Adding and Multiplying Algebraic Fractions

The rules you already know

Algebraic fractions add and multiply by the same rules as number fractions. To add, rewrite both fractions over a common denominator, then add the numerators. To multiply, multiply the numerators together and the denominators together.

Take $\frac{x}{2}+\frac{x}{3}$x/2 + x/3. Halves and thirds are pieces of different sizes, so their numerators cannot be added as they stand. Both fit into sixths, because 6 is a multiple of 2 and of 3.

Let each whole bar stand for x. Cut the half and the third into sixths: half of x is 3 sixths and a third of x is 2 sixths, so together they are 5 sixths of x.

Over a common denominator

Rewrite each fraction over 6 by multiplying its numerator and its denominator by the same number. $\frac{x}{2}=\frac{3x}{6}$x/2 = 3x/6, multiplying both by 3, and $\frac{x}{3}=\frac{2x}{6}$x/3 = 2x/6, multiplying both by 2. Now the pieces are the same size, and the numerators add: $\frac{3x}{6}+\frac{2x}{6}=\frac{5x}{6}$3x/6 + 2x/6 = 5x/6.

Subtraction works the same way: $\frac{x}{2}-\frac{x}{3}=\frac{3x}{6}-\frac{2x}{6}=\frac{x}{6}$x/2 − x/3 = 3x/6 − 2x/6 = x/6.

Adding the denominators is the usual slip

Adding the numerators and the denominators, $\frac{x}{2}+\frac{x}{3}=\frac{2x}{5}$x/2 + x/3 = 2x/5, is the most common mistake. A denominator is the size of the pieces, and sizes do not add. No fifths appear anywhere: the pieces are sixths.

Substitution catches it. Put x = 6: $\frac{x}{2}+\frac{x}{3}=3+2=5$x/2 + x/3 = 3 + 2 = 5, and $\frac{5x}{6}=5$5x/6 = 5 as well. But $\frac{2x}{5}=\frac{12}{5}=2.4$2x/5 = 12/5 = 2.4, which is less than $\frac{x}{2}=3$x/2 = 3 on its own.

Multiplying goes straight across

Multiplying needs no common denominator. Multiply the numerators, then the denominators: $\left(\frac{x}{2}\right)\times \left(\frac{x}{3}\right)=\frac{\left(x\times x\right)}{\left(2\times 3\right)}=\frac{x^2}{6}$(x/2) × (x/3) = (x × x)/(2 × 3) = x²/6.

A square shows why. Take a square of side x, whose area is $x^2$x². Cut its top side into halves and its left side into thirds. The square is now cut into 2 × 3 = 6 equal pieces, and one piece is $\frac{x}{2}$x/2 wide and $\frac{x}{3}$x/3 tall. Its area is $\left(\frac{x}{2}\right)\times \left(\frac{x}{3}\right)$(x/2) × (x/3), and it is one sixth of $x^2$x², so $\left(\frac{x}{2}\right)\times \left(\frac{x}{3}\right)=\frac{x^2}{6}$(x/2) × (x/3) = x²/6.

A square of side x cut into halves across and thirds down. One piece is $\frac{x}{2}$x/2 by $\frac{x}{3}$x/3, and it is one of 6 equal pieces of $x^2$x², so its area is $\frac{x^2}{6}$x²/6.

Letters in the denominators

A denominator can contain a letter too. The common denominator is still a multiple of both denominators, and when they share no factor, their product will do.

For $\frac{1}{x}+\frac{1}{2}$1/x + 1/2 the common denominator is 2 × x = 2x. Multiply the numerator and the denominator of each fraction by the factor its denominator is missing: $\frac{1}{x}=\frac{2}{\left(2x\right)}$1/x = 2/(2x) and $\frac{1}{2}=\frac{x}{\left(2x\right)}$1/2 = x/(2x). Then $\frac{1}{x}+\frac{1}{2}=\frac{\left(2+x\right)}{\left(2x\right)}$1/x + 1/2 = (2 + x)/(2x).

Check with x = 4: $\frac{1}{4}+\frac{1}{2}=\frac{3}{4}$1/4 + 1/2 = 3/4, and $\frac{\left(2+4\right)}{\left(2\times 4\right)}=\frac{6}{8}=\frac{3}{4}$(2 + 4)/(2 × 4) = 6/8 = 3/4.