Algebraic Fractions

A fraction with letters in it

An algebraic fraction is a fraction with a letter in its numerator, its denominator or both, such as $\frac{3x}{6x}$3x/6x or $\frac{\left(x+3\right)}{2}$(x + 3)/2. It is simplified the same way as a number fraction: divide the numerator and the denominator by a factor they share. $\frac{6}{8}$6/8 simplifies to $\frac{3}{4}$3/4 because 6 = 2 × 3 and 8 = 2 × 4 share the factor 2.

Take $\frac{3x}{6x}$3x/6x. Think of 6x as six equal parts, each worth x. The numerator, 3x, is three of those parts.

The bar is 6x, cut into six parts of x each. 3x is three of the six parts, which is half the bar, whatever the value of x.

Write both as factors, then cancel

Three parts out of six is half, whatever size a part is. To show it without a picture, write the numerator and the denominator as products of factors: 3x = 3 × x and 6x = 2 × 3 × x.

Both contain the factors 3 and x. Divide the numerator and the denominator by 3x, and $\frac{3x}{6x}=\frac{1}{2}$3x/6x = 1/2. Dividing by a factor of both is what "cancel" means.

This is only allowed when x is not 0, because 6x must not be 0 for the fraction to have a value. The same is true of every algebraic fraction: the letter cannot take a value that makes the denominator 0.

Only a factor of the whole cancels

In $\frac{\left(x+3\right)}{3}$(x + 3)/3, it is tempting to cross out the two 3s and write x. But the 3 in the numerator is a term, added to x. It is not a factor of the whole numerator, x + 3, so it cannot be canceled.

Substitution shows the mistake. Put x = 6: $\frac{\left(6+3\right)}{3}=\frac{9}{3}=3$(6 + 3)/3 = 9/3 = 3, but crossing out the 3s gives 6. The two answers differ, so crossing out a term changed the value.

A factor multiplies the whole numerator or the whole denominator. A term is added to other terms. Only factors cancel.

Factor first, then cancel

When the numerator is a sum, factor it first to see whether it has a factor in common with the denominator. In $\frac{\left(2x+6\right)}{2}$(2x + 6)/2, the numerator has a common factor of 2: 2x + 6 = 2(x + 3). Now 2 is a factor of the whole numerator, so it cancels: $\frac{\left(2x+6\right)}{2}=\frac{2\left(x+3\right)}{2}=x+3$(2x + 6)/2 = 2(x + 3)/2 = x + 3.

2x + 6 is two equal rows of x + 3. Half of it is one row, so $\frac{\left(2x+6\right)}{2}=x+3$(2x + 6)/2 = x + 3, not x + 6.

A letter as the shared factor

The same works with a letter. In $\frac{\left(x^2+5x\right)}{x}$(x² + 5x)/x, the numerator factors as x(x + 5), and x is a factor of both, so $\frac{\left(x^2+5x\right)}{x}=x+5$(x² + 5x)/x = x + 5. Check it at x = 2: $\frac{\left(4+10\right)}{2}=7$(4 + 10)/2 = 7, and 2 + 5 = 7.

Halving only the first term is the usual slip here. $\frac{\left(2x+6\right)}{2}$(2x + 6)/2 is not x + 6: each term of the numerator is divided by 2, which is what taking out the factor 2 does.