continued product of the divisors and quotients, will give the multiple required. EXAMPLES. 1. What is the least common multiple of 4, 5, 6 and 10? Operation, x5)4 5 6 10 5x2x2x3=60 Ans. 2. What is the least common multiple of 6 and 8. Ans. 24. 3. What is the least number that 3, 5, 8 and 12 wil measure ? Ans. 120. 4. What is the least number that can be divided b the 9 digits separately, without a remainder ? Ans. 2520. REDUCTION OF VULGAR FRACTIONS, IS the bringing them out of one form into another, order to prepare them for the operation of Addition, Sul traction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE. 1. Find a common measure, by dividing the grea term by the less, and this divisor by the remainder, and on, always dividing the last divisor by the last remaind till nothing remains; the last divisor is the common me * ure 2. Divide both of the terms of the fraction by the co mon measure, and the quotients will make the fraction quired. * To find the greatest common measure of more than two numbers, 1 must find the greatest common measure of two of them as per rule abo then, of that common measure and one of the other numbers, and so through all the numbers to the last; then will the greatest common m ure last round be the answer. OR, í you choose, you may take that easy method in Problem I. (page 74.) EXAMPLES. 48 1 48 Ans. The Ans. ii Ans. 1. Reduce 18 to its lowest terms. Operation. common mea. 8) = Ans. Rem. 2. Reduce to its lowest terms. 3. Reduce 18 to its lowest terms. 4. Reduce 3798 to its lowest terms. CASE II. To reduce a mixed number to its equivalent improper fraction. RULE. Multiply the whole number by the denominator of the given fraction, and to the product add the numerator, this sum written above the denominator will form the fraction required. EXAMPLES. 360 1. Reduce 457 to its equivalent improper fraction. 45X8+7=397 Ans 2. Reduce 191to its equivalent improper fraction. Ans. 3.5_2 3. Reduce 1610. to an improper fraction. Ans. 2018 4. Reduce 611 to its equivalent improper fraction. Ans. 22.085 RULE. quotient will be the value sought. EXAMPLES. 1. Find the value of 5)48(9; Ans. 2. Find the value of 33 Ans. 191% 3. Find the value of 933 4. Find the value of '1785 Ans. 61 5. Find the value of yo Ans. & Ans. 8401 CASE IV. To reduce a whole number to an equivalent fraction, having a given denominator. RULE. Multiply the whole number by the given denominator ; place the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 7 to a fraction whose denominator will be 9 Thus, 7x9=63, and 63 the Ans. 2. Reduce 18 to a fraction whose denominator shall be 12. 3. Reduce 100 to its equivalent fraction, having 90 for a denominator. Ans. "O =j=' CASE V. To reduce a compound fraction to a simple one of equal value. RULE. 1. Reduce all whole and mixed nunbers to their equ.va. lent fractions. 2 Multiply all the numerators together for a new numerator, and all the denoninators for a new denominator ; and they will form the fraction required. EXAMPLES. 1. Reduce 1 off of of 1 to a simple question. 1X2X3X4 == Ans. 2x3x4X10 2. Reduce $ of of to a single fraction. Ans. 24% 3. Reduce of it of to a single fraction. Ans. 936 4. Reduce of of 8 to a simple fraction. Ans. 1233 5. Reduces of 12 42} to a simple fraction. Ans. 1389°=21, NOTE.--If the denominator of any member of a com pound fraction be equal to the numerator of another mem ber thereof, they may both be expunged, and the other members continually multiplied (as by the rule) will produce the fraction required in lower terms. 6. Reduce of of to a simple fraction. Thus 2x5 == Ans. 4X7 7. Reduce of f of of into a simple fraction. Ans. = CASE VI. To reduce fractions of different denominations to equivalent fractions having a common denominator. RULE I. 1. Reduce all fractions to simple terms. 2. Multiply each numerator into all the denominators except its own, for a new numerator ; and all the denominators into each other continually for a common denominator ; this written under the several new numerators will give the fractions required. EXAMPLES. 1. Reduce to equivalent fractions, having a common denominator. } + t =24 common denominator. 24 24 24 denominators. 2. Reduce Z to and 11 to a common denominator. Ans. 84: 994 and 99% 3Reduce and I to a common denominator. Ans. 4 and 153 4. Reduce and so to a common denominator 800 300 400 -and -=% % and 1=16 Ans. 1000 1000 1000 3. Reduce it and 12} to a common denominator. Ans. 1972 888 6. Reduce si and ã of 1 to a common denominator. Ans. 348 3 vit The foregoing is a general Rule for reducing fractions to common denominator ; but as it will save much labour to keep the fractions in the lowest terms possible, the following Rule is much preferable. RULE II For reducing fractions to the least common denominator. (By Rule, page 155) find the least common multiple of all the denominators of the given fractions, and it will be the common denominator required, in which divide each particular denominator, and multiply the quotient by its own numerator for a new numerator, and the new numerators being placed over the common denominator, will express the fractions required in their lowest terms. EXAMPLES 1. Reduce } and to their least common denominator. 4)2 4 8 1 1 1 4X2=8 the least com. denominator, 8:2x1=4 the 1st. numerator. 8:8X5=5 the 3d. numerator. These numbers placed over the denominator, give the answer equal in value, and in much lower terms than the general Rule, which would produce 2. Reducers and to their least common denomina, Ans. 43 44 41 |