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 ×2)4 1 6 2 X2 1x3 1 5 × 2 × 2 × 3=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 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, Subtraction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE. 1. Find a common measure, by dividing the greater term by the less, and this divisor by the remainder, and so on, always dividing 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 measure last found be the answer. OR, If you chuse, you may take that easy method in Problem I. (page 74.) 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. 1. Reduce 45% to its equivalent improper fraction. 45x8+7=387 Ans. 2. Reduce 1913 to its equivalent improper fraction. 3. Reduce 161 to an improper fraction. Ans. 354 18 Ans. 1618 100 4. Reduce 611 to its equivalent improper fraction. CASE III. To find the value of an improper fraction. RULE. Divide the numerator by the denominator, and the quotient will be the value sought. 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 shall be 9. Thus, 7x9-63, and 63 the Ans. 2. Reduce 18 to a fraction whose denominator shall be Ans. 216 3. Reduce 100 to its equivalent fraction, having 90 for denominator. Ans. 9000-900-100 12. FCASE V. 4 To reduce a compound fraction to a simple one of equal value. RULE.' 1. Reduce all whole and mixed numbers to their equiva lent fractions. 2. Multiply all the numerators together for a new nu merator, and all the denominators for a new denominator; and they will form the fraction required. EXAMPLES. 1. Reduce of of of to a simple fraction. 1x2x8x4 2. Reduce of of to a single fraction. Ans. 3. Reduce of 1 of 1 to a single fraction. Ans. 8361 4. Reduce of 5 of 8 to a simple fraction. 2 1500 Ans. S 5. Reduce of 12 423 to a simple fraction. Ans. 1660-21 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 pro duce the fraction required in lower terms. 6. Reduce of of to a simple fraction. 7. Reduce of of of to a simple fraction. ៖ Ans. $311 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 numerator: and all the denomi nators into each other continually for a common denomi nator; this written under the several new numeraters will give the fractions required.. EXAMPLES. 1. Reduce to equivalent fractions, having a common denominator. + + =24 common denominator. 6. Reduce and 4 of 11 to a common denominator. Ans. 768 2592 1980 70% The foregoing is a general Rule for reducing fractions to a 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 denomina tor. 4)2 4 8 2)2 1 2 1 1 1 4x2=8 the least com. denominatør. 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. Reduce and to their least common denomi Bater. Ans. |