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of the theory of vulgar fractions. A concise account of the principal rules and processes will therefore be given in the present article.

In most systems of arithmetic, fractions are treated under the separate heads of proper and improper, simple and compound fractions. When the numerator is less than the denominator, as, the fraction is said to be proper: that which may properly be termed a fraction, or part of a unit. When the numerator exceeds the denominator, as in, the expression is of greater value than a single unit it represents the whole of one unit and part of another such unit, and in this sense, is called an improper fraction. As the numerator will contain the denominator once, or more than once, in all instances of improper fractions, we may perform the divisions as far as possible, and in place of the improper fraction, write an equivalent whole number and proper fraction, or mixed number as it is frequently styled. ths thus operated on, becomes 1 and, usually written 13. A fraction having the same number for its numerator and denominator, as 11, is also called an improper fraction. In every instance of this kind, we may cancel the fractional expression and substitute 1. If a unit be divided into any number of equal parts, and all those parts be taken, the whole unit must be taken.

A simple fraction, consists of one fraction only, proper or improper: for example,,.

A compound fraction, is a fraction of a fraction, or a fractional part of some whole number with or without a fraction attached; of; of 21, are compound fractions.

When the numerator or the denominator of a fraction is itself a fraction, or when both numerator and denominator are fractions, or whole numbers and fractions joined, the expression is termed a complex fraction.

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Except for the sake of convenience in designating the various forms of fractions, it is better not to attend to these distinctions, as the rules for operating are the same under whatever aspect a fraction may be presented. Besides, every fraction, or whole number and fraction, may be reduced to the form of a simple fraction, and be thus the more easily managed in practice.

It is useful to bear in mind the following general truths with regard to fractional expressions:

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Any fraction, as g, may be looked upon as signifying the part or proportion which 5 is of 8: or generally what part or proportion the numerator is of the denominator. The expression, when read in the usual way, itself tells us what part of 8 is 5-five-eighths of it. 1 is the one-eighth of 8, therefore, 5 must be five-eighths of it. Again, any fraction, as g, expresses how often,-how many times or parts of a time, 5 contains 8; or generally, how often the numerator contains the denominator. In this case not a whole time or once; only fiveeighths of a time. If the fraction were an improper one, as it is called, that is, if its numerator were greater than its denominator, as in 11, one might for simplicity divide out as far as possible, and say that 11 contains 7, once and four-sevenths (14ths) of a time.

The phrase, parts of a time, used in this sense, implies that the unit of measurement is greater than the object to be measured; that 7 is too large a number to measure 4, or to be wholly contained in the number 4. As 4 is the 4ths of 7, a

line 7 feet long if taken 4ths of its own length, or 4ths of a time, would exactly coincide with a line 4 feet long.

It follows from this view of fractional expressions, that any fraction, as may be regarded as expressing either five-eighths of 1, or one-eighth of 5. Similarly may be regarded either as seven-tenths of a unit, or as one-tenth of 7 such units. To understand this, it is only necessary to consider, that gths of 1 is five times as much as 4th of 1, and that 4th of 5 is also five times as much as 4th of 1, consequently the values of gths of 1 and 4th of 5 must be equal. gths of a shilling is 74d, and so is the 4th of 5 shillings. In operating on fractions it will often be found of advantage to attend to these identities of value.

Although the form of a fraction is thus commonly employed to represent the process of division—the division of the upper number by the lower—yet a fraction or part of any quantity should not be confounded with the symbol of an unexecuted division, a thing which conveys no distinct idea to the mind, like the word fraction, and is merely an indication, not a result.

The principle of greatest importance and most extensive application in the arithmetic of fractions, is the following:

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The value of a fraction is not altered-its magnitude is not rendered less or greater by multiplying or dividing both its terms by the same number.

40

Multiply the numerator and denominator of by 3, still preserving its fractional form: the result is 21. Again, multiply both terms of by 5: the result is 38. Now it is easy to see that the three fractions 1, 2, and 35 are of equal value; that while the absolute numbers composing each fraction differ from one another, all three fractions express the same portion of a unit. On the other hand, divide both terms of 38 by 5, and of 21 by 3; the result in each instance is the original fraction

Draw three lines of equal length. Divide the first into 8 equal parts and mark off 7 of them: the second into 24 equal parts and mark off 21 of them; and the third into 40 equal parts and mark off 35 of them. It will be found that precisely the same fraction of the line is arrived at by each of these methods of division.

