Adding and subtracting with monomial denominators

Fractions can be added or subtracted only when they have the same denominator. Use the lowest common denominator (LCD): the lowest common multiple of the denominators. Recap from Secondary 1: \(\dfrac{x}{3}+\dfrac{x}{4}=\dfrac{4x+3x}{12}=\dfrac{7x}{12}\).

Finding the LCD of monomial denominators

Why: multiplying the numerator and the denominator of a fraction by the same expression gives an equivalent fraction, so each fraction can be rewritten over the LCD without changing its value. A bracket such as \(y-3\) is treated like a letter.

Method

1. Factorise any denominator that has a common factor, for example \(3y-9=3(y-3)\).
2. Find the LCD.
3. Rewrite each fraction over the LCD, multiplying its numerator by the same factor as its denominator.
4. Write one fraction, keeping each numerator in brackets so a minus sign applies to the whole numerator.
5. Expand, collect like terms and simplify if possible.

Example

Express \(\dfrac{5}{4x}-\dfrac{x-3}{6x^2}\) as a single fraction.

\(\begin{aligned}\dfrac{5}{4x}-\dfrac{x-3}{6x^2}&=\dfrac{3x(5)}{12x^2}-\dfrac{2(x-3)}{12x^2} \\ &=\dfrac{15x-2(x-3)}{12x^2} \\ &=\dfrac{15x-2x+6}{12x^2} \\ &=\dfrac{13x+6}{12x^2},\qquad x\ne0\end{aligned}\)

Check: \(-2(x-3)=-2x+6\); writing \(-2x-6\) is the most common error. Also, \(\dfrac{2}{3a}\) is not the same as \(\dfrac23a\): the letter is in the denominator.