How to Solve: Equations with Decimals

How to Solve: Equations with Decimals

Introduction

"If you can solve equations with decimals, you can calculate discounts, adjust recipes, or even figure out how much gas you’ll need for a road trip—without getting ripped off. Let’s master this in 10 minutes."

What You Need To Know First

Before tackling equations with decimals, you must already understand: 1. Solving one-step and two-step linear equations (e.g., 2x + 3 = 7). 2. Operations with decimals (adding, subtracting, multiplying, dividing). 3. Clearing fractions (multiplying both sides by the denominator).

If you’re shaky on any of these, pause here and review them first.

Key Vocabulary

Formulas To Know

No new formulas—just one golden rule: Multiply both sides of the equation by a power of 10 to eliminate decimals. - MEMORIZE THIS: The power of 10 you choose = the number of decimal places in the longest decimal in the equation.

Example: - 0.3x + 1.2 = 2.7 → Longest decimal has 1 decimal place → Multiply by 10. - 0.05y – 0.2 = 1.3 → Longest decimal has 2 decimal places → Multiply by 100.

Step-by-Step Method

Follow these 5 steps for every equation with decimals.

1. Count the decimal places.
2. Find the term with the most decimal places.

3. Example: In 0.4x + 1.5 = 3.28, 3.28 has 2 decimal places.

4. Multiply every term by 10, 100, or 1000 (based on Step 1).

5. This turns decimals into whole numbers.

6. Example: Multiply 0.4x + 1.5 = 3.28 by 100 → 40x + 150 = 328.

7. Solve the new equation (now with whole numbers).

8. Use inverse operations (add/subtract first, then multiply/divide).

9. Example:

   • 40x + 150 = 328
   • Subtract 150: 40x = 178
   • Divide by 40: x = 178 ÷ 40 = 4.45

10. Check your answer by plugging it back into the original equation.

11. Example: 0.4(4.45) + 1.5 = 3.28 → 1.78 + 1.5 = 3.28 ✓

12. Write the solution clearly.

13. Example: x = 4.45

Worked Example Using the Steps

Equation: 0.6x – 1.2 = 3.0

1. Count decimal places: 0.6 (1), 1.2 (1), 3.0 (1) → 1 decimal place.
2. Multiply by 10: 10(0.6x) – 10(1.2) = 10(3.0) → 6x – 12 = 30.
3. Solve:
4. Add 12: 6x = 42
5. Divide by 6: x = 7
6. Check: 0.6(7) – 1.2 = 3.0 → 4.2 – 1.2 = 3.0 ✓
7. Solution: x = 7

Worked Examples

Example 1 – Basic

Equation: 0.5x = 2.0

1. Decimal places: 1 → Multiply by 10.
2. 10(0.5x) = 10(2.0) → 5x = 20
3. Solve: x = 20 ÷ 5 = 4
4. Check: 0.5(4) = 2.0 ✓
5. Solution: x = 4

What we did and why: We eliminated the decimal by multiplying by 10, turning 0.5 into 5 (easier to divide). Always check by plugging back in.

Example 2 – Medium

Equation: 0.25y + 0.3 = 1.8

1. Decimal places: 0.25 (2) → Multiply by 100.
2. 100(0.25y) + 100(0.3) = 100(1.8) → 25y + 30 = 180
3. Solve:
4. Subtract 30: 25y = 150
5. Divide by 25: y = 6
6. Check: 0.25(6) + 0.3 = 1.8 → 1.5 + 0.3 = 1.8 ✓
7. Solution: y = 6

What we did and why: 0.25 has 2 decimal places, so we multiplied by 100. This avoids messy fractions and keeps numbers whole.

Example 3 – Exam Style

Equation: 0.3(a + 2) = 1.5

1. Decimal places: 1 → Multiply by 10.
2. 10[0.3(a + 2)] = 10(1.5) → 3(a + 2) = 15
3. Solve:
4. Divide by 3: a + 2 = 5
5. Subtract 2: a = 3
6. Check: 0.3(3 + 2) = 1.5 → 0.3(5) = 1.5 ✓
7. Solution: a = 3

What we did and why: We distributed the multiplication first (10 × 0.3), then solved step-by-step. Always simplify inside parentheses first if possible.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. To solve equations with decimals: 1. Find the longest decimal—count how many digits are after the dot. 2. Multiply every term by 10, 100, or 1000 to turn decimals into whole numbers. 3. Solve the new equation like normal—add/subtract first, then multiply/divide. 4. Check your answer by plugging it back into the original equation. 5. Write it clearly—no messy decimals in your final answer!

Examiners love to hide decimals in coefficients or parentheses. Always multiply first to avoid mistakes. You’ve got this—go ace that exam!

Teacher Notes for Recording:

• Pacing: Spend 30 seconds on the hook, 1 minute on prerequisites/vocab, 2 minutes on the method, 3 minutes on examples, 1 minute on mistakes/traps, and 30 seconds on the recap.
• Visuals: Write each step on screen as you say it. Highlight the decimal places in red.
• Engagement: Ask students to pause and solve a practice problem after Example 2.
• Props: Use a whiteboard or digital pen to show the multiplication step clearly.

Final Tip: If a student struggles, have them write out every single step—no shortcuts! Decimals are just whole numbers in disguise.