EXAMPLES: 1. What is the least common multiple of 4, 5, 6 and 109 Operation, x504 5 6 10 5x2*2*3=60 Ans. 2. What is the common multiple of 6 and 8? Ans. 24. 3. What is the least number that 3, 5, 8 and 12 will measure ? Ans. 120. 4. What is the least number that can be divided by the 9 digits separately, without a remainder? Ans. 2520. REDUCTION OF VULGAR FRACTIONS, IS the bringing them out of one form into another, in order to prepare them for the operation of Addition, Subo traction, &c. CASE I. To abbreviate or reduce fractions to their lowest termsi Rute.-1. Find a common measure, by dividing the greater lerm by the less, and this divisor by the remainder, and go on, always di. viding the last divisor by the last remainder, till nothing remains; the last divisor is the common measure.* 2. Divide both of the terms of the fraction by the common measure, and the quotients will make the fraction required. * To find the greatest common measure of more than two numbers, you must find the greatest common measure of two of them as per rule above; then, of that common measure and one of the other numbers, and so on through all the numbers to the last; then will the greatest common mea. sure last found be the answer. Or, if you choose, you may take that easy method in Problem I. (page 69.) EXAMPLES. 1. Reduce to its lowest terms. Operation. coinmon measure, 2)18= Ans. 48 8): 8( Rem. 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. 18 1. Reduce 451 to its equivalent improper fraction. 45*8 :-7=3. Ans. 2. Reduce 1913 to its equivalent improper fraction. Ans. 35 4 3. Reduce 16,1% to an improper fraction. Ans, 16.18 4. Reduce 6113 to its equivalent improper fraction. Ans. 2 20 8 5 CASE III. To find the value of an improper fraction. RULE.—Divide the numerator by the denoininator, and the quo. tient will be the value sought. 100 360 CASE IV. To reduce a whole number to an equivalent fraction, hav ing 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. Ans. ? 3. Reduce 100 to its equivalent fraction, having 90 for a denominator. Ans. "48° ==' CASE V. To reduce a compound fraction to a simple one of equal value. Rule.-1. Reduce all whole and mixed nurnbers to their equiva. lont fractions, 2. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator; and they will form the fraction required. EXAMPLES. 287 1. Reduce of fofof ta to a simple fraction. 1x2x3 x4 fit=. Ans. 2x3x4x10 2. Reduce of off to a single fraction. Ans. 3. Reduce of li of j} to a single fraction. Ans. 836 4 Reduce of of 8 to a simple fraction. Ans. 43 5. Reduce of ll of 42} to a simple fraction. Ans. '60°=21 Note.--If the denominator of any member of a com pound fraction be equal to the numerator of another men 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. b Thus 2 x 5 == Ans. 4X7 7. Reduce of sof of il to a simple fraction. Ans. ;=ki CASE VI. To reduce fractions of different denominations to equiva lent 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 numera ior; and all the denominators into each her continually for a common denonıinator ; this written under the several new numerators will give the fractions required. EXAMPLES. 1. Reduce ], }, ,, to equivalent fractions, having a common denominator. i + s + =24 common denominator, 24 24 24 denominators. Ans. 47, 78' and Ans. 34*, !, hii, and 4. Reduce , m, and to, to a common denominator 800 300 400 --and- -=and n=1 Ans. 1000 1000 1000 5. Reduce , and 123, to a common denominator. Ans., 49, 6. Reduce , i, and of 1, to a common denominator Ans. . The foregoing is a general rule for reducing fractions to a coinmon denominator ; but as it will save much labour to keep the fractions in the lowest terms possible, the follow, ing Rule is much preferable. 7 72 73 ! RULE II. For reducing fractions to the least common denominator, (By Rule, page 143) find the least common multiple of all the denominators of the given fractions, and it wili he 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 48 1 1 1 4X2=8 the least comn. denominator, 8:-2X]=4 the 1st numerator. 8:8 x 5=5 the 3d numerator. These numbers placed over the denominator, give the answer Gel equal in value, and in much lower terms than the general Rule would produce 2. Reduce , 5, and 75, to their least common denominatòr: Ans. i. 11,3 4 8 |