Algebra, as you know, is a branch of mathematics that is dedicated to the manipulation of the letter and their coefficient.

Fractions that have algebra at its denominator and numerator are called algebraic fractions.

More appropriately, an algebraic fraction is one with algebra in the numerator and/or the denominator.

Adding algebraic fractions is straightforward. As with common fractions, we take the L.C.D so that the denominator becomes the same.

Example 1
Solve $\frac{1}{a}+\frac{2}{b}$

Solution:
By taking the L.C.D
$\frac{1(b)+2(a)}{ab}$
$\frac{b+2a}{ab}$
$\frac{2a+b}{ab}$

Example 2
Solve $\frac{4}{3y}-\frac{1}{y}$

Solution:
By taking the L.C.D
$\frac{4(1)-1(3)}{3y}$
$\frac{4-3}{3y}$
$\frac{1}{3y}$

Example 3
Solve the algebraic fractions $\frac{1}{x+2}+\frac{3}{x+1}$

Solution:
Taking the L.C.D
$\frac{1(x+1)+3(x+2)}{(x+2)(x+1)}$
$\frac{x+1+3x+6}{(x+2)(x+1)}$
$\frac{4x+7}{(x+2)(x+1)}$

Example 4
Solve $\frac{1}{x+2}+\frac{3}{x+1}$

Solution:
Taking the L.C.D
$\frac{1(x+1)+3(x+2)}{(x+2)(x+1)}$
$\frac{x+1+3x+6}{(x+2)(x+1)}$
$\frac{4x+7}{(x+2)(x+1)}$

Example 5
Solve $\frac{t}{t^2-1}+\frac{2}{t-1}$

Solution:
$\frac{t}{(t+1)(t-1}+\frac{2}{t-1}$

Taking the L.C.D
$\frac{t(1)+2(t-1)}{(t+1)(t-1)}$
$\frac{t+2t-2}{(t+1)(t-1)}$
$\frac{3t-2}{(t+1)(t-1)}$

Example 6
Solve $\frac{1}{x-2}+\frac{5}{x^2+x-6}$

Solution:
$x^2+x-6$ can be factorized to $(x-2)(x+3)$, therefore.
$\frac{1}{x-2}+\frac{5}{(x-2)(x+3)}$

Taking the L.C.D
$\frac{1(x+3)+5(1)}{(x-2)(x+3)}$
$\frac{x+3+5}{(x-2)(x+3)}$
$\frac{x+8}{(x-2)(x+3)}$

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Example 7
Solve $\frac{2}{w-1}+\frac{w-11}{w^2+3w-4}$

Solution:
$w^2+3w-4$ can be factorized to $(w-1)(w+4)$, therefore
$\frac{2}{w-1}+\frac{w-11}{(w-1)(w+4)}$

Taking the L.C.D
$\frac{2(w+4)+1(w-11)}{(w-1)(w+4)}$
$\frac{2w+8+w-11}{(w-1)(w+4)}$
$\frac{3w-3}{(w-1)(w+4)}$

But, this is not the final answer as this can be further simplify
$\frac{3(w-1)}{(w-1)(w+4)}$
$\frac{3\require{cancel}\bcancel{(w-1)}}{\require{cancel}\bcancel{(w-1)}(w+4)}$
$\frac{3}{w+4}$

We would continue our series on algebraic fractions in this next post where we will learn to subtract algebraic fraction.

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