Fractions are subject to the same operation as whole numbers; they may be added, subtracted,. multiplied, or divided.

ADDITION OF FRACTIONS.

Suppose several fractions such as †,†,†, which we must add together; this operation is indicated as follows: ++3.

We shall give out the process first, and then the explanation. Begin with bringing the several fractions to the same denominator, which is done by multiplying both terms of each fraction by the product of the denominators of all the others. By this operation, the above fractions become + + 18They all have the same denominator, which could not fail, since each denominator is the product of all the denominators of the above fractions, Now, to add them up, take the sum of all the numerators, for it is evident that 24+36-68, and that +49=48 the last fraction being the sum of the three proposed fractions,

4 8

100

Before I proceed, I observe, that 100 is not properly a fraction; for, from the definition, a fraction is a quantity less than a unit; but we call this a fractionary number, and, if the division be effected, we shall have for result two units, more the fraction.

Now, let us analyse the foregoing operation: I said, first, that we must bring all the fractions to the same denominator. Why so? because we cannot add together things of various kinds; and it is the denominator which designates the kind of the fractionary unit. No demonstration is necessary to show that it is impossible to add with, whereas there is no difficulty in adding up with, which evidently gives but in order to prove in a sensible manner what I have advanced, viz. : that it is the denomina◄

for of a fraction which designates the kind of the fractionary unit, let us take an example, and suppose that the fathom be the principal unit, and that we have to add the two fractions & and; the fathom being equal to six foot, it is plain that the express five foot; likewise the fathom being equal to 72 inches, the of a fathom will be worth 7 inches; consequently in the question, as above stated, we had feet to add with inches, which is impossible, and shows very sensibly that it is the denominator which marks the kind of the fractionary unit. With respect to the operation for bringing the fractions to the same denominator, it is evident, that by multiplying both numerator and denominator by the same number, we do not alter the value of the fraction, because the numerator and denominator represent the two terms of a division, and we know that the quotient is not changed, if after making the dividend a certain number of times greater, we also make the divisor greater in the same proportion.

From what has been said it follows, that in prac tice, the addition of fractions consists in multiplying the numerator of each fraction by the product of the denominators of all the others; then adding all these numerators, and dividing that sum by the product of all the denominators.

If we had a whole number to add with one or several fractions, this particular case would be brought under the general rule, by supposing that the whole number has 1 for denominator.

EXAMPLE. ++, reduced to the same de60x8+3 71

nominator, give me

12

, a fractionary

12

number resulting from the addition of the whole number 5, with the two fractions ‡+1.

D

Hence, to add a whole number with one fraction only, you must multiply the whole number by the denominator of the fraction, to this product add the numerator of the same fraction, and divide the whole of it by its denominator.

N. B. That the general rule just given to bring fractions to the same denominator, would induce into lengthy calculations, which may often be avoided by an abridged method which comprises three different

cases.

First Case. When the greatest denominator among all the fractions, is found to be a multiple of each of the other denominators.

EXAMPLE. ++++; 12, which is the greatest denominator, may be taken for the common denominator; in fact, by multiplying the denominator of the first fraction by 6, that of the second by 3, of the third by 2, &c. 12 would be the common denominator; but then in order not to change the value of the fractions, you must multiply their numerator by the same number by which you have multiplied their denominator; thus the fractions will become 6+9+10+8+1 34

12

-, result very simple; which

12

4896

by the general method would have been found 998.

Second Case. When the greatest denominator being taken 2, 3, 4, &c. times, gives a number multiple of the denominator of each fraction.

20

60

5

EXAMPLE. +++; I see, that taking three times the greatest denominator 20, I have 60 which I can take for common denominator, since it is a multiple of all the denominators. Write this

number above the fractions, as you see in the example, in order to facilitate the operation to be performed on the numerators. The proposed fractions, having 60 for common denominator, are reduced to 12+9+50+28

60

99

60

-; by the general method the re

sult would have been found 850.

Third Case. When the denominators of fractions, though not falling under either of the two preceding cases, are nevertheless composed of some common factors.

45

EXAMPLE. tátitá+; I write these fractions under the following form, resolving their denominators into factors as nearly alike as possible, 4X3X3 X5

·+ : Now, in or4 X 5

·+. + + 4. 4X3 3 X 3 3 X3 X5 der to have an abridged expression of the denominator, I compose it as it is seen on the top of these fractions, taking first the denominator of the 1st fraction such as it is: then the 3 of the 2d. then one 3 of the 3d. then the 5 of the 4th, and none of the factors of the denominator of the 5th, because they had already been written.

It is evident, that the number expressed by 4+3+3+5, which is equal to 180, may serve as a common denominator, since it is (as may be seen by its factors) a multiple of all the denominators. Multiplying, now, each of the numerators, by the same number by which its denominator should be multi

plied to produce the common denominator 180, the fractions become 135+75+40+16+63 329

which, by the general method, would have been 718868.

SUBTRACTION OF FRACTIONS.

This operation presents no difficulties. The fractions, as in addition, must he brought to the same denominator, and the subtraction be executed on the Numerators.

EXAMPLE.

Suppose are to be subtracted from , which is represented in this manner 2-3; brought to the same denominator, these two fractions become

15-8

20

; then the subtraction being effected on the

numerators, will give

Another Example, more complex: -+equal to 72-96+100--36

144

172-132

40

If we had a fraction to subtract from a whole number, we should multiply the whole number by the denominator of the fraction, subtract from it its numerator, and divide the result by its denominator. This process will be easily understood, by supposing the unit as a denominator to the whole number.