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continued product of the divisors and quotients, will give the multiple required.

1. What is the least common multiple of 4, 5, 6 and 10,3 Operation, x5)4 5 6 10

EXAMPLES.

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5 X 2 X 2 X 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 reinain.ler, and so on, always dividing the last divisor by the last remainder, till nothing remains ; the last divisor is the common

2. Divide both of the terms of the fraction by the common measure, and the quotients will make the fraction required.

measure.

* 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 pumbers, and so on through all the mimbers to the last, then will the greatest common measure ast found be the answer

OR, If you chuse, you may take that easy method in Problem I. (page 74.)

EXAMPLES.

48/56 (1

48 08

8) (Rem.

1. Reduce to its lowest terms.
8

Operation.

common mea. 8)3=Ans. 2. Reduce to its lowest terms.

Ans. S. Reduce 1:2 to its lowest terms.

Ans. 4. Reduce 3798 to its lowest terms. Ans.

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. 1. Reduce 457 to its equivalent improper fraction.

45X8+7=567 Ans. 2. Reduce 19H to its equivalent improper fraction.

Ans. 354

18 3. Reduce 16106 to an improper fraction.

Ans. 4618 4. Reduce 6144 to its equivalent improper fraction.

Ans. ****
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.

EXAMPLES. 1. Find the value of

5)48(98 Ans. 2. Find the value of 35

Ans. 19H 3. Find the value of u

Ans. 844 4. mail the value 0.1s

Ans. 6117 wind the valus ir

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 denominawr, and it will form the fraction required.

EXAMPLES. 1. Reduce 7 to a fraction whose denominator shall be 9.

Thus, 7x9=63, and 13 the Ans. 2. Reduce 18 to a fraction whose denominator shall be 12.

ins. 216

12 3. Reduce 100 to its equivalent fraction, having 90 for a denominator.

Ans. 999°='0=100

CASE V. To reduce a compound fraction to a simple one of equal

value.

RULE. 1. Rerluce al' whole and mixed numbers to their equivalent fractions.

2. Multiply all the numerators together for a new nu. merator, and ail the lenuminators for a new denominatur; and they will form the fraction required.

EXAMPLES.

1. Reduces of of of it to a simple fraction

1x2x3x4

2X3 X4 X10 2. Reduce, of 1 of 1 to a single fraction. Ans. S. Reduce of 11 of it to a single fraction.

Ans. 4. Reduce of of 8 to a simple fraction.

Ans. <3 5. Reduce of 42to a simple fraction.

Ans. 12680=21 to Notr.-If the denominator of any member of a com Donnd fraction be equal to the numerator of another mon

Der thereof, they may both be expunged, and the otser members continually multiplied (as by the rule) 'vili produce the fraction required in lower terins. 6. Reduce for up to a simple fraction.

Thus 2 x5

== Ans,

4X7 7. Reduce or off of ii to a simple fraction.

Ans. ****

CASE VI. To reduce fractions of uifferent denominations to equiva lent fractions having a comipon denominator.

RULE I. 1. Reiluce 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 nuinerators will give the fractions required.

EXAMPLES.

1. Reduce to equivalent fractions, having a com mon denominator.

☆ + i + 24 common denominator.

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24 24 24 denominators. 2. Reduce I io and iż to a common denominator.

Ans. 4 and; it 3. Reduce } { and } to a common denominator

Ans. Hiti and HI

4. Reduce any and into a common denominator. 800 300 400

and =115 and 1=14 Ans. 1000 1000 1000 5. Reduce and 124 to a common denominatur.

Ans. 44 6. Reduce $ $ and f of H to a common denoininator.

Ans. 1036 41% 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.

(Ry 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

2)2 1 2

1 1 1 4.x258 the least com. denominator.

8=-2x1=4 the 1st. numerator.
8:4x3=6 the 2d. numerator.

8-;-8X5=5 the 3d. nunierator. These numbers placed over the denominator, give the answer of equal in value, and in much lower terms than the general Rule, which would produce 11 *** 2. Reduce ff and í to their least common denomi

Ans.

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