Solving Equations with Fractions

Equations with fractions appear on both Foundation and Higher papers and are a common source of lost marks. The key idea is simple: clear the fractions first by multiplying every term by the lowest common denominator (LCD). Once the fractions are gone, you solve the equation using the techniques you already know. This page teaches you the method step by step and shows you how to handle cross-multiplication too.

What Is Solving Equations with Fractions?

An equation with fractions is any equation where the variable appears in a fraction, or where fractional coefficients are used. Examples include x/3 + 2 = 5 and (2x + 1)/4 = (x − 3)/2. The goal is to find the value of x that makes the equation true.

Key Formulas

Step-by-Step Method

1. Identify every denominator in the equation.
2. Find the LCD of all denominators.
3. Multiply every term on both sides by the LCD. This eliminates all fractions.
4. Expand any brackets that result from the multiplication.
5. Solve the resulting equation using standard methods (collect terms, isolate x).
6. Check your answer by substituting back into the original equation.

Worked Example 1 — Foundation Level

Question: Solve x/3 + 2 = 5.

Working:

• The only denominator is 3. LCD = 3.
• Multiply every term by 3: 3 × (x/3) + 3 × 2 = 3 × 5.
• This gives: x + 6 = 15.
• Subtract 6: x = 9.
• Check: 9/3 + 2 = 3 + 2 = 5 ✓

Answer: x = 9

Worked Example 2 — Higher Level

Question: Solve (2x + 1)/4 = (x − 3)/2.

Working:

• Denominators are 4 and 2. LCD = 4.
• Multiply every term by 4: 4 × (2x + 1)/4 = 4 × (x − 3)/2.
• This gives: (2x + 1) = 2(x − 3).
• Expand: 2x + 1 = 2x − 6.
• Subtract 2x from both sides: 1 = −6.
• This is a contradiction, so the equation has no solution.

Alternatively, using cross-multiplication: (2x + 1) × 2 = (x − 3) × 4, giving 4x + 2 = 4x − 12, which again gives 2 = −12 — no solution.

Answer: No solution

Worked Example 3 — Exam Style

Question: Solve (x + 1)/3 − (x − 2)/5 = 1.

Working:

• Denominators are 3 and 5. LCD = 15.
• Multiply every term by 15: 15 × (x + 1)/3 − 15 × (x − 2)/5 = 15 × 1.
• This gives: 5(x + 1) − 3(x − 2) = 15.
• Expand: 5x + 5 − 3x + 6 = 15.
• Simplify: 2x + 11 = 15.
• Subtract 11: 2x = 4.
• Divide by 2: x = 2.
• Check: (2 + 1)/3 − (2 − 2)/5 = 3/3 − 0/5 = 1 − 0 = 1 ✓

Answer: x = 2

Common Mistakes

• Forgetting to multiply every term by the LCD. Students often multiply the fractions but forget to multiply whole-number terms on the other side. Every term must be multiplied.
• Sign errors when expanding brackets. Watch out for minus signs before brackets, such as −3(x − 2) = −3x + 6, not −3x − 6.
• Using cross-multiplication when there are more than two fractions. Cross-multiplication only works when you have one fraction equal to another fraction. Use the LCD method for anything more complex.

Exam Tips

• Always show the LCD multiplication step clearly — examiners award method marks for this.
• If a question says "solve algebraically," you must show working, not trial-and-improvement.
• Substitute your answer back in to check, especially when there are multiple fractions.
• On Higher papers, you may get x in the denominator. Multiply through carefully and note any values that make the denominator zero (these are excluded).

Practice Questions

Q1 (Foundation): Solve x/5 + 3 = 7.

Q2 (Foundation): Solve (x + 4)/2 = 6.

Q3 (Higher): Solve (3x − 1)/4 + (x + 2)/6 = 2.

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Related Topics

• Solving Linear Equations
• Equations with Unknowns on Both Sides
• Adding and Subtracting Fractions

Summary

• Clear fractions by multiplying every term on both sides by the LCD.
• Cross-multiplication is a shortcut when one fraction equals another fraction.
• Always expand brackets carefully, paying close attention to negative signs.
• Substitute your answer back into the original equation to verify it is correct.
• This skill is essential for algebraic fractions and simultaneous equations on the Higher paper.