How to write a decimal as a fraction in simplest form (Grade 6).

📚 Understanding Decimals and Fractions

Decimals and fractions are two different ways of representing parts of a whole. Converting between them is a fundamental skill in mathematics. Let's explore how to express decimals as fractions in their simplest form.

📜 A Little History

The concept of fractions dates back to ancient civilizations, with Egyptians using them for measurements and land division. Decimals, on the other hand, emerged later, becoming more widely used in the 16th century with the development of a clear notation system. Together, they provide powerful tools for expressing numbers beyond whole numbers.

➗ The Key Principles

• 🔍 Identify the Decimal's Place Value: Determine the place value of the last digit in the decimal (tenths, hundredths, thousandths, etc.). For example, in 0.25, the last digit (5) is in the hundredths place.
• 📝 Write as a Fraction: Write the decimal as a fraction with the decimal number as the numerator and the place value as the denominator. For 0.25, this would be $\frac{25}{100}$.
• ➗ Simplify the Fraction: Find the greatest common divisor (GCD) of the numerator and denominator and divide both by the GCD to simplify the fraction.

✏️ Step-by-Step Example: 0.75 to a Fraction

1. Identify the Place Value: In 0.75, the 5 is in the hundredths place.
2. Write as a Fraction: $0.75 = \frac{75}{100}$
3. Simplify: The GCD of 75 and 100 is 25. Divide both by 25: $\frac{75 ÷ 25}{100 ÷ 25} = \frac{3}{4}$

💡 More Examples

• 🧪 Example 1: Convert 0.5 to a fraction. $0.5 = \frac{5}{10}$. Simplifying gives $\frac{1}{2}$.
• 🌍 Example 2: Convert 0.125 to a fraction. $0.125 = \frac{125}{1000}$. Simplifying gives $\frac{1}{8}$.
• 📈 Example 3: Convert 0.6 to a fraction. $0.6 = \frac{6}{10}$. Simplifying gives $\frac{3}{5}$.

✍️ Expressing Repeating Decimals as Fractions

Converting repeating decimals to fractions requires a slightly different approach. Here's the basic idea:

1. Let $x$ equal the repeating decimal.
2. Multiply $x$ by a power of 10 such that only the repeating part remains after the decimal point.
3. Subtract the original equation from the new equation to eliminate the repeating part.
4. Solve for $x$.

For example, to express $0.\overline{3}$ as a fraction:

1. Let $x = 0.\overline{3}$
2. $10x = 3.\overline{3}$
3. Subtract the first equation from the second: $10x - x = 3.\overline{3} - 0.\overline{3}$ which simplifies to $9x = 3$
4. Solve for x: $x = \frac{3}{9} = \frac{1}{3}$

🔑 Conclusion

Converting decimals to fractions, especially in simplest form, is a crucial mathematical skill. By understanding place values and simplification techniques, you can confidently navigate between these two representations. Remember to practice and apply these concepts to real-world scenarios!