0.15 Recurring As A Fraction

Decoding 0.15 Recurring: A Comprehensive Guide to Converting Repeating Decimals to Fractions

Understanding how to convert repeating decimals, like 0.15 recurring (often written as 0.151515...), into fractions is a fundamental skill in mathematics. This seemingly simple task involves a clever application of algebra and a deep understanding of place value. This article provides a comprehensive walkthrough, addressing the underlying principles, offering step-by-step solutions, exploring related concepts, and answering frequently asked questions. By the end, you'll not only know how to convert 0.15 recurring to a fraction, but also possess the tools to tackle any repeating decimal.

Understanding Repeating Decimals

Before diving into the conversion process, let's clarify what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. The repeating digits are indicated by a bar placed above them. For example:

• 0.333... is written as 0.3̅
• 0.142857142857... is written as 0.142857̅
• 0.151515... is written as 0.15̅

Our focus is on 0.15 recurring (0.15̅), which means the digits "15" repeat endlessly. Understanding this notation is crucial for the conversion process.

Step-by-Step Conversion of 0.15 Recurring to a Fraction

The conversion of a repeating decimal to a fraction relies on algebraic manipulation. Here's a step-by-step approach:

Step 1: Assign a Variable

Let's represent the repeating decimal with a variable, say 'x':

x = 0.15̅

Step 2: Multiply to Shift the Repeating Block

Multiply both sides of the equation by a power of 10 that shifts the repeating block to the left by one full repeating cycle. Since the repeating block "15" consists of two digits, we multiply by 100:

100x = 15.15̅

Step 3: Subtract the Original Equation

Subtracting the original equation (x = 0.15̅) from the equation obtained in Step 2 (100x = 15.15̅) eliminates the repeating decimal part:

100x - x = 15.15̅ - 0.15̅

This simplifies to:

99x = 15

Step 4: Solve for x

Now, solve for 'x' by dividing both sides of the equation by 99:

x = 15/99

Step 5: Simplify the Fraction

Finally, simplify the fraction by finding the greatest common divisor (GCD) of the numerator (15) and the denominator (99). The GCD of 15 and 99 is 3. Dividing both the numerator and the denominator by 3, we get the simplified fraction:

x = 5/33

Therefore, 0.15 recurring is equal to 5/33.

Mathematical Explanation and Proof

The method used above is based on the concept of infinite geometric series. A repeating decimal can be expressed as the sum of an infinite geometric series. For example, 0.15̅ can be written as:

0.15 + 0.0015 + 0.000015 + ...

This is a geometric series with the first term (a) = 0.15 and the common ratio (r) = 0.01. The sum of an infinite geometric series is given by the formula:

Sum = a / (1 - r) (where |r| < 1)

Substituting the values for our series:

Sum = 0.15 / (1 - 0.01) = 0.15 / 0.99 = 15/99 = 5/33

This confirms our result obtained through the algebraic method.

Extending the Technique: Converting Other Repeating Decimals

The method described above can be applied to any repeating decimal. The key is to identify the repeating block and multiply by the appropriate power of 10 to shift the repeating block. Let's look at a few examples:

• 0.7̅: Let x = 0.7̅. Multiply by 10: 10x = 7.7̅. Subtract x from 10x: 9x = 7. Therefore, x = 7/9.

• 0.285714̅: Let x = 0.285714̅. Multiply by 1,000,000: 1,000,000x = 285714.285714̅. Subtract x: 999,999x = 285714. Therefore, x = 285714/999999, which simplifies to 2/7.

• 0.36̅: Let x = 0.36̅. Multiply by 100: 100x = 36.36̅. Subtract x: 99x = 36. Therefore, x = 36/99 = 4/11.

Notice that the denominator is always a number with as many 9's as there are digits in the repeating block. This is a useful shortcut, but remember to always simplify the resulting fraction.

Dealing with Non-Repeating Parts: Mixed Repeating Decimals

Sometimes, you encounter decimals with a non-repeating part followed by a repeating part, such as 0.23̅. The approach remains similar, but requires a slightly modified strategy.

Example: Converting 0.23̅ to a fraction:

1. Let x = 0.23̅

2. Multiply by 10 to isolate the repeating part: 10x = 2.3̅

3. Multiply by 100 to shift the repeating block: 100x = 23.3̅

4. Subtract 10x from 100x: 90x = 21

5. Solve for x: x = 21/90

6. Simplify: x = 7/30

This method systematically handles the non-repeating initial digits before dealing with the repeating sequence.

Frequently Asked Questions (FAQ)

Q1: What if the repeating block is longer than two digits?

A: The process remains the same. Multiply by 10 raised to the power of the number of digits in the repeating block. For instance, for 0.123̅, you would multiply by 1000.

Q2: Can all repeating decimals be converted to fractions?

A: Yes, all repeating decimals can be expressed as fractions. This is a fundamental property of rational numbers.

Q3: What about decimals that don't repeat?

A: Non-repeating, non-terminating decimals (like pi) are irrational numbers and cannot be expressed as fractions.

Q4: Is there a quicker way to convert simple repeating decimals?

A: For simple repeating decimals like 0.7̅ or 0.3̅, you can use the shortcut of placing the repeating digit over as many 9s as there are repeating digits. For example, 0.7̅ = 7/9 and 0.3̅ = 3/9 = 1/3. However, this shortcut doesn't apply to more complex repeating decimals.

Conclusion

Converting repeating decimals to fractions is a valuable mathematical skill with applications in various fields. By understanding the underlying principles of infinite geometric series and employing the systematic steps outlined in this article, you can confidently tackle any repeating decimal and express it as a fraction. Remember, the key is to manipulate the equation algebraically to eliminate the repeating part and solve for the unknown variable. Practice is crucial to mastering this technique, and with consistent effort, you'll find it becomes second nature. The ability to confidently work with decimals and fractions solidifies your mathematical foundation and opens doors to more advanced mathematical concepts.