Officers may readily devise additional illustrations of the truth of this principle, for their own satisfaction.

To reduce a fraction to lower terms.-RULE.-Divide the numerator and denominator by any small number that will obviously divide them both without remainder the quotient again, if possible, by the same or any other number, and so on, until a result is reached which appears to admit of no further reduction.

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24

136

Examples.-(1), if continually divided by 2 is 444 | 4 & In this case, after the first division by 2, it will be readily perceived that the quotient 24 is divisible by 8, which gives the final result at one operation. (2). 1170, admits of division by 3, 5, 7, 7, successively: thus

136

2205

1470 490
2205 735

It must be evident that the method of reduction here exemplified, is an application of the principle, that the value of a fraction suffers no change if both its terms be divided by the same number, while the fraction itself is by such a process rendered much more manageable, should it be necessary to operate with it, and is also presented in a form that enables the mind to judge more nearly of its value or magnitude than when its terms consist of large numbers. In the arithmetical

questions given at examinations, it is always expected that fractional results will be reduced to the simplest form attainable by the use of obvious and easily-applied divisors. It would not be considered good work, for instance, to leave a fraction in such terms as 15, when it is so manifest that division by 3 will reduce it to g In order to discover with quickness and certainty, by what small numbers the terms of any fraction are divisible, it may be of advantage to note, that a number is divisible by 2, if its last figure be an even number or 0; by 3 and 9, if the sum of its figures be divisible by 3 or 9; by 4, if the number formed by its last two figures be divisible by 4; by 8, if the number formed by its last three figures be divisible by 8; by 5, if its last figure be either 5 or 0, &c. Thus, 54 is divisible by 2, as 4 is divisible by 2; 783 by 3 and 9, as 18, the sum of 7, 8, and 3 is divisible by 3 and 9; 628 by 4, as 28 is divisible by 4; 6152 by 8, as 152 is divisible by 8; 95 and 60 by 5, as they end in 5 and 0 respectively. Mere inspection, however, will in most cases, point out the available divisors without the assistance of rules.

When fractions occur which resist exact division by each of the smaller numbers, it is necessary to find the greatest common measure of the numerator and denominator, and to divide them by that measure, if it be desired to reduce the fraction to its lowest terms, or to discover whether it is capable of being in any degree simplified. But, in practice it would be extremely laborious to ascertain for this purpose the greatest common measure of the terms of every fraction, unless indeed that were the special object of the calculation. On all ordinary occasions it is sufficient to make trial of the first few numbers, and if these fail to divide the fraction, no further trouble need be taken.

Least common multiple.-The rule usually given for transforming fractions with different denominators into other fractions of equal value with the same or a common denominator, directs us, to multiply each numerator by the product of all the denominators except its own, for the new numerators, and to multiply all the denominators together for the common denominator. But this rule, if literally followed, would involve in many instances a very tedious process, and afford results expressed in inconveniently high numbers. A much better method is that supplied by the use of the least common multiple, which enables us to effect the required reduction with great facility, and presents the results at once in the lowest, or nearly the lowest possible terms.

One number is said to be a multiple of another when it can be divided by that other without remainder when it consists of an exact number of repetitions of that other. Thus, 12 is a multiple of each of the numbers 1, 2, 3, 4, 6, and 12, since it is divisible by each of those numbers without remainder. A common multiple of two or more numbers is one which exactly contains each of them; and the least such number is their least common multiple. Thus 12 is the least common multiple of 1, 2, 3, 4, 6, and 12, as no number less than 12 will contain each of them without remainder. It is not usual, however, to consider

any number as a multiple of unity or itself.

To determine, in the most convenient manner, the least common multiple of any series of numbers. RULE.-Write down the numbers in a row, separating them by some mark, such as a comma. Divide all those which are seen to have an obvious common measure by that measure. Write the results in a line below, and bring down also the undivided numbers. Repeat this process so long as two or more of

The meaning of the term "greatest common measure" and the process of finding the divisor so named, are described in every elementary treatise on arithmetic.

the numbers have any common measure.

