1.5Equations with Fractions
Fractions in equations look intimidating, but one move eliminates them entirely — multiply both sides by the LCD. Once the fractions are gone, you're back to familiar territory.
Why This Matters
Fractions appear constantly in science and everyday life — from mixing solutions in Chemistry to calculating unit rates in economics. Clearing fractions efficiently is a skill you'll use in every math course through Calculus.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What single step eliminates all fractions from an equation — and why does multiplying by the LCD work?
Lesson Overview
An equation with fractions can always be converted into an equivalent equation with no fractions by multiplying every term on both sides by the Least Common Denominator (LCD). The LCD is the smallest number divisible by every denominator in the equation. When you multiply a fraction by its denominator, the denominator cancels — leaving only whole-number coefficients. After clearing fractions, the equation becomes a standard one-step, two-step, or multi-step equation. Always check your answer by substituting back into the original equation with fractions.
Worked Examples
Solve: x/2 + 3 = 7
LCD = 2. Multiply every term by 2:
2(x/2) + 2(3) = 2(7)
x + 6 = 14
Subtract 6: x = 8
Check: 8/2 + 3 = 7 ✓
Solve: x/3 + x/4 = 7
Denominators 3 and 4 → LCD = 12
12(x/3) + 12(x/4) = 12(7)
4x + 3x = 84
7x = 84
x = 12
Check: 12/3 + 12/4 = 4 + 3 = 7 ✓
Solve: (2x)/5 − 1 = 3
LCD = 5. Multiply every term by 5:
2x − 5 = 15
Add 5: 2x = 20
Divide by 2: x = 10
Check: 2(10)/5 − 1 = 3 ✓
Solve: x/2 + x/3 = 5
Denominators 2 and 3 → LCD = 6
3x + 2x = 30
5x = 30
x = 6
Check: 6/2 + 6/3 = 3 + 2 = 5 ✓
Solve: (x + 1)/4 = (x − 2)/3
Denominators 4 and 3 → LCD = 12
3(x + 1) = 4(x − 2)
3x + 3 = 4x − 8
Subtract 3x: 3 = x − 8
Add 8: x = 11
Check: 12/4 = 3; 9/3 = 3 ✓
Guided Practice
Answers are in the Answer Key section.
Solve: x/5 + 2 = 6
Hint: LCD = 5. Multiply every term by 5: x + 10 = 30. Then subtract 10.
Solve: x/4 + x/2 = 9
Hint: Denominators are 4 and 2. LCD = 4. Multiply every term by 4: x + 2x = 36.
Solve: (3x)/8 = 6
Hint: LCD = 8. Multiply both sides by 8: 3x = 48. Then divide by 3.
Solve: x/3 − x/9 = 4
Hint: Denominators 3 and 9 → LCD = 9. Multiply every term by 9: 3x − x = 36.
Solve: (x + 2)/3 = 5
Hint: LCD = 3. Multiply both sides by 3: x + 2 = 15. Then subtract 2.
Key Vocabulary
Least Common Denominator (LCD)
The smallest number that is a multiple of every denominator in the equation.
Example: For denominators 3 and 4, LCD = 12.
Clearing Fractions
Multiplying every term in an equation by the LCD to eliminate all denominators.
Example: Multiply x/3 + 5 = 8 by 3: x + 15 = 24.
Equivalent Equations
Two equations that have the same solution. Multiplying both sides by the same nonzero number produces an equivalent equation.
Example: x/2 = 5 and x = 10 are equivalent.
Denominator
The bottom number of a fraction. It tells how many equal parts the whole is divided into.
Example: In 3/4, the denominator is 4.
Practice Questions
Interactive Practice — 5 Questions
Solve: x/3 + 4 = 10
What is the LCD for an equation with denominators 4, 6, and 8?
Solve: x/4 + x/2 = 6
A student solved x/5 + 3 = 8 and got x = 25. Is this correct?
Solve: (x + 1)/4 = (x − 2)/3
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Solve: x/6 + 4 = 9
Solve: x/3 + x/6 = 8
Solve: (5x)/6 = 10
Solve: x/4 − x/8 = 3
Solve: (x + 5)/2 = 9
Common Mistakes
Multiplying only the fraction terms by the LCD and skipping the whole-number terms.
Multiply every single term on both sides by the LCD — including constants and whole numbers.
Finding the LCD incorrectly by just multiplying the denominators instead of finding the LCM.
Find the LCM of the denominators. For 1/4 and 1/6, the LCD is 12, not 24.
Canceling incorrectly — e.g., (12)(x/4) = 3 instead of 3x.
After multiplying, simplify the fraction: (12)(x/4) = 3x. Keep the variable.
Forgetting to distribute the LCD to both terms when a side has addition or subtraction.
If the right side is 3 + x/2 and LCD = 6, multiply each term: 6(3) + 6(x/2) = 18 + 3x.
Math Tips
Multiply every term — including whole-number terms — by the LCD. Missing even one term is the most common error.
After multiplying, double-check that no fractions remain before solving.
Mixed numbers (e.g., 1½) must be converted to improper fractions (3/2) before finding the LCD.
If the equation has variables on both sides, clear fractions first — then collect variable terms.