FRACTIONS.

It is not our intention to write a treatise on the part of arithmetic which stands at the head of this article, or to enter into the reasons why so many persons, who can solve a simple question in which there are nothing but whole numbers, are puzzled by anything which contains fractions. Our object is, to give some slight notions on this part of the subject to those who are already able to work the four rules in whole numbers.

When we add any two numbers together, it is understood that both of them have the same unit, or that both are some number of times the same thing. Thus, that two and three make five, means that two yards and three yards make five yards, or that two pounds and three pounds make five pounds, and so on. We do not in that case say anything of two yards and three feet, or of two pounds and three shillings. The following questions might arise:—If we have a distance which is neither six yards nor seven yards, but something between the two, how are we to represent this in numbers, and form rules for adding and subtracting this length to or from others of the same kind, without introducing a new measure, or talking of any other length except a yard? The answer to this will bring us, as we shall see, to the common meaning of the word fraction, and the way of representing a fraction. As we cannot measure anything exactly, we must first decide what degree of accuracy is necessary. This will vary in different operations, but we will suppose, for example’s sake, that a line may be rejected as insignificant, of which it would take more than a hundred to make a yard. If then we divide a yard into one hundred equal parts, and first remove the six whole yards which the above-mentioned distance contains, we have a remainder which does not contain all the hundred parts just mentioned, 286since it is less than one yard. Suppose that, on measuring the remainder, we find it to contain more than 53 and less than 54 of the hundred parts: if then we call it 53 parts out of a hundred of a yard, the error committed will be less than one part out of a hundred; that is, by what was supposed above, it will be sufficient to say that the length of the whole is 6 yards and 53 of the hundred equal parts which would compose another yard, or 53 hundredths of a yard. If we were inventing a system of arithmetic, we might choose among many different ways of representing this. For example, 6 yards 53₁₀₀ of a yard; 6 yards and 53÷100 of a yard; and so on. The common method is the following, 6⁵³⁄₁₀₀ yards, it being always understood that when we write two numbers under one another with a line between, the unit of which we speak, be it a yard, pound, acre, or any other, is cut into as many equal parts as are shown by the lower number, and as many of them are taken as is shown by the higher number. Thus, ⅞ of a mile is the length obtained by cutting a mile into 8 equal parts, and taking 7 of them, being of course less than the whole mile by one of these parts.

Such a fraction as we have described is less than the unit of which it is a part; but a whole number of units and a fraction may be represented together by the same method. If, in the preceding example, we had divided each of the six yards into 100 parts, there would have been 600 such parts, which, with the 53 parts furnished by the fraction, would have made 653, not of yards, but of the hundredth parts of yards. This we should represent by ⁶⁵³⁄₁₀₀, denoting that each of a succession of yards has been divided into 100 parts, out of which collection of parts 653 have been taken. The term fraction is applied equally to all cases; and with this extension of meaning, the unit itself maybe represented as a fraction, for one yard is ²⁄₂ yards, or ³⁄₃ yards, or ⁴⁄₄ yards, and so on.

The lower line of a fraction is called the denominator, and the upper the numerator: these are Latin words, which may be literally translated by the namer and the numberer; the first tells what sort of parts is taken, and the second how many of them are taken. The following propositions will serve for consideration, and also to familiarize the reader with the use of these terms. When the numerator is less than the denominator, the fraction is less than a unit. When the numerator is greater than the denominator, the fraction is greater than the unit. Of two fractions which have the same denominator, that is the greater which has the greater numerator. Of two fractions which have the same numerator, that is the greater which has the less denominator. It is usual to distinguish fractions which are less than the unit from those which are greater by calling the former proper, and the latter improper, fractions.

