Equations with Fractions

Learn to solve equations that contain fractions by clearing denominators and using inverse operations.

Intermediate25 minLesson
Printable worksheet

Definition

An equation with fractions contains one or more fractions with variables. To solve these equations, we often clear the denominators by multiplying every term by the Least Common Denominator (LCD).
General Strategy:
  1. 1.Find the LCD of all fractions
  2. 2.Multiply every term by the LCD
  3. 3.Simplify (fractions disappear!)
  4. 4.Solve the resulting equation
  5. 5.Check your answer
Example:
x2+13=56\frac{x}{2} + \frac{1}{3} = \frac{5}{6}
LCD of 2, 3, and 6 is 6. Multiply every term by 6:
6⋅x2+6⋅13=6⋅566 \cdot \frac{x}{2} + 6 \cdot \frac{1}{3} = 6 \cdot \frac{5}{6}
3x+2=53x + 2 = 5
3x=33x = 3
x=1x = 1

Try it now

What is the first step to solve x5=4\frac{x}{5} = 4?

Worked Examples

Solve: x4=3\frac{x}{4} = 3
1

Identify what operation is applied to x

xx is divided by 4 → Division by 4
2

Apply the inverse operation

Multiply both sides by 4 → 4⋅x4=4⋅34 \cdot \frac{x}{4} = 4 \cdot 3
3

Simplify

x=12x = 12
4

Check

124=3\frac{12}{4} = 3 ✓ → Solution verified

Common Mistakes

Multiplying only some terms by the LCD

Why it's wrong: When clearing denominators, you must multiply EVERY term (including whole numbers) by the LCD to keep the equation balanced.

Correct: In x2+x3=5\frac{x}{2} + \frac{x}{3} = 5, multiply ALL terms by 6, including the 5: 6⋅x2+6⋅x3=6⋅56 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 5 gives 3x+2x=303x + 2x = 30, not 3x+2x=53x + 2x = 5

Forgetting to distribute after clearing fractions

Why it's wrong: When you multiply x+14\frac{x + 1}{4} by 12, you get 3(x+1)3(x + 1): the 3 must multiply the whole numerator, both xx and 11.

Correct: 12⋅x+14=3(x+1)=3x+312 \cdot \frac{x + 1}{4} = 3(x + 1) = 3x + 3, not 3x+13x + 1

Using wrong LCD

Why it's wrong: A bigger common multiple (like 24 for 4 and 6) still works but makes the numbers larger; a number that is not a common multiple leaves fractions in the equation.

Correct: LCD of 4 and 6 is 12, not 24. Find the LEAST common multiple, not just any common multiple.

Not checking the answer

Why it's wrong: Clearing fractions involves several multiplications, so a slip is easy. When a variable is in a denominator, a value can even make a denominator 0 (an extraneous solution).

Correct: Always substitute your answer back into the original equation to verify it works.

Interactive Visual

Equation Solver

Solve for xx:

x4=3\frac{x}{4} = 3
1
x4=3\frac{x}{4} = 3
Multiply both sides by the LCD, 44Now there are no fractions left

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

Balance Scale

(1/4)x=3
(1/4)x
3
Apply to both sides:

Solution: x = 12

Interactive Sandbox

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Practice Problems

17 problems
Problem 1 of 17
Easy
What is the first step to solve x5=4\frac{x}{5} = 4?

Why It Matters

Equations with fractions appear constantly in real life:
  • Cooking: Scaling a recipe for 6 people down to 4 means using 23\frac{2}{3} of every amount. If the smaller batch needs 2 cups of flour, solving 23x=2\frac{2}{3}x = 2 gives the original amount: x=3x = 3 cups
  • Finance: Calculating how long to save when putting away 14\frac{1}{4} of your income monthly
  • Science: Rate problems like dt=r\frac{d}{t} = r (distance, time, rate)
  • Construction: Scaling blueprints where 150\frac{1}{50} scale means solving x50=12\frac{x}{50} = 12
Mastering this skill opens doors to solving proportions, rational equations, and real-world word problems!

Real World Applications

Recipe Scaling

Chefs often need to adjust recipes when cooking for different numbers of people.

Example:

A recipe uses 34\frac{3}{4} cup of flour to make 12 cookies. How much flour for 20 cookies? Solve 3/412=x20\frac{3/4}{12} = \frac{x}{20}.

1Try It Yourself

A smoothie recipe calls for 23\frac{2}{3} cup of berries for 2 servings. You want to make 5 servings.

How many cups of berries do you need?

Step 1: Write the mathematical expression

Set up the proportion: 2/32=x5\frac{2/3}{2} = \frac{x}{5}

Shared Work Problems

When two people work together, their combined work rate determines how fast they finish.

Example:

If Maria can paint a room in 4 hours and John in 6 hours, working together they complete 14+16\frac{1}{4} + \frac{1}{6} of the room per hour.

2Try It Yourself

Pipe A fills a tank in 3 hours. Pipe B fills it in 6 hours. With both pipes open, what fraction of the tank fills in 1 hour?

What fraction of the tank is filled per hour?

Step 1: Write the mathematical expression

Add the rates: 13+16\frac{1}{3} + \frac{1}{6}

Speed and Distance

The formula $d = rt$ (distance = rate times time) often involves fractions when times are not whole numbers.

Example:

If you travel 34\frac{3}{4} of an hour at 60 mph, you cover 60×34=4560 \times \frac{3}{4} = 45 miles.

3Try It Yourself

A cyclist travels 15 miles in 23\frac{2}{3} of an hour.

What is the cyclist's average speed in miles per hour?

Step 1: Write the mathematical expression

Use r=dtr = \frac{d}{t}: r=152/3r = \frac{15}{2/3}

Key Takeaways

  • 1To solve equations with fractions, find the LCD and multiply every term to clear denominators
  • 2The reciprocal method works well when a fraction multiplies the variable: multiply both sides by the reciprocal
  • 3Always distribute carefully when clearing fractions with expressions in the numerator
  • 4Check your answer by substituting back into the original equation

Frequently Asked Questions

Multiplying by the LCD eliminates all denominators at once, converting the fraction equation into a simpler equation with whole numbers. This makes solving much easier.
Multiplying by the LCD eliminates all denominators at once, converting the fraction equation into a simpler equation with whole numbers. This makes solving much easier.
Cross-multiplication works when you have a proportion (one fraction equals another fraction). For equations with more terms, use the LCD method.
That is perfectly fine! Many equations with fractions have fractional solutions. Just verify by substituting the fraction back into the original equation.

Glossary

LCD (Least Common Denominator)
The smallest number that is a multiple of all denominators in an equation
Reciprocal
The flip of a fraction; the reciprocal of ab\frac{a}{b} is ba\frac{b}{a}
Clear denominators
Multiply every term by the LCD to eliminate all fractions
Extraneous solution
A solution that emerges from solving but does not satisfy the original equation

Formula Card

LCD Method

Multiply all terms by LCD

Clears all fractions at once

Reciprocal Method

If abx=c\frac{a}{b}x = c, then x=c⋅bax = c \cdot \frac{b}{a}

Multiply by the reciprocal of the coefficient

Cross-Multiplication

If ab=cd\frac{a}{b} = \frac{c}{d}, then ad=bcad = bc

Only for proportions

Today's challengeOne problem, three tries

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