Addition of Fractions

145. In adding fractions, we may either write them one after the other, with their signs, as in the addition of integers, or we may incorporate them into a single fraction, by the following rule:
Reduce the fractions to a common denominator, make the signs before them all positive, and then add their numerators.

The common denominator shows into what parts the integral unit is supposed to be divided ; and the numerators show the number of these parts belonging to each of tlte fractions (Art 131.) Therefore the numerators taken together sliov the whole number of parts in all the fractions.

Tnus, $\frac{2}{7}=\frac{1}{7}+\frac{1}{7}$. And $\frac{3}{7}=\frac{1}{7}+\frac{1}{7}+\frac{1}{7}$.

Therefore, $\frac{2}{7}+\frac{3}{7}=\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}+\frac{1}{7}=\frac{5}{7}$.

The numerators are added, according to the rules for the addition of integers. It is obvious that the gum is to be placed over the common denominator. To avoid the perplexity which might be occasioned by the signs, it will be expedient to make those prefixed to the fractions uniformly positive. But in doing this, care must be taken not to alter the value. This will be preserved, if all the signs in the numerator are changed at the same time with that before the fraction. (Art. 141)

Ex.1 Add $\frac{2}{16}$ and $\frac{4}{16}$ of a pound. Ans. $\frac{2+4}{16}$ or $\frac{6}{16}$.

It is as evident that $\frac{2}{16}$, and $\frac{4}{16}$ of a pound, are $\frac{6}{16}$ of a pound, as that $2$ ounces and $4$ ounces, are $6$ ounces.

2. Add $\frac{a}{b}$ and $\frac{c}{d}$. First reduce them to a common denominator. They will then be $\frac{ad}{bd}$ and $\frac{bc}{bd}$, and their sum $\frac{ad+bc}{bd}$.

3. $\frac{a}{d}$ and $-\frac{b-m}{y}=\frac{a}{b}+\frac{(-b+m)}{y}=\frac{(ay-bd+db)}{dy}$.

4. $\frac{a}{a+b}$ and $\frac{b}{a-b}=\frac{aa-ab+ab+bb}{aa+ab-ab-bb}=\frac{aa+bb}{aa-bb}$.

146. For many purposes, it is sufficient to add fractions in the same manner As integers are added, by writing them one after another with their signs.

Thus the sum of $\frac{a}{b}$ and $\frac{3}{y}$ and $\frac{-d}{2m}$, is $\frac{a}{b}+\frac{3}{y}-\frac{d}{2m}$.

In the same manner, fractions and integers may be added.

The sum of $a$ and $\frac{d}{y}$ and $3m$ and $\frac{-h}{r}$, is $a+3m+\frac{d}{y}-\frac{h}{r}$.

147. Or the integer may be incorporated with the fraction, by converting the former into A fraction, and then adding Iho numerators. See Art 138.

The sum of $a$ and $\frac{b}{m}$, is $\frac{a}{1}+\frac{b}{m}=\frac{am}{m}+\frac{b}{m}=\frac{(am+b)}{m}$.

Incorporatmg an integer with a fraction, is the same as reducing a mixed quantity to an improper fraction. For a mixed quantity is an integer and a fraction. In arithmetic, these are .generally placed together, without any sign between them. But in algebra, they are distinct tenns. Thus $2$ and $\frac{1}{3}$ is the same as $2+\frac{1}{3}$.

Ex. 1. Reduce $a+\frac{1}{b}$ to an improper fraction. Ans. $\frac{ab+1}{b}$.

2. Reduce $1+\frac{d}{b}$. Ans. $\frac{b+d}{b}$.

3. Reduce $b + \frac{c}{d-y}$.

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