principal and the product is divided by 100, producing the sum £16 8 6, as the interest for 1 year or 12 months;

.

of which the fourth part £ 4.. 2 .. 14 is the interest for 3 months, as required.

At 4 per cent. required the interest of £ 5683.. 12, for the months of May, June, July and August.

Days

May 31

June 30

July 31

£ 227

Augt. 31

123

20

4546

Sh

12(

6 .. 10

Here 4 being the twenty-fifth part of 100, we divide the given principal by 25, and obtain the interest for a whole year, or 365 days; but as the given months contain only 123 days, we multiply this interest by 123 and divide the product by 365, whence we have £76. 12... for the interest required.

Again what is the Discount on a Bill for £ 580, for 90

days at 24 per cent ?

4|0)580(

£14.. 10

90

As 24 is the 40th part

of 100 we divide the giv

en principal by 40, to ob

tain the interest for 1 year, Sh. d. which multiplied by the

365) 1305. 00 (3 .. 11.6. given number of days, and

1095

By a Fraction in arithmetic is meant a quantity less than a given unit; thus one-half is a Fraction of a whole, that is, it means, as the term imports, a part broken off from a whole. To have a just conception of fractions we must suppose a unit to consist of a certain number of equal parts, of which parts one or more being taken, and one or more left, the part or parts taken and the part or parts left are equally fractions of that unit. Thus if we divide a pound of tea into 16 equal parts or ounces, each ounce, or any number of ounces less than 16, will be a fraction of the pound; and

one

one ounce would be called one-sixteenth part, five ounces would be five-sixteenth parts of a pound, &c.

Fractions are expressed in different ways; one is to give these fractional parts particular names, and then to employ them as integers; thus 12 ounces, although with regard to a pound as the integer, they may be considered as twelvesixteenth parts of a pound, may also be expressed as twelve units, when an ounce is the integer. In the same manner 12 shillings are a fraction, when a pound is the integer; but they are 12 units when the shilling is the integer.

The other mode of expressing a Fraction is to write under a line the number of parts into which the unit is divided, and above the line the number of those parts of which the fraction consists; thus 12 shillings would be expressed in a fraction by writing the number of shillings in a pound, 20, under a line, and the number of shillings in the given fraction, 12, aboye the line, thus, to be read twelve-twentieth parts, or briefly twelve-twentieths of a pound: and means that the unit is divided into two equal parts of which one is the given fraction, to be read one-half, Hence in expressing a sum of money, sixteen shillings eight pence and two farthings, for instance, it is customary to write it in this way 16sh. 84d, because four farthings forming one penny, two farthings must be one half of a penny: and five shillings and nine-pence three-farthings would be written 5sh. 9 d.

In this mode of representing a fraction the number under the line is termed the Denominator, because it denotes the number of parts into which the unit or integer is divided; and the number above the line is termed the Numerator, because it shows how many of such parts are contained in the given fraction.

In working with fractions it often happens that we obtain fractional numbers of which the numerator is greater than the denominator, as, or five quarters of a yard: this however

is not properly a fraction, but rather a mode of representing a number composed of an integer and a fraction, as 14 yard, that is one yard and one quarter: and the value of such an improper fraction is obtained by dividing the numerator by the denominator. In the same manner 27 is a way of expressing 53, and is obtained by multiplying the integer by the denominator of the fractional part, and to the product adding the given numerator.

10

Fractions of the same real value may be expressed in various ways according to the number of parts into which the integer is supposed to be divided: thus one-half, twofourths, three-sixths, five-tenths, all equally express the same fraction of any given integer: for if we divide any integer, a pound of tea, for instance, into 2 equal parts and take 1 of them, or into 4 parts and take 2, into 6 parts and take 5, or into 10 parts and take 5 of them, it is evident that we still take precisely the same quantity, that is, one-half of the pound of tea. We may therefore increase or diminish the numerator of any fraction as much as we choose, without in the least affecting the value of the fraction, provided the denominator be increased or diminished in precisely the same proportion: thus and are equal because both the numerator and the denominator of the first fraction are multiplied by the same number 5; and and are also equal, because the numerator and denominator of the first fraction are both divided by the same number 3, giving for quotients 3.

10

1st. In operations with what are called Vulgar Fractions, such as have just been described, the first thing to be known is how to reduce a mixed number to an improper fraction for example, to reduce 53, which is a mixed number consisting of an integer or whole number and a fraction, to an improper fraction, we place the figures as in the mar

gin,

3

making in all 17 for a numerator to the new fraction, under which we draw a line and write the former denominator 3, as a denominator to the new fraction: hence we have 17 an improper fraction of equal value with the mixed number 53. The reason of this is, that by the fractional part, we observe the integer to be divided into 3 equal parts, in which case 5 integers must contain 15 of such parts, and adding the 2 third parts of the given fraction, we have 17 third parts as the value of the whole mixed number given. In the same way the mixed number 251 will be reduced to the improper fraction 3, by multiplying the integral part 25 by the denominator of the

25

311 Numr.

12 Denr.

fractional part 12, and adding the numerator 11 to the product for a new numerator 311, and placing the former denominator 12 under it, as a denominator to the improper fraction thus obtained.

1 2

311

2d. To reduce an improper fraction to its equivalent mixed or whole number, as, for instance, to bring back the improper fraction of the foregoing example 11, to its equivalent whole or mixed number, we divide, as in the margin, the numerator 311 by the denominator 12, and the quotient 25 is the integral part of the required number, while the remainder 11 assumes the fractional form, by writing it as a numerator, and the divisor 12 as a denominator, making

12)311(25

24

71

60

11

12.

altogether