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 3263 by 15%. Here the divisor 15% is reduced to the improper fraction 125, and the dividend 3263 to 25; 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, and 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 5 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. 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 munerator 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 1 of 4 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 1 or 2 of a Ton for the value of 6 ounces. 246 179 1 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, &c. 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 decimal 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 one thousand 808 ten thousandth parts 1st. To reduce Vulgar fractions to Decimal fractions. To the numerator add a number of noughts, separated from the the integers by a point, and divide it by the given denominator. Reduce to a decimal fraction. before the fourth figure the distinguishing comma or point, as is done here; and the result of the operation is that the decimal fraction ,7794 is of equal value with the Vulgar fraction; but not precisely, because there is a remainder in the division, which is lost: but if the division be earried on, by the addition of more noughts to the dividend, the quotient will approach gradually nearer to the truth. The reason of this operation is that to reduce a vulgar to a decimal fraction is in fact only to find the proportion between two fractions, of which the first has any given denominator and the second has the constant denominator unity or 1. The example here given may therefore be thus stated as 68 the given denominator to its numerator 53, so the constant denominator 1, to its numerator :-68 : 53 :: 1 : where if the middle term 53 be multiplied by 1, and the product divided by 68, the quotient will be the fourth proportional required: but as multiplying by 1 produces no effect on the middle term, the operation is abridged by at once dividing this term, with the addition of some noughts, by the given denominator 68; and the quotient may be represented as a vulgar fraction, to correspond to that given in the question; thus equal to 7, or 7794. 190009 |