What Is The Fraction For The Following Decimal Expansion

Decoding Decimal Expansions: A Comprehensive Guide to Converting Decimals to Fractions

Converting decimal expansions to fractions might seem daunting, but with a systematic approach and a solid understanding of the underlying principles, it becomes a manageable and even enjoyable mathematical exercise. This comprehensive guide will walk you through various methods for converting decimals to fractions, covering everything from simple terminating decimals to more complex repeating decimals. We'll explore the theory behind the conversions, provide step-by-step examples, and offer tips and tricks to streamline the process.

Understanding Decimal Expansions

Before diving into the conversion methods, let's solidify our understanding of decimal expansions. A decimal expansion is simply a way of representing a number using a base-10 system. The digits to the left of the decimal point represent whole numbers, while the digits to the right represent fractions of a whole. Each digit to the right of the decimal point represents a power of 10 in the denominator:

• The first digit after the decimal point represents tenths (1/10).
• The second digit represents hundredths (1/100).
• The third digit represents thousandths (1/1000), and so on.

For instance, the decimal 0.75 represents 7 tenths + 5 hundredths, which is equivalent to 7/10 + 5/100 = 75/100. This understanding forms the foundation of converting decimals to fractions.

Converting Terminating Decimals to Fractions

Terminating decimals are those that have a finite number of digits after the decimal point. Converting these to fractions is relatively straightforward. The process involves:

1. Identify the place value of the last digit: This determines the denominator of the fraction. For example, if the last digit is in the hundredths place, the denominator is 100.

2. Write the digits after the decimal point as the numerator: This forms the numerator of the fraction.

3. Simplify the fraction: Reduce the fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD.

Example 1: Convert 0.75 to a fraction.

The last digit (5) is in the hundredths place, so the denominator is 100. The numerator is 75. Therefore, the fraction is 75/100. Simplifying this fraction by dividing both numerator and denominator by 25 (their GCD), we get 3/4.

Example 2: Convert 0.625 to a fraction.

The last digit (5) is in the thousandths place, so the denominator is 1000. The numerator is 625. The fraction is 625/1000. Simplifying by dividing by 125 (their GCD), we obtain 5/8.

Example 3: Convert 0.12 to a fraction.

The last digit (2) is in the hundredths place, so the denominator is 100. The numerator is 12. The fraction is 12/100. Simplifying by dividing by 4 (their GCD), we get 3/25.

Converting Repeating Decimals to Fractions

Repeating decimals, also known as recurring decimals, have a digit or a sequence of digits that repeat infinitely. Converting these to fractions requires a slightly more sophisticated approach. The key is to use algebra to eliminate the repeating part of the decimal.

Method 1: Using Algebra (for single repeating digits)

1. Let x equal the repeating decimal.

2. Multiply x by a power of 10 to move the repeating block to the left of the decimal. The power of 10 will be equal to the number of digits in the repeating block.

3. Subtract the original equation (x) from the new equation (10x, 100x, etc.). This will eliminate the repeating digits.

4. Solve for x. This will give you the fraction representation of the repeating decimal.

Example 4: 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... This simplifies to 9x = 3.

4. Solve for x: x = 3/9 = 1/3

Method 2: Using Algebra (for multiple repeating digits)

This method extends the above, accommodating multiple repeating digits. The power of 10 used in step 2 will correspond to the length of the repeating sequence.

Example 5: Convert 0.142857142857... (0.142857 recurring) to a fraction.

1. Let x = 0.142857142857...

2. Multiply by 1,000,000 (since there are six repeating digits): 1,000,000x = 142857.142857...

3. Subtract: 1,000,000x - x = 142857.142857... - 0.142857... which simplifies to 999,999x = 142857

4. Solve for x: x = 142857/999999. This fraction can be simplified to 1/7.

Method 3: Using the Place Value Method (for simple cases)

For very simple repeating decimals, you can directly use the place value method. For example, 0.111... can be seen as 1/10 + 1/100 + 1/1000 + ... This is a geometric series that sums to 1/9.

Handling Mixed Numbers and Decimals

When you have a whole number combined with a decimal fraction, you first convert the decimal part to a fraction as described above. Then, you add the whole number part to the fractional part. This forms an improper fraction. You can also convert the mixed number to an improper fraction.

Example 6: Convert 3.75 to a fraction.

1. Convert 0.75 to a fraction: 0.75 = 3/4

2. Add the whole number part: 3 + 3/4 = 15/4

Advanced Cases and Considerations

While the methods above cover a wide range of decimal-to-fraction conversions, some scenarios might require more advanced techniques, such as dealing with non-repeating, infinite decimals (irrational numbers) which cannot be expressed as simple fractions. These often require approximations or the use of continued fractions, a topic beyond the scope of this introductory guide.

Conclusion: Mastering Decimal-to-Fraction Conversions

Converting decimal expansions to fractions is a crucial skill in mathematics. By understanding the principles of place value and applying the appropriate algebraic methods, you can effectively convert both terminating and repeating decimals into their fractional equivalents. Mastering this skill opens doors to deeper mathematical understanding and problem-solving capabilities. Remember to practice regularly to reinforce your understanding and improve your efficiency in performing these conversions. With consistent effort, converting decimals to fractions will become second nature.