plier is a whole number, it shows how many times the multiplicand is to be repeated; but the term multiplication, extended to fractional expressions, does not always imply augmentation, as in the case of whole numbers. To comprehend in one statement every possible case, it may be said, that to multiply one number by another is, to form a number by means of the first, in the same manner as the second is formed, by means of unity. In reality, when it is required to multiply by 2, by 3, &c. the product consists of twice, three times, &c. the multiplicand, in the same way as the multiplier consists of two, three, &c. units; and to multiply any number by a fraction, for example, is to take the fifth part of it, because the multiplier, being the fifth part of unity, shows that the product ought to be the fifth part of the multiplicand*.

Also, to multiply any number by is to take out of this number or the multiplicand, a part, which shall be four fifths of it, or equal to four times one fifth.

Hence it follows, that the object in multiplying by a fraction, whatever may be the multiplicand, is, to take out of the multiplicand a part, denoted by the multiplying fraction; and that this operation is composed of two others, namely, a division and a multiplication, in which the divisor and multiplier are whole numbers.

Thus, for instance, to take of any number, it is first necessary to find the fifth part, by dividing the number by 5, and to repeat this fifth part four times, by multiplying it by 4.

We see, in general, that the multiplicand must be divided by the denominator of the multiplying fraction, and the quotient be multiplied by its numerator.

The multiplier being less than unity, the product will be smaller than the multiplicand, to which it would be only equal, if the multiplier were 1.

67. If the multiplicand be a whole number divisible by 5,

for

* We are led to this statement, by a question which often presents itself; namely, where the price of any quantity of a thing is required, the price of the unity of the thing being known. The question evidently remains the same, whether the given quantity be greater or less than this unity.

instance, 35, the fifth part will be 7; this result, multiplied by 4, will give 28 for the of 35, or for the product of 35 by . If the multiplicand, always a whole number, be not exactly divisible by 5, as, for instance, if it were 32, the division by 5 will give for a quotient 63; this quotient repeated 4 times will give 24. This result presents a fraction in which the numerator exceeds the denominator, but this may be easily explained. The expression, in reality denoting 8 parts, of which 5, taken together, make unity, it follows, that is equivalent to unity added to three fifths of unity, or 13; adding this part to the 24 units, we have 25% for the value of of 32.

68. It is evident, from the preceding example, that the fraction contains unity, or a whole one, and 3, and the reasoning, which led to this conclusion, shows also, that every fractional expression, of which the numerator exceeds the denominator, contains one or more units, or whole ones, and that these whole ones may be extracted by dividing the numerator by the denominator; the quotient is the number of units contained in the fraction, and the remainder, written as a fraction, is that, which must accompany the whole ones.

The expression 307, for instance, denoting 307 parts, of which 53 make unity, there are, in the quantity represented by this expression, as many whole ones, as the number of times 53 is contained in 307; if the division be performed, we shall obtain 5 for the quotient, and 42 for the remainder; these 42 are fifty third parts of unity; thus, instead of 307, may be written 533.

Examples for practice.

Reduce the fraction to its equivalent whole number.

Ans. 2.

Reduce to its equivalent whole or mixed number. Ans. 31. Reduce to its equivalent whole or mixed number.

Ans. S.

Reduce 482 to its equivalent whole or mixed number.

Ans. 24.

Ans. 121.

Reduce 2 to its equivalent whole or mixed number.

Reduce 513 to its equivalent whole or mixed number.

Ans. 10.

69. The expression 543, in which the whole number is given, being composed of two different parts, we have often occasion to convert it into the original expression 307, which is called, reducing a whole number to a fraction.

To do this, the whole number is to be multiplied by the denomi nator of the accompanying fraction, the numerator to be added to the product, and the denominator of the same fraction to be given to the

In this case, the 5 whole ones must be converted into fiftythirds, which is done by multiplying 53 by 5, because each unit must contain 53 parts; the result will be with the second, 3, the answer will be 307.

Examples for practice.

; joining this part

70. We now proceed to the multiplication of one fraction by another.

If, for instance, were to be multiplied by ; according to article 66, the operation would consist in dividing by 5, and multiplying the result by 4; according to the table in article 65, the first operation is performed by multiplying 3, the denominator of the multiplicand, by 5; and the second, by multiplying 2, the numerator of the multiplicand, by 4; and the required product is thus found to be 13.

It will be the same with every other example, and it must consequently be concluded from what precedes, that to obtain the product of two fractions, the two numerators must be multiplied, one by the other, and under the product must be placed the product of the denominators.

Examples.

Multiply by. Ans. 33.

Multiply by 2. Ans. 783•

71. It may sometimes happen that two mixed numbers, or whole numbers joined with fractions, are to be multiplied, one by the other, as, for instance, 3 by 4%. The most simple mode of obtaining the product is, to reduce the whole numbers to fractions by the procees in article 69; the two factors will then be expressed by 26 and ^^, and their product, by 1144 or 1818, by extracting the whole ones (68).

44

72. The name fractions of fractions is sometimes given to the product of several fractions; in this sense we say, of. This expression denotes of the quantity represented by of the original unit, and taken in its stead for unity. These two fractions are reduced to one by multiplication (70), and the result,, expresses the value of the quantity required, with relation to the original unit; that is, of the quantity represented by of unity is equivalent to of unity. If it were required to take of this result, it would amount to taking 7 of of, and these fractions, reduced to one, would give for the value of the quantity sought, with relation to the original unit.

56

135

73. The word contain, in its strict sense, is not more proper in the different cases presented by division, than the word repeat in those presented by multiplication; for it cannot be said that the dividend contains the divisor, when it is less than the latter; the expression is generally used, but only by analogy and extension.

To generalize division, the dividend must be considered as having the same relation to the quotient, that the divisor has to unity, because the divisor and quotient are the two factors of the dividend (36). This consideration is conformable to every case that division can present. When, for instance, the divisor is 5, the dividend is equal to 5 times the quotient, and, consequently, this last is the fifth part of the dividend. If the divisor be a fraction, for instance, the dividend cannot be but half of the quotient, or the latter must be double the former.

The definition, just given, easily suggests the mode of proceeding, when the divisor is a fraction. Let us take, for ex

ample,. In this case the dividend ought to be only of the quotient; but being of, we shall have of the quotient, by taking of the dividend, or dividing it by 4. Thus knowing of the quotient, we have only to take it 5 times, or multiply it by 5, to obtain the quotient. In this operation the dividend is divided by 4 and multiplied by 5, which is the same as taking of the dividend, or multiplying it by, which fraction is no other than the divisor inverted.

This example shows, that, in general, to divide any number by a fraction, it must be multiplied by the fraction inverted.

For instance, let it be required to divide 9 by ; this will be done by multiplying it by 4, and the quotient will be found to be 36 or 12. Also 13 divided by will be the same as 13 multiplied by or 2. The required quotient will be 181, by extracting the whole ones (68).

It is evident that, whenever the numerator of the divisor is less than the denominator, the quotient will exceed the dividend, because the divisor in that case, being less than unity, must be contained in the dividend a greater number of times, than unity is, which, taken for a divisor, always gives a quotient exactly the same as the dividend.

74. When the dividend is a fraction, the operation must be performed by multiplying the dividend by the divisor inverted (70). Let it be required to divide by ; according to the preceding article, must be multiplied by 3, which gives 2.

It is evident, that the above operation may be enunciated thus; To divide one fraction by another, the numerator of the first must be multiplied by the denominator of the second, and the denominator of the first, by the numerator of the second.

If there be whole numbers joined to the given fractions, they must be reduced to fractions, and the above rule applied to the results.