To reduce a mixed number to its equivalent improper fraction. RULE.* Multiply the whole 'number by the denominator of the fraction, and add the numerator to the product, then that sum written above the denominator will form the fraction required. * All fractions represent a division of the numerator by the de. nominator, and are taken altogether as proper and adequate expressions for the quotient. Thus the quotient of 2 divided by 3 is; from whence the rule is manifest; for if any number is multiplied and divided by the same number, it is evident the quotient must be the same as the quantity first given. I 2. Reduce 183 to its equivalent improper fraction. 3. Reduce 514 to an improper fraction. 4. Reduce 100 to an improper fraction. Ans. 3-848 2 8229 1 5 5319 5. Reduce 47to an improper fraction. Ans. 1947. CASE III. To reduce an improper fraction to its equivalent whole or mixed number. RULE.* Divide the numerator by the denominator, and the quo tient will be the whole or mixed number required. EXAMPLES. 1. Reduce to its equivalent whole or mixed number. 16 81 6 16)981(61 21 5 Or, 981÷1661, the answer. 2. Reduce to its equivalent whole or mixed number. Ans. 73. Reduce to its equivalent whole or mixed num Ans. 56. 4. Reduce 343 to its equivalent whole or mixed num *This rule is plainly the reverse of the former, and has its reason in the nature of common division.' CASE IV. To reduce a whole number to an equivalent fraction, having given denominator.. RULE. a Multiply the whole number by the given denominator, and place the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 7 to a fraction, whose denominator shall be 9. 7X963, and the answer. And 6397 the proof. 2. Reduce 13 to a fraction, whose denominator shall be 12. Ans. 3. Reduce 100 to a fraction, whose denominater shall be 90. Ans. 200 90 CASE V. To reduce a compound fraction to an equivalent single one. RULE.† Multiply all the numerators together for the numerator, and all the denominators together for the denominator, and they will form the single fraction required. If * Multiplication and division are here equally used, and consequently the result is the same as the quantity first proposed. That a compound fraction may be represented by a single one is very evident, since a part of a part must be equal to some part of the whole. The truth of the rule for this reduction may be shewn as follows. Then Let the compound fraction to be reduced be of 4. of 4431, and consequently of 44X2 as by the rule, and the like will be found to be true in all cases. 8 the same If If part of the compound fraction be a whole or mixed number, it must be reduced to a fraction by one of the for mer cases. When it can be done, divide any two terms of the frac tion by the same number, and use the quotients instead thereof. EXAMPLES. 8 1. Reduce of of of to a single fraction. I I To reduce fractions of different denominators to equivalent fractions, having a common denominator. RULE I.* Multiply each numerator into all the denominators, except its own, for a new numerator; and all the denomi nators continually for the common denominator. EXAMPLES. If the compound fraction consist of more numbers than 2, the two first may be reduced to one, and that one and the third will be the same as a fraction of two numbers; and so on. * By placing the numbers multiplied properly under one another, it will be seen, that the numerator and denominator of every frac EXAMPLES. 1. Reduce, and 4, to equivalent fractions, having a common denominator. IX5X7 35 the new numerator for 4. 3X2X7=42 4×2×5=40 do. do. for for 4. 2X5X770 the common denominator. Therefore the new equivalent fractions are 5, 4, and 48, the answer. 2. Reduce, and, to fractions, having a common denominator. Ans. I 4 4 19 2 240 252 258 288 288 288 3. Reduce 3, 4 of 3, 51 and 5, to a common denomi❤ nator. 3. 9 370 370 5709570' 4. Reduce, of 14, and to a common de nominator. Ans. 13552 150 15 13 104 1 1 4 4 0 16016 TO 16' 16010' 16010 RULE I To reduce any given fractions to others, which shall have the least common denominator. 1. Find the least common multiple of all the denominators of the given fractions, and it will be the common de nominator required. 2. Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, and the product will be the numerator of the fraction required. EXAMPLES. tion are multiplied by the very same number, and consequently their values are not altered. Thus in the first example : In the 2d rule, the common denominator is a multiple of ali the denominators, and consequently will divide by any of thera; it is manifest, therefore, that proper parts may be taken for all the pumerators as required. |