 3 216 X 30 to be 6, and therefore D= -1980. Again the greatest common divi 6 1080X32 sor of 1080 and 32 is 8, therefore E = = 4320. Farther, the greatest 4320 X 36 common divisor of 4320 and 36 is 36, whence F= -4320. In like 36 4320 X 40 4320 X45 manner 6 = = 4320, and u= 4320, and lastly k = 40 45 4320 X 48 = 4320 = the least common multiple of the given numbers. 48 8 2x3 1 Ex. 2. Let 2, 3, 4, 5, 6, 7, 8, 9, 10 be the given numbers. Here c= 420 x 8 *26, D= =12, E= 60, F= =60,6= =420, = 2 1 6 840 x9 2320x10 =840, x= =2520, and lastly = 220 = the least common 3 10 multiple required. This general rule may be expressed as follows. Divide the first by the greatest common measure of the first and second, and multiply the quotient by the second, and call the product c: divide c by the greatest common measure of c and the third given number, and multiply the quotient by the third, call this product p : in like manner proceed with n and the fourth given nunber, and the last product will be the least common multiple required. 9. What * This is the same as the Rule of Reduction in whole numbers from one denotaination to another. : 9. What is the value of 1 of an acre ? 10. What is the value of 7 of a day? Ans. 3 ro. 20 po. Ans. 7 hrs 12 min. CASE VIII. To Reduce a Fraction from one Denomination to another. * Consider how many of the less denomination make one of the greater; then multiply the numerator by that number, if the reduction be to a less name, but multiply the denoniDator, if to a greater. EXAMPLES. 1. Reduce of a pound to the fraction of a penny. 용 fXY XY=**°='50, the Answer. 2. Reduce of a penny to the fraction of a pound. X X go o the answer. 3. Reduce il to the fraction of a penny. Ans. d. 4. Reduce q to the fraction of a pound Ans. 5. Reduce cwt to the fraction of a lb. 6. Reduce dwt to the fraction of a lb troy. 7. Reduce crown to the fraction of a guinea Ans. 8. Reduce half-crown to the fract. of a shilling. Ans. 1 9. Reduce 2s 6d to the fraction of a £. Ans. 10. Reduce 178 7d 39 to the fraction of a £. 1 3400 Ans. a a a ADDITION OF VULGAR FRACTIONS. If the fractions have a common denominator ; add all the numerators together, then place the sum over the common denominator, and that will be the sum of the fractions required. † If the proposed fractions have not a common denominator, they must be reduced to one. Also compound fractions must be reduced to simple ones, and fractions of different denomiuations to those of the same denomination. Then add the numerators as before. As to mixed numbers, they may either be reduced to improper fractions, and so added with the others; or else the fractional parts only added, and the integers united afterwards. + Before fractions are reduced to a common denominator, they are quite dissi. milar, as much as shillings and pence are, and therefore cannot be incorporated with one another, any more than these can. But when they are reduced to a common denominator, and made parts of the same thing, their sum, or difference, may then be as properly expressed by the sum or difference of the numerators, as the sum or difference of any two quantities whatever, by the sum or difference of their individuals. Whence the reason of the Rule is manifest, both for Addition and Subtraction. When When several fractions are to be collected, it is commonly best first to add two of them together that most easily reduce to a common denominator; then add their sum and a third, and so on. EXAMELES. 30 36 1. To add and together. 号 Here } + =j=1}, the Answer, 2. To add and á together. +1 = 1 + 35 = 13 = 11, the Answer, 3. To add and 7 and of together. + 7 + of i=++=+*+==87. 4. To add 7 and together. Ans. 14. 5. To add and k together. Ans. 118 6. Add and is together. Ans. 7. What the sum of sand and ? Ang. 1143 8. What is the sum of į and 1 and 21 ? Aos. 338 9. What is the sum of and of } and 976? Ans. 10.6. 10. What is the sum of of a pound and of a shilling ? Ans. '5s or 138 10d 279. 11. What is the sum of of a shilling and is of a penny ? Ans. id or 7d 1}}9. 12. What is the sum of ļof a pound, and of a shilling, and of penoy? Aus. 12:s or 3s 1d 1319. 103 1008 SUBTRACTION OF VULGAR FRACTIONS. PREPARE the fractions the same as for Addition, when necessary; then subtract the one numerator from the other, and set the remainder over the common denominator, for the disference of the fractions sought. EXAMPLES Here 1-1== }, the Answer. 37 . 3. What |