Lesson 4 · from §64 of the 1910 text
Adding and Subtracting Fractions
Fractions with the same denominator are easy: add or subtract the numerators and keep the denominator. \frac{a}{7} + \frac{2a}{7} + \frac{3a}{7} = \frac{6a}{7}, \qquad \frac{a}{x} - \frac{b}{x} = \frac{a - b}{x}.
With different denominators, first change to equivalent fractions with a common denominator. In arithmetic, \frac12 + \frac23 + \frac34 = \frac{6}{12} + \frac{8}{12} + \frac{9}{12} = \frac{23}{12}. In algebra, \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad}{bd} + \dfrac{bc}{bd} = \dfrac{ad + bc}{bd}.
Example 1. \dfrac{6x + 1}{3x^2} + \dfrac{3x - 5y}{4xy}. The L.C.D. is 12x^2y: \frac{4y(6x + 1)}{12x^2y} + \frac{3x(3x - 5y)}{12x^2y} = \frac{24xy + 4y + 9x^2 - 15xy}{12x^2y} = \frac{9x^2 + 9xy + 4y}{12x^2y}. Check with x = y = 1: \frac73 - \frac24 = \frac{22}{12} and \frac{9 + 9 + 4}{12} = \frac{22}{12}. ✓
Example 2. \dfrac{2x - 3}{x^2 - 9} - \dfrac{3x - 4}{x^2 - 5x + 6} + \dfrac{-2}{x + 3}. Factor the denominators: the L.C.D. is (x + 3)(x - 3)(x - 2). Then \frac{(2x - 3)(x - 2) - (3x - 4)(x + 3) - 2(x - 3)(x - 2)}{(x + 3)(x - 3)(x - 2)} = \frac{-3x^2 - 2x + 6}{(x + 3)(x - 3)(x - 2)}. Notice each numerator goes in parentheses with the sign of its fraction in front. That's where most mistakes happen.
Example 3. 3x - 4 - \dfrac{x^2 - 5x}{x + 2}. Think of 3x - 4 as \dfrac{3x - 4}{1}: \frac{(3x - 4)(x + 2) - (x^2 - 5x)}{x + 2} = \frac{3x^2 + 2x - 8 - x^2 + 5x}{x + 2} = \frac{2x^2 + 7x - 8}{x + 2}.
When you check by substituting numbers, pick values that don't make any denominator zero. In Example 2, x = 2, 3, or -3 won't work.
Rule. Change the fractions to equivalent fractions with the lowest common denominator. Write the numerators over the L.C.D., each in parentheses with its fraction's sign in front. Remove the parentheses, combine like terms, and reduce to lowest terms if you can.
Quick check
\displaystyle \frac{2}{x}+\frac{3}{x+1}
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Find the algebraic sum, as a single fraction in lowest terms.
\displaystyle \frac{2x}{3}+\frac{3x}{5}
\displaystyle \frac{2a}{3}-\frac{5a}{6}+\frac{a}{4}
\displaystyle \frac{3c}{7}+\frac{c}{21}-\frac{1}{3}
\displaystyle \frac{9a}{16}-\frac{12m}{8}-\frac{5m}{24}
\displaystyle \frac{x-3}{4}+\frac{2x-5}{6}
\displaystyle \frac{5a+7}{14}-\frac{9a+x}{21}
\displaystyle \frac{5c}{3}-\frac{3c-x}{5}+\frac{x-3c}{10}
\displaystyle \frac{4m-3}{6}-\frac{7-9m}{9}-\frac{3a+5m-4}{27}
Letters in the denominators now.
