Fractions in an equation are allowed, but whole numbers are easier to work with. Since you may multiply both sides by the same nonzero number, you can choose a number that clears every fraction at once: the least common denominator, which you know from least common multiples.
Why it works
Multiplying a side by $6$ means multiplying the whole side, and by the distributive property that means multiplying every term: $6\left(\frac{x}{3} + \frac{1}{2}\right) = 6 \cdot \frac{x}{3} + 6 \cdot \frac{1}{2} = 2x + 3$. Because $6$ is a multiple of every denominator, each term comes out as a whole number. Decimals work the same way: a decimal is a fraction with denominator $10$, $100$ or $1000$.
| Term | Times 6 | Result |
|---|---|---|
| $\frac{x}{3}$ | $6 \cdot \frac{x}{3}$ | $2x$ |
| $\frac{1}{2}$ | $6 \cdot \frac{1}{2}$ | $3$ |
| $\frac{5}{6}$ | $6 \cdot \frac{5}{6}$ | $5$ |
Key idea
Multiply every term on both sides by the least common denominator (or by $10$, $100$, … for decimals). The equation that remains has the same solution and no fractions.
Worked examples
Worked example: Single fractions
Solve $\frac{x}{3} + \frac{1}{2} = \frac{5}{6}$.
- The denominators are $3$, $2$ and $6$, so the least common denominator is $6$.
- Multiply every term by $6$: $2x + 3 = 5$.
- Subtract $3$: $2x = 2$. Divide by $2$: $x = 1$.
- Check: $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$.
Answer: $x = 1$
Worked example: Numerators with two terms
Solve $\frac{x + 1}{4} - \frac{x - 2}{6} = 1$.
- The least common denominator of $4$ and $6$ is $12$. Multiply every term by $12$, keeping each numerator in parentheses.
- $12 \cdot \frac{x + 1}{4} = 3(x + 1)$, $12 \cdot \frac{x - 2}{6} = 2(x - 2)$, and $12 \cdot 1 = 12$: so $3(x + 1) - 2(x - 2) = 12$.
- Distribute. The minus sign multiplies both terms of $2(x - 2)$: $3x + 3 - 2x + 4 = 12$.
- Combine: $x + 7 = 12$, so $x = 5$.
Answer: $x = 5$
Worked example: Decimals
Solve $0.05x + 1.2 = 0.2x - 0.3$.
- The most decimal places is two, so multiply every term by $100$: $5x + 120 = 20x - 30$.
- Subtract $5x$: $120 = 15x - 30$. Add $30$: $150 = 15x$.
- Divide by $15$: $x = 10$.
Answer: $x = 10$
Common mistake
Multiplying only the fraction terms. In $\frac{x}{3} + 2 = \frac{5}{6}$, multiplying by $6$ gives $2x + 12 = 5$, not $2x + 2 = 5$: the whole number $2$ must be multiplied too. Check by putting your answer back into the original equation, fractions and all.