DIVISION OF VULGAR FRACTIONS. Division is performed by multiplying the numerator of the dividend by the denominator of the divisor, for a numerator to the quotient, and the denominator of the dividend by the numerator of the divisor, for a denominator to the quotient. For example, let it be required to divide by, we multiply the numerator of the dividend 3, by the denominator of the divisor 3, and the product 9 is the numerator of the quotient: again the numerator of the divisor 2 multiplied into the denominator of the dividend 4 gives 8 for the denominator of the quotient, which then becomes the improper fraction equal to 1. In this operation the object being to discover how often are contained in 4, it is evident that two thirds will be contained in any quantity only half the number of times that one third would be contained in it: we therefore divide the dividend by 2 and multiply the quotient by 3, or in other words, we take 3 times the half of the dividend; for the quotient required by the question proposed. Thus in the example given; bring the given dividend to some equivalent fraction that will admit of division by 2, as; the half of this is and 3 times the quotient is equal to 1, as before found. This will be illustrated if we suppose the fractions given to be parts of a foot for instance, in which the divisor will be 8 inches and the dividend will be 9 inches and here it is evident that 8 will be contained in 9, once with one over; or the quotient will be 14 as before. In dividing an integer by a fraction, or a fraction by an integer, you must bring the integer into a fractional form by writing 1 under it, for a denominator, and then working as in the preceding example: thus to divide 5 by the operation would be performed as in the margin, where the quotient by a fraction, you multiply the integer by the denominator of the fraction and divide the product by the numerator, and when a fraction is to be divided by an integer, you multiply the integer into the denominator of the dividend, for a new denominator to the numerator of the dividend. When mixed numbers are given either in the divisor or the dividend, or in both, they must be reduced to equivalent improper fractions; and then the division is performed as here shown. Divide 326 by 15. Here the divisor 15 is reduced to the improper fraction 135, and the dividend 3263 to 230; then multiplying alternately as directed, we have for a quotient, an improper fraction equal to the mixed number 209. In common accounts shillings and pence may be considered as fractions; that is the pence as fractions of a shilling and both pence and shillings as fractions of a pound: in the same manner lower denominations of any kind may be considered as fractions of higher denominations, aud operations where different denominators occur, may be performed by expressing the higher as integers, and the lower as fractions, to be worked with as in the preceding examples. Thus the lower denomination becomes a fraction by placing it as a numerator, with the value of the higher as the denominator, 2 K 2 minator, 5 pence for instance will be of 1 shilling and 12 2-0 2 of equal to of a pound. Again the value in lower denominations of the fraction of a higher denomination, is found by reduction, that is by multiplying the numerator by the units in the next inferior denomination and dividing the product by the denominator of the given fraction: thus the value of of a pound will be found to be 18sh. 4d. 11 20 12)220( 18 .. 4 In calculations it often happens that a quantity may be expressed as the fraction of a fraction; and this manner of expression may be carried through any number of stages. To reduce all these fractions to one of the same value, we multiply all the numerators together for a new numerator, and all the denominators together for a new denominator, thus as in the preceding example, where 5 pence are expressed as of, of a pound, Here multiplying the numerator 5 by 1, we have 5, and the denominators 12 by 20, we have 240 to form the new fraction of a pound. In the same way 6 ounces will be represented as of of of of a Ton, and multiplying all the numerators together for a numerator, and all the denominators together for a denominator, we have the fraction 35 or 120 of a Ton for the value of 6 ounces. Σ 6 4 Having thus shown the method of calculation by Vulgar Fractions, it remains to give some explanation of the nature and uses of what are called Decimal fractions. In money, weight, capacity, dimensions, &e. the unit or integer has by common consent been divided into various numbers of smaller parts: but as these divisions have been optional optional and arbitrary, and that calculations by them are frequently tedious and consequently liable to mistake; it has been agreed upon to suppose integers of all kinds to be divided into ten equal parts, which are hence termed tenths or decimals from the Latin word decem signifying ten. Each of these decinal parts is again divided into ten other equal parts called hundredth parts; these into ten others called thousandth parts; and so on indefinitely. These decimal parts with all their subdivisions, are represented by the same numerical characters or cyphers as integers; but they are distinguished from integers by having a comma placed before them, or on their left hand; thus 5 without any point, represents five integers; but,5 with a comma before it stands for five tenth parts or decimals of an integer and 8,5 would be read eight integers and five tenths. The numeration of Decimals is just the reverse of Integers; for as these last increase in value in proportion as they recede from the right hand to the left; so those diminish in value in proportion as they recede from the left hand to the right. From the nature of a decimal fraction it is evident that the first place after the point of separation between them and integers must be that of tenths, the next to the right hand that of hundredth parts, as being tenths of tenths; the third place is that of thousandths; and so on, as in the following Table. Here the first figure after the point is 1 tenth of an unit; the second is 2 hundredth parts, to which adding the 1 tenth, which is equal to ten hundredth parts, we have, 12 representing 12 hundredths: the third figure 3 is 3 thousandth parts, and joined to the preceding figures we have, 123 thousandths; the fourth figure is 4 tenthousandth parts which being joined to the preceding figures, we have,1234 ten thousandth parts of an integer: so that it is to be remembered in general that, although the value of decimals diminishes in a tenfold proportion with respect to an integer, as they recede to the right hand from the point of separation, yet in reading them, their value relatively to themselves, is reckoned from right to left as in numeration of integers. When a nought is placed on the right hand of an integer, the value of the integer is increased tenfold; thus 5 stands for five, but 50 for fifty; but one or more noughts on the left hand have no effect on the value: on the contrary, one or more noughts placed on the right hand of a decimal fraction have no effect on its value; but for every nought on its left hand, the value is diminished ten fold: thus ,5, ,50, or ,5000, are all but five tenths; whereas,,05 will represent five hundredth parts, ,005, five thousandth parts, and ,000005 will be five hundred thousandth parts. For example the characters 1808 will have very different values according to the position of the separating point: thus 1808, are 1 thousand 8 hundred and 8. 180,8 1 hundred and 80, and 8 tenths 18,08 eighteen and 8 hundredth parts 1,808 one and 808 thousandth parts. ,1808 one thousand 808 ten thousandth parts 1808 hundred thousandth parts ,001808 1808 millionth parts 1st. To reduce Vulgar fractions to Decimal fractions. To the numerator add a number of noughts, separated from the |