\displaystyle \frac{2}{a}+\frac{3}{4a^2}
\displaystyle \frac{3}{x^2}+\frac{4}{x}-\frac{8}{3x^3}
\displaystyle \frac{x}{y}-\frac{5}{2y^3}+\frac{a}{4}
\displaystyle \frac{a}{b}+\frac{b}{c}-\frac{c}{a}
\displaystyle \frac{7}{m^2}-\frac{5}{mn}+\frac{6}{n^2}
\displaystyle \frac{3}{xy}-\frac{5}{2x^2}-\frac{6}{5xy^3}
\displaystyle \frac{3x-1}{6x^2}-\frac{7-2x^2}{4x^3}+\frac{5x^3-8}{9x^4}
\displaystyle \frac{4c^2-9}{2cd}-\frac{6-c}{5c^2}-\frac{c^2-4}{cd^2}
\displaystyle \frac{3+x}{x-3}+\frac{3}{4}
\displaystyle \frac{3}{x-5}-\frac{2}{x+5}
\displaystyle \frac{5x}{x^2+xy}-\frac{7}{x}
\displaystyle \frac{3a-b}{a^2-b^2}+\frac{2}{a-b}
Factor every denominator first, as in Example 2.
\displaystyle \frac{10}{25-m^2}-\frac{1}{m^2+16m+55}
\displaystyle \frac{3}{a^2-16}+\frac{5}{a^2-6a+8}
\displaystyle \frac{2x+1}{x^2-1}+\frac{4}{x^2-3x+2}
\displaystyle \frac{c-5}{c^2-6c}+\frac{2c-3}{c^2-8c+12}
\displaystyle \frac{x^2-3xy+y^2}{9-6x+x^2}-\frac{5x-3y}{9-3x}
\displaystyle \frac{x+2}{x^2+x}+\frac{1}{x}+\frac{3-x}{x^2+2x+1}
\displaystyle \frac{a+b}{a^2-ab}+\frac{a-2b}{a^2-2ab+b^2}-\frac{2a-b}{a^2-b^2}
\displaystyle \frac{m-2n}{m^2+mn+n^2}-\frac{m^2-3n^2}{m^3-n^3}+\frac{3m-n}{m-n}
\displaystyle \frac{c^2+cd+d^2}{c^2-cd+d^2}-\frac{c-d}{2c+2d}-\frac{2c^2+5cd}{c^3+d^3}
\displaystyle \frac{x+y}{x^3-y^3}-\frac{x-2y}{(x-y)^3}
Whole expressions mixed with fractions: write the whole part over 1, as in Example 3.
\displaystyle R^2+\frac{R^2}{4}
\displaystyle x-3-\frac{x+3}{5}
\displaystyle \frac{m^2+n^2}{m-n}+n+m
\displaystyle 3a+b-\frac{a^2-3b^2}{3a-b}
\displaystyle x^2+y^2-\frac{3x^3-y^3}{x+y}-xy
\displaystyle m^2-\frac{m^4-2m^2}{m^2+m+1}+1-m
\displaystyle \frac{m^3+8n^3}{m^2+2mn+4n^2}+m-2n
\displaystyle a^2-\frac{a^4+2a^2b^2-b^4}{a^2+ab+b^2}-ab+b^2
\displaystyle x^3+x^2y-\frac{x^4+3y^4}{x-y}+xy^2+y^3
\displaystyle \frac{2c^2+d^2}{c^2-d^2}-\frac{c}{c+d}+5
\displaystyle 6+\frac{6r^2-s^2}{r^2-9rs+14s^2}-\frac{3r-5s}{r-7s}+5
Remove the parentheses first (mind the signs), then combine.
\displaystyle \left(x-3-\frac{3}{x+3}\right)-\left(2x-7-\frac{4}{x+4}\right)
\displaystyle \left(5m+\frac{6m}{2n}\right)-\left(m-\frac{m-3n}{n}\right)
\displaystyle \left(7c-\frac{2d}{3de}\right)-\left(5c+\frac{c}{3de}\right)
\displaystyle \left(\frac{2a}{a-b}-5a\right)-\left(2a-\frac{4a}{a+b}\right)
\displaystyle \left(2a-3b+\frac{3a}{2a+b}\right)-\left(3a+2b-\frac{5a}{2a-b}\right)
\displaystyle \left(-\frac{4r^2}{2r+s}+2r-s\right)-\left(-\frac{4s^2}{2s-3r}+2s+3r\right)