As yet we have only considered fractions of the unit; and it is always understood that a simple fraction, such as ⅞, is a fraction of the unit, or it is one yard or one pound which is divided into 8 parts. Fractions of other numbers are written by placing the number to be divided after the fraction of it which is to be taken, thus—¾ of 7, which means that 7 is to be divided into 4 parts, of which parts, 3 are taken. We now ask, what fraction of the unit is ¾ of 7, or into how many parts must one yard be cut, and how many times must one of those parts be repeated, so as to give the same length which arises from cutting seven yards into 4 parts, and taking 3 of them? It is obvious that ¾ of 7 yards is 7 times as much as ¾ of 1 yard, or simply ¾ and 3 quarters of a yard repeated 7 times is 21 quarters or ²¹⁄₄. Similarly ⅖ of 8 is ¹⁶⁄₅ of 1, or ¹⁶⁄₅. Hence it follows that ¼ of 3 is ¾, ⅑ of 13 is ¹³⁄₉, and so on. If therefore we take the eighth part of nine, we get the same as if we had repeated the eighth part of the unit nine times. We may therefore consider a fraction, such as ⅚, in two ways, either as the sixth part of five, or as the sixth part of unity repeated 5 times. It may sometimes be necessary to take a fraction of a fraction, such as ⅖ of ⅞, or having found ⅞ of 1, to divide it into five parts, and take two of them. We ask, what fraction of the unit would the result of this double operation give? The answer is multiply the two numerators together, and also the two denominators, which gives ¹⁴⁄₄₀, or two-fifths of seven-eighths of a yard is fourteen parts out of forty. To see the reason, let us first take the more simple case ⅕ of ⅛. It is plain that if we divide one yard into eight equal parts, and afterwards divide each of these parts into 5 equal parts, we have divided the whole yard into 8 times 5, or 40 equal parts. Consequently the fifth part of an eighth part is one fortieth of the whole, or ⅕ of ⅛ is ¹⁄₄₀. But one fifth of seven eighths will be 7 times as much as ⅕ of one eighth, and will therefore be ⁷⁄₄₀; again, two fifths of ⅞ will be twice as much as one fifth of ⅞, and will therefore be ¹⁴⁄₄₀, or ⅖ of ⅞ is ¹⁴⁄₄₀, according to the rule. In the same way ³⁄₇ of ¹¹⁄₁₀ is ³³⁄₇₀. This rule corresponds to the multiplication of whole numbers, and is therefore called multiplication of fractions. The connexion is not obvious at first, owing to a little difference in our manner of speaking about whole numbers and fractions. But if we were in the habit of saying that 2 multiplied by 6 is six of 2, in the same way as we say “six of them,” “six of his men,” it would appear natural to call those rules which tell us how many units there are in six of two, and what fraction of unit there is in ⅖ of ⅞, by the same name. By this rule all questions of fractions are solved, which would have required multiplication if they had been in whole numbers. For example, if 1 pound cost 2 shillings, 6 pounds will cost 6 times 2 shillings; similarly, if 1 pound costs ⅞ of a shilling, ⅖ of a pound will cost ⅖ of ⅞ of a shilling.

The most important proposition relating to fractions, being the one on which the rules most materially depend, is the following: If the numerator and denominator be either both multiplied or both divided by the same number, the value of the fraction is not altered. For example, take ⅗ and multiply its numerator and denominator by 4, which gives ¹²⁄₂₀. In the second fraction we cut the unit into four times as many parts as in the first, consequently each part of the unit signified in the second fraction is the fourth of that signified in the first. But in the second fraction, four times as many parts are taken as in the first, by which the balance is restored. Let us suppose that two yards of cloth are to be measured by a foot measure. The foot being ⅓ of the unit, and 6 of these being necessary, ⁶⁄₃ will be the fraction in yards, representing not only the number of yards measured, but in what parts of yards they were measured. No one would object to an inch measure, which is ¹⁄₁₂ of a foot, provided 12 times as many inches were given as there were feet in the first case. But one inch is ¹⁄₃₆ of a yard, and 12 times 6 is 72; and in this way of measuring ⁷²⁄₃₆ would represent the number of yards given, which is derived from ⁶⁄₃ by multiplying the numerator and denominator by 12. Similarly, one shilling, the unit being a pound, is ¹⁄₂₀; and 12 pence, the unit being also a pound, is ¹²⁄₂₄₀; and ¹⁄₂₀ and ¹²⁄₂₄₀ only differ in that the numerator and denominator of the first must be multiplied by 12 in order to make the second.

