than an integer, that is to say, a numerator greater than the denominator: a case which has already presented itself, and which deserves attention. We have found for example (Art.111.) that the sum of the 5 fractions,,,,, was 213, and have observed that this sum may be expressed 3 and 3 or 1. In all these cases we have only to divide the numerator by the denominator, and to the quotient add the remainder, in the form of a fraction, which should be reduced to its lowest

terms.

117. In nearly the same manner are added together quantities composed of fractions and whole numbers, which are usually called mixed numbers. The fractions are first added together, and if the sum produces one or more integers, these must be added to the other integers. If, for example, it were required to add together 31 and 2, we should first add to, or to, of which the sum is 71, and then 3+2+1=61.

118. Since an improper fraction is thus reduced to a mixed number, by dividing the numerator by the denominator and retaining the remainder in the form of a fraction, it will be easily seen, that in order to express a mixed number fractionally, we have only to multiply the integer by the denominator of the fraction and adding the numerator to the product to place the denominator under the sum; and by this method we have another way of finding the value of the above mixed numbers, 34 +2. For 3 reduced to an improper fraction is, and 2 is . These fractions reduced to a common denominator become 1 and 2, the sum of which is 376, as before.

CHAP. XII.

OF THE MULTIPLICATION AND DIVISION OF FRACTIONS.

119. THE rule for the multiplication of a fraction by a whole number, is to multiply the numerator only by that number, without changing the denominator, thus

120. We may, however, instead of this rule, adopt that of dividing the denominator of the fraction by the given number, and where this can be done, it is the better method, as it tends very much to abridge the calculation.

If it were proposed for instance to multiply by 3, if we multiply the numerator by 3 we shall obtain, which by reduction becomes. But, if instead of multiplying the numerator we had divided the denominator at once by the integer, we should immediately have obtained the quotient as before. In like manner 4 multiplied by 6, produces or 34. (e)

121. Ge

(e) The truth of this reasoning may perhaps appear more evident if we consider attentively the nature of a fraction as pointed out in Art. 98, viz. that the larger the denominator of a fraction be, the smaller is the fraction itself, or the quotient represented by it. Now if it be required to divide the fraction by 5, it is evident that the resulting fraction must be three times as small as, and therefore that the denominator must be three times as great, and consequently the resulting fraction will evidently be. In other words, the fraction must be divided into three equal parts, of which the resulting fraction or quotient shall be one. Now this must evidently be done by multiplying the denominator 9 by 3 ;for as in the first

instance

121. Generally, then, the product of the multiplication of

a

b

a c

any fraction by c, is; and we may remark that when the whole number is exactly equal to the denominator of the fraction, the product will be equal to the numerator.

a

2

3.

a

And in general if the fraction be multiplied by b, the pro. duct will be a, as we have already seen; for since represents the quotient of the division of the dividend a by the divisor b, and since it has been demonstrated that the quotient multiplied by the divisor gives the dividend, it is evident that xb must produce a.

122. Having seen how a fraction is to be multiplied by a whole number, we now proceed to consider how it may be divided. This consideration is necessary before we proceed to the multiplication of fractions by fractions. Now it is clear that if we have to divide the fraction by 2, the quotient will be, and that the quotient of divided by 3 is 4. We have therefore only to divide the numerator by the whole number without changing the denominator; as the fractions

was

instance unity or 1 was divided into 9 parts, of which the fraction one part, so in the second instance it is required to subdivide that part into 3 times as many parts, and to take one of those parts for the quotient, and as divided into 3 equal parts is evidently, so one of those parts or is the quotient sought.

But if it be required to multiply the fraction. by 3, it is clear that the product must be three times as great as, or that the denominator must be three times as small, viz., that 9 must be divided by 3, and consequently that the resulting fraction will be .

123. This

123. This rule may be practised without any difficulty, when the numerator is divisible by the number proposed, but it is frequently not so divisible, and we must therefore observe, that as a fraction may be changed to an infinite variety of other fractions without altering its value, we cannot fail to arrive at some, of which the numerator may be divisible by the number sought. If, for example, it were required to divide by 2, we might change the fraction into §, and then dividing the numerator of this fraction by 2, we should obtain for the quotient sought.

ac

a

124. In general, therefore, if it were proposed to divide the fraction by c, we might transform this fraction into and then dividing the numerator by c, we should obtain for the quotient sought. From which it appears that where any fraction is to be divided by a whole number c, we have only to multiply the denominator of the fraction by the whole number, without altering the numerator, and the resulting fraction will be the quotient sought. As for instance, divided by 3 makes, and divided by 5 makes.

a

125. It should be remembered, however, that this expedient is only to be resorted to when the numerator itself is not divisible by the given whole number, as otherwise the rule given in Art. 122. will be found the most expeditious method of arriving at the true quotient. For example, divided by 3, would by this rule produce, but by the first rule, which may in this case be applied, we should obtain immediately the quotient, which, although its value be the same as is expressed in a simpler form.

a

126. We are now able to comprehend how one fraction is to be multiplied by another fraction as 7. For this purpose

C

с

we have only to observe that the expression signifies that c is divided by d. We shall therefore first multiply the frac

a

tion by c, by which we obtain the product, which being

а в

divided by d gives

b d

We therefore obtain the following

rule for the multiplication of fractions.

Multiply the numerators together for a new numerator, and the denominators for a new denominator.

127. It remains to show how one fraction is to be divided by another. Now we must remark, first, that if the two fractions have a common denominator, the division takes place only with their numerators, for it is evident that are coǹtained as many times in as 3 is contained in 9, viz. 3 times. And in like manner to divide by, we have only to divide 8 by 9, which gives . We shall also have;

in

3 times; in 47 times; in, 4, &c., or generally

128. But if the fractions have not equal denominators, it becomes necessary to have recourse to the method before shown of reducing them to a common denominator.

a

b

If, for example, the fraction is to be divided by, we must reduce them to a common denominator, by which we shall obtain to be divided by and it is then clear from

ad

b d

b c

the above examples that the quotient will be following rule.

Multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor; the first product will be the numerator of the quotient, and the second product the denominator.

129. Fol