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The value of a fraction increases or lessens, directly as the numerator is increased or lessened, (the denominator remaining unchanged) or it increases or lessens as the denominator is lessened or increased, (the numerator remaining unchanged) thus of any one whole thing, is evidently twice as much as of the same, but is also evidently twice as great as of the same whole thing.

The value of a fraction remains unaltered, if both its terms are multiplied by the same number; that value depending altogether on the comparative magnitude of its terms, and not on their absolute greatness or smallness. Thus, the fraction is equal to the fraction or or respectively, for comparing any one with the other, as the fractions and in the latter, the whole thing is conceived to be divided into 10 times as many equal parts as in the former, each of which is, therefore, ten' times less than the former; consequently, if we take ten times as many of them as of the former, we shall take just the same quantity of the whole; for example, as a twelfth part of a foot being one inch, of a foot is 6 inches, but that is just half a foot, or the fraction, or one-half; and if both the terms of a fraction be divided by a common divisor, the value of the fraction remains still unaltered, as in the former case.

A simple fraction contains one or many parts of some one whole, as one-fourth of a yard, three-fourths of a pound, five sixths of a foot, &c.

A compound fraction is otherwise called a fraction of a fraction, as two-thirds of three-fourths of a pound, or twenty shillings; which is the same as to say two-thirds of fifteen shillings.

A complex fraction is that which has a fraction either in its numerator or denominator, or in each of them. Thus,

3,, 5 52

and are complex fractions.

63 9 $

The same operations can be performed on fractional as on integral quantities. Before entering on these operations, it is proper to show how such quantities may be modified, without changing their value, so as to fit them for the several operations to be performed on them, for the different uses to which they are to be applied. This will constitute reduction of fractions.

For a demonstration of this rule, and other observations on fràetions, see John Walker's Philosophy of Arithmetic, pages 41 to 44.

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REDUCTION OF FRACTIONS.

First preparatory problem.

To find the greatest common measure of two numbers, that is, the greatest number which will evenly divide each of them, RULE.

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Divide the greatest number by the least, if there is no remainder, the least is the greatest common measure, as no number greater than itself can be a measure of the same number; but if there is a remainder after the first division, then divide the divisor by that remainder; and so on, the last divisor by the last remainder, until a remainder is found, which will measure, or exactly divide the last divisor; this will be the greatest common measure required. But if 1 is the last remainder, one is the greatest common measure, and the numbers are said to be prime to each other, and are already in their lowest terms.

Examples.

What is the greatest common measure of 64 and 1'44.

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Here 16 is the greatest common measure, being the last remainder which will measure the former divisor.

Find the greatest common measure of the following pairs of numbers, each pair respectively.

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For a demonstration of this rule, see John Walker's Philosophy of

Arithmetic, page 87.

To find the least common

Second preparatory problem. multiple of two or more numbers, that is the least number into which each of the given numbers may be evenly divided.

Place the numbers in a row, after each other, with a point or comma between each; then see what divisor will measure the greatest number of them; divide the numbers thereby, and place the quotients respectively under each number so divided, and bring down any number or numbers which the first divisor did not measure orderly in a row, with the quotients of the numbers already divided; divide this row as before, by the divisor, which will measure the greatest number of them, and proceed as before; and so on, until there be no two numbers which can be divided by a common divisor, then the continual product of all the divisors, and the last quotients, will give the least common multiple required.

What is the least common multiple of the numbers, 4, 6, and 8, and 3, 5, 6, 10 and 12, each set respectively?

2)4, 6, 8.

2)3, 5, 6, 10, 12

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Find the least common multiple of the following sets of

numbers, each set respectively.

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5, 9, 13

and 17

3. 2, 4, and 8

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5. 5, 7, 42 and 10

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NOTE.-When no two of the given numbers can be divided by a common divisor, the

continual product of the given numbers will be the least common multiple.

REDUCTION OF FRACTIONS.

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CASE 1st. To reduce fractions to their least terms; or to express them by the least whole numbers possible.

Find the greatest common measure of the numerator and denominator, and divide both the terms of the fraction thereby, and the quotients will be the terms required, which will express the given fraction in its lowest terms.

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57

C. M. 3 = the Answer.
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Reduce the following fractions to their lowest terms.

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Though the rule already given is general for all fractions whatever, yet it will not be always necessary to find the greatest common measure, as many fractions may be reduced to their lowest terms by inspection, as in the first six examples; and whenever the sum of the digits of any term of a fraction, or any number whatever, can be divided by 3 or by 9, the whole term or number may be divided by 3 or 9. When both terms are even numbers, they may be each divided by 2; and if the sum of the digits of an even number can be divided by 3 or 9, this number may be also divided by 6 or 18. When the terms of a fraction, or any number, ends with a cipher or 5, the whole can be divided by 5; and if the sum of the digits of any number ending with a cipher or 5 can be divided by 3 or 9, this num

aber may be also divided by 15 or 45. When the two last digits of any number can be evenly divided by 4, the whole can be divided by 4; and when the 3 last digits can be evenly divided by 8, the whole may be divided by 8. By the proper application of this information, much labour, may be often saved.* Cramples.

Let be reduced to its lowest terms.

Here the sum of the digits in each term of the fraction, is 9, therefore, the total of each term may be divided by 9; which being done, the fraction becomes 44, the terms of which not falling under any of the prescribed conditions, we may reasonably conclude the fraction is reduced to its lowest terms.

The first rule, by the common measure, always reduces the fraction to its lowest terms; but when the terms of a fraction are large, and the divisors not such as have been specified, it is sometimes doubtful whether the fraction is in its lowest terms or not; it will be, therefore, advantageous to carry the reduction by the second rule, as far as it will go, and then apply the first rule to the result.

Example.

Let 4299 be reduced to its lowest terms.

Here we instantly perceive that 6 is a common measure, and dividing the terms of the fraction thereby, it becomes proceeding then by the first rule, we discover that 143 is the greatest common measure, and that is the fraction in its lowest terms.

Let the following fractions be reduced to their lowest terms each respectively..

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* For a farther elucidation of those curious properties in numbers, and for a demonstration, I refer the reader to John Walker's Philosophy of Arithmetic, pages 25 to 29:

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