Equations with Fractions
Clear every denominator in one move and never work with fractions again.
The explanation
Do not fight the fractions. Remove them.
Find the least common denominator of all fractions in the equation, then multiply *every term* by it — including terms without fractions.
For x/3 + 1/4 = 5/6, the LCD is 12:
12(x/3) + 12(1/4) = 12(5/6)
4x + 3 = 10
4x = 7, so x = 7/4.
The only real risk is missing a term. Multiply every single one, or the equation is no longer balanced.
Multiplying both sides by the LCD is an application of the multiplication property of equality, valid because the LCD is a nonzero constant. Each fraction's denominator divides the LCD exactly, so every denominator clears in a single step.
With numerical denominators the process is entirely safe. With variable denominators it is not: multiplying by an expression such as (x − 3) is only valid when that expression is nonzero, so x = 3 must be excluded from the domain in advance. This is why rational equations require checking for extraneous solutions — a root that the algebra produces but that makes an original denominator zero is not a solution of the original equation.
The same LCD technique clears decimals: multiplying by an appropriate power of ten converts 0.25x + 1.5 = 3 into 25x + 150 = 300. Integers are less error-prone to work with, which is the whole motivation.
Worked example
Solve (2x + 1)/5 − x/2 = 3/10.
- LCD of 5, 2, 10 is 10.
- Multiply every term: 2(2x + 1) − 5x = 3.
- Distribute: 4x + 2 − 5x = 3.
- Combine: −x + 2 = 3, so −x = 1.
Answer: x = −1
Common mistakes
- Multiplying only the fraction terms and skipping the whole numbers.
- Forgetting that the LCD multiplies the entire numerator, so (2x+1)/5 becomes 2(2x+1), not 2x+1.