How to Solve Equations with Fractions

Grade: 7-8 | Topic: Algebra

What You Will Learn

After this lesson you will be able to solve linear equations that contain fractions. You will master the "clear the fractions" technique using the Least Common Denominator (LCD), handle equations with fraction coefficients, and know when cross-multiplication is a shortcut. These skills combine your fraction knowledge with your equation-solving abilities.

Theory

Why fractions make equations harder

Fractions do not change the rules of solving equations — you still use inverse operations and keep both sides balanced. However, fractions introduce extra arithmetic that can lead to mistakes. The most reliable strategy is to eliminate the fractions first, then solve the resulting whole-number equation.

The LCD method — clearing fractions

The fastest way to remove fractions from an equation is to multiply every term by the Least Common Denominator (LCD) of all the fractions in the equation.

Given an equation like:

$\frac{x}{3} + \frac{2}{5} = \frac{7}{15}$

1. Find the LCD of all denominators (3, 5, and 15). The LCD is 15.
2. Multiply every term on both sides by 15:

$15 \cdot \frac{x}{3} + 15 \cdot \frac{2}{5} = 15 \cdot \frac{7}{15}$

$5x + 6 = 7$

3. Now solve the whole-number equation using standard methods.

When to use cross-multiplication

If the equation is a proportion — one fraction equals another fraction — you can use cross-multiplication as a shortcut:

$\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c$

This only works when each side is a single fraction. For equations like $\frac{x}{3} + 2 = \frac{x}{5}$, you must use the LCD method.

Equations with fraction coefficients

Sometimes the fraction multiplies the variable directly, like $\frac{2}{3}x = 10$. To isolate $x$, multiply both sides by the reciprocal of the fraction coefficient:

$\frac{2}{3}x = 10 \implies x = 10 \times \frac{3}{2} = 15$

The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. Multiplying a fraction by its reciprocal always gives 1, which isolates $x$.

Worked Examples

Example 1: Simple equation with one fraction (easy)

Problem: Solve $\dfrac{x}{4} + 3 = 7$

Step 1: Subtract 3 from both sides.

$\frac{x}{4} = 4$

Step 2: Multiply both sides by 4.

$x = 16$

Answer: $x = 16$

Check: $\frac{16}{4} + 3 = 4 + 3 = 7$ ✓

Example 2: Equation with two fractions — LCD method (medium)

Problem: Solve $\dfrac{x}{2} + \dfrac{x}{3} = 10$

Step 1: Find the LCD of 2 and 3. The LCD is 6.

Step 2: Multiply every term by 6.

$6 \cdot \frac{x}{2} + 6 \cdot \frac{x}{3} = 6 \cdot 10$

$3x + 2x = 60$

Step 3: Combine like terms.

$5x = 60$

Step 4: Divide both sides by 5.

$x = 12$

Answer: $x = 12$

Check: $\frac{12}{2} + \frac{12}{3} = 6 + 4 = 10$ ✓

Example 3: Proportion — cross-multiplication (medium)

Problem: Solve $\dfrac{x + 1}{4} = \dfrac{3}{2}$

Step 1: Cross-multiply (each side is a single fraction).

$(x + 1) \times 2 = 4 \times 3$

$2(x + 1) = 12$

Step 2: Distribute the 2.

$2x + 2 = 12$

Step 3: Subtract 2 from both sides.

$2x = 10$

Step 4: Divide both sides by 2.

$x = 5$

Answer: $x = 5$

Check: $\frac{5 + 1}{4} = \frac{6}{4} = \frac{3}{2}$ ✓

Example 4: Fraction coefficient (medium)

Problem: Solve $\dfrac{3}{5}x - 6 = 9$

Step 1: Add 6 to both sides.

$\frac{3}{5}x = 15$

Step 2: Multiply both sides by the reciprocal $\frac{5}{3}$.

