0.64 As A Fraction In Simplest Form

0.64 as a Fraction in Simplest Form: A Comprehensive Guide

Converting decimals to fractions might seem daunting at first, but with a structured approach, it becomes a straightforward process. This comprehensive guide will walk you through converting 0.64 into its simplest fraction form, explaining the underlying principles and offering practical tips for similar conversions. We'll delve into the steps, explore the rationale behind each step, and offer additional examples to solidify your understanding. By the end, you'll not only know the answer but also possess the skills to tackle any decimal-to-fraction conversion confidently.

Understanding Decimal to Fraction Conversion

Before jumping into the conversion of 0.64, let's establish the fundamental principles. Decimals represent parts of a whole number, expressed as tenths, hundredths, thousandths, and so on. Fractions, on the other hand, represent a part of a whole using a numerator (top number) and a denominator (bottom number). Converting a decimal to a fraction involves expressing the decimal's value as a ratio of two integers.

The key to this conversion lies in recognizing the place value of the last digit in the decimal. In 0.64, the last digit, 4, is in the hundredths place. This indicates that the decimal represents 64 hundredths.

Converting 0.64 to a Fraction: A Step-by-Step Guide

Here's how to convert 0.64 to a fraction, breaking it down step-by-step:

Step 1: Write the decimal as a fraction with a denominator of 1.

This is the initial step to formalize the conversion process. We write 0.64 as:

0.64/1

Step 2: Multiply both the numerator and the denominator by a power of 10 to eliminate the decimal point.

Since the decimal 0.64 has two digits after the decimal point, we multiply both the numerator and the denominator by 10². This effectively shifts the decimal point two places to the right, removing it entirely.

(0.64 × 100) / (1 × 100) = 64/100

Step 3: Simplify the fraction to its lowest terms.

This is crucial for expressing the fraction in its simplest form. To simplify, we find the greatest common divisor (GCD) of the numerator (64) and the denominator (100). The GCD is the largest number that divides both 64 and 100 without leaving a remainder.

Finding the GCD can be done through various methods, including prime factorization or the Euclidean algorithm. Let's use prime factorization:

• Prime factorization of 64: 2 x 2 x 2 x 2 x 2 x 2 = 2⁶
• Prime factorization of 100: 2 x 2 x 5 x 5 = 2² x 5²

The common factors are two 2's (2²). Therefore, the GCD is 4.

Now, we divide both the numerator and the denominator by the GCD (4):

64/4 = 16 100/4 = 25

Therefore, the simplified fraction is 16/25.

Verification and Alternative Methods

To verify our result, we can convert the simplified fraction back to a decimal:

16 ÷ 25 = 0.64

This confirms that our conversion is correct.

While the above method is the most common, there are alternative approaches:

Method 2: Directly Identifying the Fraction

Recognizing that 0.64 represents 64 hundredths, we could directly write it as 64/100 and then simplify as shown in Step 3. This method is quicker if you immediately recognize the place value of the decimal.

Method 3: Using a Calculator (For Checking)

While not a primary method for understanding the process, a calculator can be used to verify the result after simplifying the fraction. Most calculators have a function to convert decimals to fractions.

Expanding Your Understanding: More Examples

Let's apply the same principles to a few more examples:

Example 1: Converting 0.75 to a fraction:

1. Write as a fraction: 0.75/1
2. Multiply by 100: (0.75 × 100) / (1 × 100) = 75/100
3. Simplify (GCD = 25): 75/25 = 3; 100/25 = 4. The simplest form is 3/4.

Example 2: Converting 0.125 to a fraction:

1. Write as a fraction: 0.125/1
2. Multiply by 1000: (0.125 × 1000) / (1 × 1000) = 125/1000
3. Simplify (GCD = 125): 125/125 = 1; 1000/125 = 8. The simplest form is 1/8.

Example 3: Converting 0.375 to a fraction:

1. Write as a fraction: 0.375/1
2. Multiply by 1000: (0.375 × 1000)/(1 × 1000) = 375/1000
3. Simplify (GCD = 125): 375/125 = 3; 1000/125 = 8. The simplest form is 3/8.

Example 4: Converting a repeating decimal

Repeating decimals require a slightly different approach. For example, to convert 0.333... (0.3 recurring) to a fraction:

1. Let x = 0.333...
2. Multiply by 10: 10x = 3.333...
3. Subtract the first equation from the second: 10x - x = 3.333... - 0.333...
4. This simplifies to 9x = 3
5. Solve for x: x = 3/9
6. Simplify: x = 1/3

Dealing with More Complex Decimals

The principles remain the same even with more complex decimals. The key is to always multiply by the appropriate power of 10 to eliminate the decimal point and then simplify the resulting fraction. For example, converting 0.00625 involves multiplying by 10,000.

Conclusion: Mastering Decimal-to-Fraction Conversion

Converting decimals to fractions is a fundamental skill in mathematics with applications across various fields. By understanding the underlying principles of place value, greatest common divisors, and the steps outlined in this guide, you can confidently convert any decimal to its simplest fraction form. Remember to practice regularly, and you'll soon master this important skill. The ability to seamlessly convert between decimals and fractions enhances your mathematical fluency and problem-solving capabilities.