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VULGAR TRACTIONS. - HAVING briefly introduced Vulgar Fractions imme

diately after reduction of whole numbers, and given some ey general definitions, and a few such problems therein as

were necessary to prepare and lead the scholar immediun ately to decimals; the learner is therefore requested to be read those general definitions in page 74.

Vulgar Fractions are either proper, improper, single, compound, or mixed.

1. A single, simple, or proper fraction, is when the numerator is less than the denominator, as i , &c.

2. An Improper Fraction, is when the numerator exceeds the denominator, as } , &c. o 3. A Compound Fraction, is the fraction of a fraction, coupled by the word of, thus, of 1, tof of 1, &c.

4. A Mixed Number, is composed of a whole number

and a fraction, thus, 8, 14 , &c. • 1 5. Any whole number may be expressed like a fraction

by drawing a line under it, and putting 1 for denominaPentor, thus, 8=, and 12 thus, 4, &c.

6. The common measure of two or more numbers, is that number which will divide each of them without a remainder; thus, 3 is the common measure of 12, 24 and 30; and the greatest number which will do this, is called the greatest common measure.

7. A number, which can be measured by two or more numbers, is called their cominon multiple : and if it be the least number that can be so measured, it is called the least common multiple: thus, 24 is the coinmon multiple of my 3 aud 4; but their least common multiple is 12.

To find the least common multiple of two or more numbers.

RULE. 1. Divide by any number that will divide two or more e of the given numbers without a remainder, and set the

quotients, together with the undivided numbers, in a line beneath.

2. Divide the second Knee as before, and so on til Shere are no two numbers that can be divided ; then the

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 10,3 Operation, X5)4 5 6 10

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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. 84. 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 measure.*

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

* To find the greatest cominon 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 througb all the mimbers to the last, then will the greatest common measure at found be the answer

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

EXAMPLES.
1. Reduce this to its lowest terms.

"Operation.
common mea. 8) = Ans.

Rem.
2. Reduce to its lowest terms.

Ans. S. Reduce 16% to its lowest terms.

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

CASE II. To reduce a mixed number to. its equivalent improper

fraction

RULL. 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. i. Reduce 45} to its equivalent improper fraction.

45x8+7=f? Ans. 2. Reduce 194 to its equivalent improper fraction.

Ans. * 3. Reduce 1646 to an improper fraction.

Ans. 4913 4. Reduce 6144 to its equivalent improper fraction.

22085 567

CASE III.
To find the value of an improper fraction.

RULE. Divide the numerator by the denominator, and the quotient wiN be the value sought.

EXAMPLES. 1. Find the value of ne

5)48(9} Ans. 2. Find the value of 3

Ans. 1947 3. Find the value of

Ans. 84 4. Man the value oi ris

Ans. 611* w wind the value of 3

CASE IV. To reduce a whole number to an equivalent fraction, her

ing a given denominator.

RULE. Multiply the whole number by the given denominator ; place the product over the said denuininaror, and it will form the fraction required.

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

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

Ans. 216 3. Reduce 100 to its equivalent fraction, haviny 90 for a denominator.

Ans. 93°='60 =100

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

value.

RULE. 1. Rerluce all whole and mixed numbers to their equivalcnt fractions.

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

EXAMPLES. 1. Reduce of of of it to a simple fraction

1x2x3x4

-=rato Ass.

2x3x4x10 2. Reduces of off to a single fraction. Ans. S. Reduce of 1 of to a single fraction.

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

Ans. 3} 5. Reduce of 13 424 to a simple fraction.

Ans. 28ofo =21' Notr.-If the denominator of any member of a com: Donnd fraction be equal to the numerator of another mo

ber thereof, they may both be expunged, and the otne: meinbers continually multiplied (as by the rule) 'vili produce the fraction required in lower terins. 6. Reduce for of to a simple fraction.

Thus 2X5

— = =*. Ans.

4x7 7. Reduce off of of it to a simple fraction.

Ans. +1 <

CASE VI. To reduce fractions of uifferent denominations to equira 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. qators into each other continually for a common denomi. nator ; this written under the several new nuinerators will give the fractions required.

EXAMPLES.

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

+ + 24 common denominator.
1 2 3
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18 new numerators.

24 24 24 denominators. 2. Reduce } % and it to a common denominator.

Ans. So if and is S. Reduce { } { and to a common denominator

Ans. Hit th it and he

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