Isolate the fraction term
Move everything that is not the fraction to the other side first: in x/3 + 1/4 = 5/6, subtract 1/4 from both sides to leave x/3 alone.
Algebraic fractions
Fraction Equations
A fraction equation is an algebraic equation whose variable sits under a bar — x/5 = 3, or x/3 + 1/4 = 5/6. The balance rule is the same as any equation: undo the arithmetic one layer at a time, doing the same thing to both sides.
- undo the bar by multiplying
- solve for x
- one step at a time
- check by substituting
Algebra addition tool
Add fractions that carry a variable
Type two terms with the same variable — the tool finds the common denominator and combines the numerators step by step.
Type the full expression
Examples: x/3 + 1/6, x/4 + x/6, 2x/5 + x/3, or 1/2 + x/8.
Derivation
x/4 + x/6
= 3x/12 + 2x/12
= 5x/12
Quick answer
Undo the bar, then the rest
In x/3 + 1/4 = 5/6, isolate the fraction first: x/3 = 5/6 − 1/4 = 7/12. Then multiply both sides by the denominator: x = 3 × 7/12 = 7/4. Substituting back confirms it — (7/4)/3 + 1/4 rebuilds 5/6.
The whole procedure is the ordinary equation balance wearing fraction clothes. Every fraction step inside — common denominators, simplifying — is the arithmetic this site practices everywhere else.
The method
Four steps: isolate, clear, handle, check
Multiply both sides by the denominator
Multiplication undoes the bar: x/3 = 7/12 becomes x = 3 × 7/12 = 7/4. Whatever you do to one side, do to the other — that is the whole equation rule.
Handle a coefficient before or together
In 2x/3 = 8, multiply by 3 (2x = 24) and divide by 2 (x = 12), or multiply both sides by 3/2 in one move. The variable is still just a number waiting to be found.
Check by substituting back
Put the answer into the original equation: x = 7/4 gives 7/4 ÷ 3 = 7/12, and 7/12 + 1/4 = 7/12 + 3/12 = 10/12 = 5/6 — the original right side, confirming the solution.
Worked example
One step: x/5 = 3
x/5 = 3 x = 5 × 3 x = 15
The bar divides x by 5, so multiplying both sides by 5 undoes it. Check: 15/5 = 3. One-step fraction equations are the division facts of algebra.
Worked example
Two steps: x/3 + 1/4 = 5/6
x/3 + 1/4 = 5/6 x/3 = 5/6 − 1/4 x/3 = 10/12 − 3/12 = 7/12 x = 3 × 7/12 = 7/4
Clear the constant first, then the bar. The fraction arithmetic inside is ordinary common-denominator work: 5/6 and 1/4 meet at twelfths (10/12 − 3/12), and multiplying by 3 finishes the isolation.
Worked example
A coefficient: 2x/3 = 8
2x/3 = 8 2x = 24 x = 12
Multiply both sides by 3 to free the numerator, then divide by the coefficient 2. Alternatively multiply both sides by 3/2 at once — both orders reach x = 12.
Worked example
Two fraction terms: x/2 + x/3 = 5
x/2 + x/3 = 5 (3x + 2x)/6 = 5 5x/6 = 5 5x = 30 x = 6
Combine the left side over the common denominator 6 first — 3x/6 + 2x/6 = 5x/6 — then clear the bar. The check: 6/2 + 6/3 = 3 + 2 = 5.
Why multiplying by the denominator is legal
An equation is a balance, and any operation applied to both sides keeps it balanced. The bar in x/5 means x divided by 5; multiplying both sides by 5 divides out that operation on the left (5 × x/5 = x) and simply scales the right side. This is the same move as subtracting 4 from both sides to undo a +4 — the inverse operation, applied fairly.
When the equation carries several fraction terms, two paths exist. Clear each bar one at a time (isolate, multiply), or clear them all at once by multiplying every term by the common denominator — in x/2 + x/3 = 5, multiplying everything by 6 gives 3x + 2x = 30 with no fractions at all. Students who dislike fraction arithmetic can lean on that clear-the-bar move; students comfortable with the rewrite can combine first. Both arrive at x = 6.
Review combining variable fractions over a common denominatorCommon mistakes when solving fraction equations
- Multiplying only one side: Clearing x/5 = 3 requires multiplying both sides by 5: x = 15. Multiplying just the fraction gives 15/5 = 3 and loses the equation entirely — whatever touches one side must touch the other.
- Adding across the bar during simplification: In x/2 + x/3, combining to 5x/6 needs the common denominator 6 with numerators 3x and 2x. Writing (x + x)/5 or (2x)/6 miscounts the pieces — the rewrite step is where fraction care pays.
- Skipping the check: A substituted check catches sign slips and lost constants instantly: x = 7/4 in x/3 + 1/4 really does rebuild 5/6. Thirty seconds of substitution outvalues a page of re-derivation.
Practice routine
Solve, then substitute to confirm
For each equation, say which layer you will undo first, solve, and check the answer in the original — the check is what turns an answer into a verified one.
TryWhat to noticeAnswer
x/4 = 5One step: multiply both sides by 4.x = 20
x/3 − 1/2 = 1/6Isolate the bar: x/3 = 1/6 + 3/6 = 4/6 = 2/3.x = 2
3x/5 = 6Coefficient first or 5/3 in one move.x = 10
x/6 + x/2 = 2Combine over 6: (x + 3x)/6 = 2.x = 3
x/2 = 3/4Cross-multiply or multiply by 2 — same result.x = 3/2
Keep learning
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Common questions
Fraction Equations FAQ
A fraction equation is an algebraic equation in which the variable appears inside a fraction bar, such as x/5 = 3 or x/3 + 1/4 = 5/6. You solve it with the same balance rule as any equation, plus the fraction arithmetic needed to combine terms — multiply both sides by the denominator to free the variable.