VULGAR FRACTIONS.

1. When a unit or single object is divided into a number of equal parts, each of these parts is a fraction.

If it be divided into two equal parts, each part is called a half, and is thus written, .

If it be divided into three equal parts, each is called a third, and is thus written, .

If the whole be separated into six equal parts, each part is called a sixth, and if into eight equal parts, an eighth, of the whole, and are thus written, 1, 1.

When more parts than one are to be expressed, the figure above the line designates their number, thus, ; by which expression, we are to understand that the unit is divided into six equal parts, and that five of these parts are included in the fraction.

The fraction therefore is used to express parts of units, and is represented by two numbers, one standing below and the other above a short horizontal line. The number below the line is called the denominator, and shows the number of equal parts into which the unit is divided. The number above the line is called the numerator, and shows how many of these equal parts are included in the fraction, or make up its value. Thus of the fraction, the lower number shows a unit to be divided into nine equal parts; and the upper number, that five of these parts are included in the fraction."

These two numbers, when spoken of collectively, are called the terms of the fraction.

2. Fractions are divided into six kinds; viz. Proper, Improper, Simple, Compound, Mixed, and Complex.

A Proper Fraction is one whose numerator is less than its denominator, as, .

An Improper Fraction is one whose numerator equals or exceeds its denominator, as, g.

A Simple Fraction consists of one expression, and is either proper or improper, as, & org.

A Compound Fraction is the fraction of a fraction, as, of of 5. It may consist of any number of simple fractions.

A Mixed Number consists of a whole number and fraction written together, as, 63, 254, &c.

A Complex Fraction is one that has a fraction in its numerator

21

4 93

or denominator, or both, as, 81'

&c.

3. The denominator shows the number of equal parts into which the unit is divided; and the numerator, how many of these parts are expressed in the fraction. Consequently, the greater the numerator, the denominator being given, the greater the value of the fraction; and the less the numerator, the less the value of the fraction. If the denominator be 8, and the numerator 1, the value expressed is, or one eighth part of a unit; if the numerator be 2, the value expressed is, or two eighth parts of a unit; if it be 4, the value is, or four eighth parts of a unit; and if it be 6, the value is g, or six eighth parts of a unit.

The value of a fraction is therefore the quotient arising from dividing the numerator by the denominator, and always increases in the same ratio as the numerator, so long as the denominator remains unaltered.

We may therefore express any value, not only less than a unit, but equal to and even greater than a unit, by a fraction. Thus, if we take 9 as the denominator of a fraction, and any number less than 9 as a numerator of the same, the value expressed is always less than a unit, as, §; or if 9 be taken as the numerator, we obtain the fraction, which, as the unit was divided into 9 parts only, is obviously equal to 1. Again, we may suppose more than a single unit of the same kind to be divided in the same manner, and their parts united in one fraction, and thus obtain fractions of any value more than a unit. If two units be thus divided into seven equal parts, and three parts of the one be united to all the parts of the other, the fraction would be 10; or if all the parts of each be united, it would be 14, which is equal to 2; or if three units were thus divided, all their parts would produce the fraction 21=3.

The only consideration which limits the value of a fraction, is the number of equal parts united in the same expression. From the preceding it is obvious that the value of a fraction is increased in the same ratio as the numerator; hence,

4. A fraction is multiplied by a whole number, by multiplying the numerator only.

In accordance with the above principle, the scholar may multiply the following examples:

From the last two examples, it is obvious that a fraction is multiplied by a number equal to its own denominator, by rejecting that denominator and retaining only the numerator.

It should always be an object with the scholar to preserve the terms of a fraction as small as is possible and express the true value. This was not regarded in the above examples. A little experience will show that to increase or diminish the value of a fraction, it is only necessary to make the numerator larger or smaller compared with the denominator. Suppose it be required to multiply the fraction by 2. By the above rule the product would be, which is equal in value to, and this is at once obtained by dividing the denominator by 2 instead of multiplying the numerator as above, thus,

1.

6+2-3 therefore,

A fraction is multiplied by a whole number, by dividing the. denominator by that number.

The following examples will illustrate this principle :

The value of a fraction may therefore be multiplied by a whole number, either by multiplying the numerator or dividing the denominator by that number.

Note. The denominator should always be divided, whenever it can be done without a remainder.

5. A fraction is divided by a whole number, by dividing the numerator by that number.

This needs no explanation. If we divide a number by 2, we take a half, and if by 3, a third of that number; that is, the divisor always shows what part of the dividend is taken; therefore, 13-2-13, and 13÷3=1.

The following examples will illustrate the operation of the above principle:

He

In this last example, the scholar will find a difficulty. cannot divide the numerator in any way, except to place it over the 7 in the form of a fraction, as will hereafter be explained; and this would make one fraction the numerator of another fraction. When, therefore, the divisor will not divide the numerator without a remainder, a more convenient mode of operating is desirable. It will be remembered, that division is the reverse of multiplication; and since we can multiply fractions by dividing the denominator, we will try the effect of dividing fractions by multiplying the denominator. Let it be required to divide by 3. By dividing as above, we obtain as the quotient, viz. 3. By the mode we propose to try, we obtain. It therefore remains to show that 36 If object be first divided into 12 equal parts, and then each of these 12 parts be divided into 3 equal parts, it is plain that the whole would be divided into 36 equal parts, and that each twelfth part would make 3 thirty-sixth parts; therefore, 23; hence, a fraction is divided by a whole number by multiplying its denominator by that number.

3

12

36

3'

3

any

This principle may be applied to the following sums:

Ex. 1. Divide by 6. Ans. 3.

By uniting the two preceding principles, we have the following more comprehensive principle, viz. :

The value of a fraction is divided by a whole number, by dividing the numerator, or by multiplying the denominator by that number.

Note. The numerator should always be divided, when it can be done without a remainder. In all other cases the denominator should be multiplied.

From the preceding remarks and illustrations we learn, that whatever operation is performed on the numerator of a fraction, the SAME OPERATION IS PERFORMED on the VALUE of the fraction; but that the effect produced on the VALUE of any fraction, is the REVERSE of the OPERATION PERFORMED ON ITS DENOMINATOR.

6. A fraction is multiplied by a fraction, by multiplying the numerators together for a new numerator, and the denominators for a new denominator.

For example, let it be required to multiply by Agreeably to the principles already explained, if I multiply the denominator of the fraction, by 4, the other denominator, I shall obtain of that quantity, viz. ; and if I multiply this quantity, viz., by 3, the other numerator, I shall make this value three times as large, that is, it will become; therefore, 12 is of, or x=

3

In accordance with the above, the scholar may multiply the following fractions :