$x = 15 \times \frac{5}{3} = \frac{75}{3} = 25$

Answer: $x = 25$

Check: $\frac{3}{5}(25) - 6 = 15 - 6 = 9$ ✓

Example 5: Multiple fractions on both sides (challenging)

Problem: Solve $\dfrac{2x}{3} - \dfrac{1}{4} = \dfrac{x}{6} + \dfrac{5}{12}$

Step 1: Find the LCD of 3, 4, 6, and 12. The LCD is 12.

Step 2: Multiply every term by 12.

$12 \cdot \frac{2x}{3} - 12 \cdot \frac{1}{4} = 12 \cdot \frac{x}{6} + 12 \cdot \frac{5}{12}$

$8x - 3 = 2x + 5$

Step 3: Subtract $2x$ from both sides.

$6x - 3 = 5$

Step 4: Add 3 to both sides.

$6x = 8$

Step 5: Divide both sides by 6.

$x = \frac{8}{6} = \frac{4}{3}$

Answer: $x = \dfrac{4}{3}$

Check: $\frac{2 \cdot \frac{4}{3}}{3} - \frac{1}{4} = \frac{8}{9} - \frac{1}{4} = \frac{32}{36} - \frac{9}{36} = \frac{23}{36}$. And $\frac{\frac{4}{3}}{6} + \frac{5}{12} = \frac{4}{18} + \frac{5}{12} = \frac{8}{36} + \frac{15}{36} = \frac{23}{36}$ ✓

Common Mistakes

Mistake 1: Multiplying only some terms by the LCD

❌ $\frac{x}{2} + 3 = \frac{x}{4} \implies x + 3 = \frac{x}{4}$ (only multiplied the first term by 2)

✅ Multiply every term by the LCD (which is 4): $2x + 12 = x$

Why this matters: Every single term — including whole numbers — must be multiplied by the LCD. Missing even one term produces a completely different (and wrong) equation.

Mistake 2: Using cross-multiplication when it does not apply

❌ $\frac{x}{3} + 2 = \frac{5}{6}$ ... then cross-multiplying $x \times 6 = 3 \times 5$

✅ This is not a proportion (the left side has two terms). Use the LCD method: multiply everything by 6 to get $2x + 12 = 5$.

Why this matters: Cross-multiplication only works when each side is a single fraction. If either side has an addition or subtraction outside a fraction, you must use the LCD approach.

Mistake 3: Forgetting to simplify the final fraction

❌ $x = \frac{8}{6}$ (left unsimplified)

✅ $x = \frac{8}{6} = \frac{4}{3}$

Why this matters: Answers should always be in simplest form. Divide numerator and denominator by their GCD.

Practice Problems

Try these on your own before checking the answers:

1. $\dfrac{x}{5} - 2 = 6$
2. $\dfrac{x}{3} + \dfrac{x}{4} = 7$
3. $\dfrac{2x + 1}{3} = \dfrac{5}{3}$
4. $\dfrac{3}{4}x + 5 = 14$
5. $\dfrac{x}{2} - \dfrac{x}{5} = 6$

Click to see answers

1. $x = 40$ — Add 2: $\frac{x}{5} = 8$, multiply by 5: $x = 40$.
2. $x = 12$ — LCD is 12: $4x + 3x = 84$, so $7x = 84$, $x = 12$.
3. $x = 2$ — Multiply by 3: $2x + 1 = 5$, subtract 1: $2x = 4$, $x = 2$.
4. $x = 12$ — Subtract 5: $\frac{3}{4}x = 9$, multiply by $\frac{4}{3}$: $x = 12$.
5. $x = 20$ — LCD is 10: $5x - 2x = 60$, so $3x = 60$, $x = 20$.

Summary

• To solve equations with fractions, multiply every term by the LCD to clear all fractions at once.
• Cross-multiplication is a shortcut that works only when the equation is a proportion (one fraction = one fraction).
• For fraction coefficients like $\frac{2}{3}x$, multiply by the reciprocal to isolate $x$.
• Always multiply every term by the LCD, including whole numbers.
• Simplify your final answer and check it in the original equation.

Related Topics

• Linear Equations — How to Solve Step by Step
• Solving Two-Step Equations Step by Step
• Fractions — Complete Guide

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