The least common multiple of the given numbers will be found by multiplying together all the divisors that have been used, and the numbers which occur in the final line.

Example. In pursuance of this rule, we arrive at the least common multiple of the numbers, 3, 5, 8, 9, and 10, by the following simple operation :—

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As the numbers in the last line have no common measure greater than 1, the least common multiple required is the product of 2, 3, 5, 4 and 3, or 360. No less number than 360 will contain each of these numbers without remainder. Where large numbers are concerned, it may not always be readily discoverable whether the results of the successive divisions possess a common measure. such cases it is necessary to apply the rule for finding the greatest common measure, and to divide by it before the final product is formed, otherwise the least possible common multiple of the given numbers will not be obtained. But, on most occasions, it is hardly worth while to seek a lower multiple than that afforded by the first few obvious divisors.

It is useful to note, that the process of finding the least common multiple may be greatly shortened by leaving out of the first or any subsequent line, such numbers as are readily seen to be measures of others of the series. Every multiple of 9, for instance,-27, 45, 81, &c. must also be a multiple of 3, which is a measure of 9; that is, every whole number of nines must also be a whole number of threes, as nine itself is a whole number of threes. On the same principle, all multiples of 10 are multiples of 2 and 5.

The result above arrived at, therefore, may be attained with less trouble by the omission of 3 and 5 from the first line, thus

2) 8, 9, 10
4, 9, 5

As 4, 9 and 5 have no common measure greater than unity, the least common multiple of the original series must be the product of the four numbers, 2, 4, 9, and 5, or 360 as before.

It is hardly possible, without the use of algebraic language, and the introduction of a mode of reasoning that would be unsuited to the character of this work, to give an intelligible proof of the generality of the method here prescribed for finding the least common multiple. But the officer may at least infer that the rule is likely to hold good on all occasions, from observing that it never fails to lead to the desired result in any instance in which he may choose to apply it for his own satisfaction.

lo reduce fractions to a common denominator. RULE.-1st. Obtain the least common multiple of the denominators. 2nd. Divide that multiple by each of the denominators successively, and multiply the result by each of the numerators. 3rd. Place that multiple as the common denominator under each of the altered

numerators.

If the terms of any of the given fractions have an obvious common measure, it

will be of advantage as a preliminary step to divide by that measure, and thus reduce the fraction to a simpler form.

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In (2) it is readily perceived that 7 respectively, to the simpler forms, instead of the given fractions.

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222768

222768 222768 222768

and 3 are reducible by division by 3 and

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and, which, therefore, should be used In (3) the least common multiple of the denominators will not be obtained, if the operator should fail to observe that 153 is exactly divisible by 17.

« Mixed numbers," or whole numbers having fractions annexed, such as 31 107 &c., should be reduced to the form of simple equivalent fractions, before operations are performed on them in common with other mixed numbers or simple fractions. To effect this reduction, multiply the whole number by the denominator of the accompanying fraction, and add in the numerator. Write the result as a numerator, and under it the given denominator. These will be the terms of a simple fractional expression equal to the mixed number. 107, thus treated, becomes &; 10 by itself is evidently so to which, if be added, the value of the whole will be 87, and so for all other cases.

The method of reducing complex fractions, such as—

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to simpler forms, must be deferred until the Division of Fractions has been explained. (See page 74.)

Let it now be required to reduce to a common denominator, a series of whole numbers, mixed numbers, and simple fractions, such as the following—

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First, change the mixed numbers 42 and 1317 into simple fractions, and write 6 and 11 as and 11* The series then presents itself as—

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Obtain the least common multiple of the denominators, 7, 20, 15, and 33, and apply it to form new numerators for all the fractions, as in the preceding examples. The results will be

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ADDITION AND SUBTRACTION OF VULGAR FRACTIONS.-To perform either of these operations on fractions that have different denominators, the RULE is, first, reduce the fractions to a common denominator. Then, add together, or subtract one from another, the numerators obtained in this process, and place the sum or difference over the common denominator.

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Any whole number may evidently be written in the form of a fraction by placing 1 under it as a denominator.

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