Hence it is allowable to multiply the numerator and denominator of a fraction by any number which is convenient, and which is called multiplicand, since that operation does not alter its value. Thus, ⅔, ⁴⁄₆, ⁶⁄₉, ⁸⁄₁₂, &c. are all of the same value, when the unit is the same in all: in common language, we should say, that two out of three is the same as four out of six, six out of nine, and so on. We are now able to remove two fractions which have different denominators, and substitute others of the same value with the same denominator. Take the fractions ⅔ and ⅘. If we ask which is the greater, no answer can at first be given, for though the second, is 4 287and the first 2, yet the second is four of the fifth parts only of unity, while the first is 2 of the third parts. But if we multiply the numerator and denominator of each fraction by the denominator of the other, the results will be ¹⁰⁄₁₅ and ¹²⁄₁₅, which have the same value ⅔ and ⅘, and also have the same denominator as each other. Hence we see that, ¹²⁄₁₅ being greater than ¹⁰⁄₁₅, ⅘ is greater than ⅔. The sum of the two is the fifteenth part of unity repeated 22 times or ²²⁄₁₅; the difference is two parts out of fifteen or ²⁄₁₅. Hence follow the common rules for addition and subtraction of fractions.

We now come to the reverse of multiplication. We have shown how to find the value of one fraction of another, such as ⅗ of ⁶⁄₁₁; we now ask, what fraction of ⅞ must be taken, to give ⅔ of 1 or simply ⅔? Into how many parts must we cut ⅞, and how many times must we repeat one of those parts, in order that the result may be the same as if we had cut unity into three parts, and taken 2 of them? Reduce the fractions ⅞ and ⅔ to other equivalent fractions having the same denominator, which are ²¹⁄₂₄ and ¹⁶⁄₂₄. If we cut ²¹⁄₂₄, which is ⅞, into twenty-one equal parts, each of these parts is ¹⁄₂₄; if we repeat ¹⁄₂₄ sixteen times, the result is ¹⁶⁄₂₄, which is ⅔: hence, if ⅞ be cut into 21 equal parts and 16 of these parts be taken, the resulting fraction is ⅔, or if we ask, what fraction of ⅞ is ⅔? the answer is ¹⁶⁄₂₁ of ⅞. By our former rule ¹⁶⁄₂₁ of ⅞ is ¹¹²⁄₁₆₈, which does not appear at first sight to be the same as ⅔, but if we examine its terms, we shall find that on dividing the numerator and denominator by 56 (which does not alter its value) it is reduced to ⅔. This rule being the reverse of multiplication is called division; the fraction which is to be cut into parts is called the divisor, that which is to be produced from it the dividend, and the fraction of the first, which it is necessary to take, in order to produce the second, is called the quotient. Thus, ¹⁶⁄₂₁ is the quotient of ⅔ divided by ⅞. The rule deduced from this reasoning is: Reverse the divisor, that is, for ⅞ write ⁸⁄₇, and proceed as in multiplication with the reversed divisor and the dividend. Thus, ⁸⁄₇ of ⅔ is ¹⁶⁄₂₁. This rule is used in every question where division would have been used, if whole numbers only had been given. Thus if 4 pounds cost 20 shillings, the price of one pound is found by dividing 20 by 4, and is 5 shillings. If ⅞ of a pound cost ⅔ of a shilling, the price of one pound is found by dividing ⅔ by ⅞ and is ¹⁶⁄₂₁ of a shilling. This might be established by independent reasoning as follows: As ⅞ of a pound costs ⅔ of a shilling, and 7 pounds cost 8 times as much as ⅞ of a pound, 7 pounds will cost ¹⁶⁄₃ of a shilling. But as the price of one pound is one-seventh of that of 7 pounds, for every third of a shilling which 7 pounds cost, one pound will cost the twenty-first part of a shilling. Hence the price of one pound is ¹⁶⁄₂₁ as before.

We shall proceed in a future number to the explanation of Decimal Fractions.