How to Solve: Equations with Fractions

How to Solve: Equations with Fractions

Introduction

"If you can solve equations with fractions, you can handle real-world problems like splitting bills, adjusting recipes, or even calculating medication doses—plus, you’ll nail those 5-6 mark exam questions that separate a B from an A!

What You Need To Know First

Before tackling equations with fractions, you must already understand: 1. Solving linear equations (e.g., 3x + 5 = 14). 2. Finding the Lowest Common Denominator (LCD) of fractions. 3. Multiplying and dividing fractions (e.g., ½ × ⅔ = ⅓).

If any of these feel shaky, pause and review them first—this guide assumes you’re solid on them.

Key Vocabulary

Formulas To Know

1. Clearing Fractions

Formula: Multiply every term in the equation by the LCD of all denominators.

What it means: - LCD = Lowest Common Denominator (smallest number all denominators divide into). - Every term = Every number, variable, or fraction in the equation.

Example: For x/2 + 3/4 = 5, the LCD is 4. Multiply every term by 4: 4 × (x/2) + 4 × (3/4) = 4 × 5 → 2x + 3 = 20

MEMORISE THIS: This is the only formula you need—everything else is just algebra.

Step-by-Step Method

Goal: Solve equations with fractions by eliminating them first.

Step 1: Identify all denominators

• List every denominator in the equation.
• Example: In x/3 + 2 = 5/6, denominators are 3 and 6.

Step 2: Find the LCD

• Find the smallest number all denominators divide into evenly.
• Example: LCD of 3 and 6 is 6.

Step 3: Multiply every term by the LCD

• This "clears" the fractions.
• Example: Multiply x/3 + 2 = 5/6 by 6: 6 × (x/3) + 6 × 2 = 6 × (5/6) → 2x + 12 = 5

Step 4: Solve the resulting equation

• Now it’s a normal linear equation.
• Example: 2x + 12 = 5 Subtract 12: 2x = -7 Divide by 2: x = -3.5

Step 5: Check your answer

• Plug x back into the original equation to verify.
• Example: x = -3.5 in x/3 + 2 = 5/6: -3.5/3 + 2 = -1.166... + 2 = 0.833... 5/6 ≈ 0.833... ✓ Correct!

Worked Example Using the Steps

Equation: 2x/5 - 1/3 = x/2

Step 1: Denominators = 5, 3, 2 Step 2: LCD = 30 (smallest number 5, 3, 2 divide into) Step 3: Multiply every term by 30: 30 × (2x/5) - 30 × (1/3) = 30 × (x/2) → 12x - 10 = 15x Step 4: Solve: Subtract 12x: -10 = 3x Divide by 3: x = -10/3 Step 5: Check: 2(-10/3)/5 - 1/3 = (-10/3)/2 → -4/3 - 1/3 = -5/3 → -5/3 = -5/3 ✓ Correct!

Worked Examples

Example 1 - Basic

Equation: x/4 + 1 = 3/2

Step 1: Denominators = 4, 2 Step 2: LCD = 4 Step 3: Multiply every term by 4: 4 × (x/4) + 4 × 1 = 4 × (3/2) → x + 4 = 6 Step 4: Solve: Subtract 4: x = 2 Step 5: Check: 2/4 + 1 = 0.5 + 1 = 1.5 3/2 = 1.5 ✓ Correct!

What we did and why: We eliminated fractions first to make the equation easier to solve. Multiplying by the LCD turns fractions into whole numbers, so we can use normal algebra.

Example 2 - Medium

Equation: (x + 1)/3 = (2x - 5)/6 + 1/2

Step 1: Denominators = 3, 6, 2 Step 2: LCD = 6 Step 3: Multiply every term by 6: 6 × (x + 1)/3 = 6 × (2x - 5)/6 + 6 × (1/2) → 2(x + 1) = (2x - 5) + 3 → 2x + 2 = 2x - 5 + 3 → 2x + 2 = 2x - 2 Step 4: Solve: Subtract 2x: 2 = -2 No solution! (This means the equation is inconsistent.)

What we did and why: We cleared fractions first, then simplified. When both sides cancel out to a false statement (2 = -2), it means no value of x works.

Example 3 - Exam Style

Equation: 3/4(x - 2) = 5/6 + x/3

Step 1: Rewrite for clarity: (3/4)(x - 2) = 5/6 + x/3 Denominators = 4, 6, 3 Step 2: LCD = 12 Step 3: Multiply every term by 12: 12 × (3/4)(x - 2) = 12 × (5/6) + 12 × (x/3) → 9(x - 2) = 10 + 4x → 9x - 18 = 10 + 4x Step 4: Solve: Subtract 4x: 5x - 18 = 10 Add 18: 5x = 28 Divide by 5: x = 28/5 (or 5.6) Step 5: Check: (3/4)(5.6 - 2) = 5/6 + 5.6/3 → (3/4)(3.6) = 0.833... + 1.866... → 2.7 = 2.7 ✓ Correct!

What we did and why: We treated the (x - 2) as a single term when multiplying by the LCD. This avoids mistakes with distribution. Always check your answer to catch errors.

Common Mistakes

Exam Traps

1-Minute Recap

"Okay, let’s lock this in. To solve equations with fractions: 1. Find the LCD of all denominators—this is your magic number. 2. Multiply every term by the LCD to clear fractions. No shortcuts—every term means every term! 3. Solve the new equation like normal algebra. 4. Check your answer by plugging it back into the original equation.

Remember: - If you get a false statement (like 3 = 5), write ‘No solution’. - If variables cancel out and you get a true statement (like 2 = 2), write ‘All real numbers’. - Always rewrite decimals as fractions first (e.g., 0.25x = x/4).

You’ve got this. Now go crush those exam questions!