Elements of Algebra

CHAP. IX. OF ALGEBRA. 29 for, to resolve this, we have only to reduce the two fractions to the same denominator. Let us take, for example, the two fractions $2/3$ and $5/7$; when reduced to the same denominator, the first becomes $14/21$, and the second $15/21$, where it is evident that the second, or $5/7$, is the greater, and exceeds the former by $1/21$. Again, if the fractions $3/5$ and $5/8$ be proposed, we shall have to substitute for them $24/40$ and $25/40$; whence we may conclude that $5/8$ exceeds $3/5$, but only by $1/40$. 99. When it is required to subtract a fraction from an integer, it is sufficient to change one of the units of that integer into a fraction, which has the same denominator as that which is to be subtracted; then in the rest of the operation there is no difficulty. If it be required, for example, to subtract $2/3$ from 1, we write $3/3$ instead of 1, and say that $2/3$ taken from $3/3$ leaves the remainder $1/3$. So $5/12$, subtracted from 1, leaves $7/12$. If it were required to subtract $3/4$ from 2, we should write 1 and $4/4$ instead of 2, and should then immediately see that after the subtraction there must remain $1\frac{1}{4}$. 100. It happens also sometimes, that having added two or more fractions together, we obtain more than an integer; that is to say, a numerator greater than the denominator: this is a case which has already occurred, and deserves attention. We found, for example [Article 96], that the sum of the five fractions $1/2$, $2/3$, $3/4$, $4/5$, and $5/6$ was $213/60$, and remarked that the value of this sum was $3\frac{33}{60}$ or $3\frac{11}{20}$. Likewise, $8/9 + 7/9$, or $\frac{8+7}{9}$, makes $15/9$, or $1\frac{6}{9}$. We have therefore only to perform the actual division of the numerator by the denominator, to see how many integers there are for the quotient, and to set down the remainder. Nearly the same must be done to add together numbers compounded of integers and fractions; we first add the fractions, and if the sum produces one or more integers, these are added to the other integers. If it be proposed, for example, to add $3\frac{1}{2}$ and $2\frac{2}{3}$; we first take the sum of $1/2$ and $2/3$, or of $3/6$ and $4/6$, which is $7/6$, or $1\frac{1}{6}$; and thus we find the total sum to be $6\frac{1}{6}$. QUESTIONS FOR PRACTICE. 1. Reduce $\frac{2x}{a}$ and $\frac{b}{c}$ to a common denominator. Ans. $\frac{2cx}{ac}$ and $\frac{ab}{ac}$.