Simplify the expression: .

Question

Homi · Accepted Answer

Understand the Question

To add two fractions with unlike denominators, first find the Least Common Multiple (LCM) of their denominators.
Convert each fraction into an equivalent fraction with the common denominator.
Add the numerators while keeping the denominator the same, then simplify the resulting fraction to lowest terms.

Step 1 · Find a Common Denominator

The denominators are $4$ and $12$ .

$\text{LCM}(4, 12) = 12$

Convert $-\dfrac{1}{4}$ to an equivalent fraction with denominator $12$ : Diagram 1

$-\dfrac{1}{4} = -\dfrac{1 \times 3}{4 \times 3} = -\dfrac{3}{12}$

Step 2 · Add and Simplify

Now add the fractions with the common denominator:

\begin{aligned} -\dfrac{3}{12} + \dfrac{5}{12} &= \dfrac{-3 + 5}{12} \\[0.6em] &= \dfrac{2}{12} \\[0.6em] &= \dfrac{1}{6} \end{aligned}

Answer

$\dfrac{1}{6}$

Common Mistakes

Adding Denominators Directly: Incorrectly adding the numerators and denominators together $\left(\text{e.g., } \dfrac{-1+5}{4+12} = \dfrac{4}{16}\right)$ instead of converting to a common denominator.
Sign Errors: Mishandling the negative sign in the numerator (computing $3 + 5 = 8$ instead of $-3 + 5 = 2$ ).
Incomplete Reduction: Forgetting to simplify $\dfrac{2}{12}$ to its simplest form $\dfrac{1}{6}$ .

Question 3