៖ 4. Reduce of of to a simple fraction. 5. Reduce of of 34 to a simple fraction. 6. Reduce of of of 4 to a simple fraction. 7. Reduce 2 and 3 of § to a fraction. CASE VI. Ans.. Ans. 3. Ans. 1. Ans. Ans. 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 that 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. 1. Reduce,, and, to a common denominator. 1 X 3 X 4 = 12 the new numerator for 2 X 3 X 4 =24 the common denominator. Therefore the equivalent fractions are 1, 1, and 1. Or the whole operation of multiplying may be best performed mentally, only setting down the results and given fractions thus: tion. 2 12 16 18 27 39 49 2. Reduce and § to fractions of a common denominator. Ans.,. 3. Reduce, 3, and, to a common denominator. 4. Reduce, 23 and 4, to a common denominator. Note I. When the denominators of two given fractions have a common measure, let them be divided by it; then * This is evidently no more than multiplying each numerator and its denominator by the same quantity, and consequently the value of the fraction is not alerid. 3. Reduce multiply the terms of each given fraction by the quotient arising from the other's denominator. Ex., and and, by multiplying the former 33 = ''; 5 by 7, and the latter by 5. 7 2. When the less denominator of two fractions exactly divides the greater, multiply the terms of that which has the less denominator by the quotient. Ex. and and, by mult. the former by 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. 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 remainder, 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, before explained. Note, by the Editor.-Fractions may be reduced to their least common deno: minator as follows. Let 24, 27, 30, 32, 36, 40, 45, 48 be the denominators: reduce each denominator into the product of the powers of its prime factors, and the given numbers become 233, 33, 2×3×5, 25, 2a ×3o, 23 X5, 3a ×5, 2a X3: now take the highest power of each prime factor and we have 2, 33, 5; the product of which 25 x33x5=32 × 27×5=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, 2×3, 7, 23, 3a, 2×5; and the highest powers of the primes are 23, 33, 5, 7, of which the product is 23 X32 X5x7=8X9X5X7=63×40=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 o: proceed in like manner with D and the fourth given number, and the last number thus found will be C the EXAMPLES. 1. What is the of 21 6s? 12. What is the value of 3 of 1l? डै By the former part of the Rule By the 2d part of the Rule, the least common multiple of the given numbers; that is, the least common denominator 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 24×27 216 c. Again, the greatest common divisor of 216 and 30 is found 3 to be 6, and therefore D 216 X 30 6 sor of 1000 and 32 is 8, therefore E 1080. Again the greatest common divi1080X32 8 common divisor of 4320 and 36 is 36, whence F 4320. Farther, the greatest 4320 X 48 48 4320 the least common multiple of the given numbers. Ex. 2. Let 2, 3, 4, 5, 6, 7, 8, 9, 10 be the given numbers. Here c= 2×3 12X5 60, F 2520, and lastly L 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 n: in like manner proceed with p and the fourth given number, and the last product will be the least common multiple required. 9. What 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 denomi nator, if to a greater. EXAMPLES. 1. Reduce of a pound to the fraction of a penny. 3X20X12=43°18°, the Answer. 2. Reduce of a penny to the fraction of a pound. = the answer. 3. Reduce to the fraction of a penny. 5. Reduce 6. Reduce 7. Reduce 8. Reduce 9. Reduce 2s 6d to the fraction of a £. 10. Reduce 17s 7d 339 to the fraction of a £. Ans. d. 1 Ans. 2400 Ans. 32. 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 This is the same as the Rule of Reduction in whole numbers from one denomination to another. Before fractions are reduced to a common denominator, they are quite dissimilar, 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 must be reduced to simple ones, and fractions of different denominations 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. and and ? of a shilling and of a penny? 12. What is the sum of 4 of a pound, and and of penny? Aus. 3139 of a shilling, 1833s or 38 1d 1109. 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 difference of the fractions sought. EXAMPLES. 1. To find the difference between and . 2. To find the difference between and §. 361 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. 3. 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