Mastering repeating decimals to fractions with algebra (8th grade tutorial)

📚 Understanding Repeating Decimals

A repeating decimal is a decimal number that has a digit or a group of digits that repeats infinitely. For example, $0.3333...$ or $0.\overline{3}$ is a repeating decimal. The bar over the 3 indicates that the digit 3 repeats indefinitely.

📜 A Little History

The concept of representing numbers as decimals has been around for centuries. Simon Stevin, a Flemish mathematician, is often credited with popularizing the use of decimal fractions in Europe in the late 16th century. Understanding how to convert repeating decimals to fractions became essential as decimal notation became more widely adopted.

🔢 Key Principles: Algebra to the Rescue!

Here’s how to convert repeating decimals to fractions using algebra:

• 🧮 Step 1: Let $x$ equal the repeating decimal.
• ➗ Step 2: Multiply $x$ by a power of 10 (i.e., 10, 100, 1000, etc.) so that one repetition of the repeating digit(s) lies to the left of the decimal point.
• ➖ Step 3: Subtract the original equation (Step 1) from the new equation (Step 2). This eliminates the repeating part.
• ➗ Step 4: Solve for $x$.
• ✏️ Step 5: Simplify the fraction if possible.

➗ Example 1: Convert $0.\overline{3}$ to a fraction

• 🧮 Let $x = 0.\overline{3}$
• ✖️ Multiply by 10: $10x = 3.\overline{3}$
• ➖ Subtract: $10x - x = 3.\overline{3} - 0.\overline{3}$ which simplifies to $9x = 3$
• ➗ Divide by 9: $x = \frac{3}{9}$
• ✏️ Simplify: $x = \frac{1}{3}$

➕ Example 2: Convert $0.\overline{27}$ to a fraction

• 🧮 Let $x = 0.\overline{27}$
• ✖️ Multiply by 100: $100x = 27.\overline{27}$
• ➖ Subtract: $100x - x = 27.\overline{27} - 0.\overline{27}$ which simplifies to $99x = 27$
• ➗ Divide by 99: $x = \frac{27}{99}$
• ✏️ Simplify: $x = \frac{3}{11}$

💯 Example 3: Convert $0.1\overline{6}$ to a fraction

• 🧮 Let $x = 0.1\overline{6}$
• ✖️ Multiply by 10: $10x = 1.\overline{6}$
• ✖️ Multiply by 100: $100x = 16.\overline{6}$
• ➖ Subtract: $100x - 10x = 16.\overline{6} - 1.\overline{6}$ which simplifies to $90x = 15$
• ➗ Divide by 90: $x = \frac{15}{90}$
• ✏️ Simplify: $x = \frac{1}{6}$

💡 Tips and Tricks

• ✔️ Ensure the repeating part is correctly identified.
• ✔️ Choose the correct power of 10 to multiply by to shift the repeating block to the left of the decimal point.
• ✔️ Always simplify the fraction at the end.

✍️ Practice Quiz

Convert the following repeating decimals to fractions:

1. ❓ $0.\overline{8}$
2. ❓ $0.\overline{45}$
3. ❓ $0.2\overline{3}$

Answers:

1. ✅ $\frac{8}{9}$
2. ✅ $\frac{5}{11}$
3. ✅ $\frac{7}{30}$

🌍 Real-World Applications

Converting repeating decimals to fractions is not just a math exercise. It has applications in various fields such as:

• 💻 Computer Science: Representing rational numbers precisely in algorithms.
• 📈 Finance: Calculating interest rates and other financial metrics.
• 📐 Engineering: Performing accurate measurements and conversions.

⭐ Conclusion

Mastering the conversion of repeating decimals to fractions using algebra is a valuable skill. By following these steps and practicing regularly, you can confidently tackle any repeating decimal problem! Keep practicing, and you’ll become a pro in no time!