Many encounter the seemingly simple task of converting a repeating decimal, like 0.737373..., into a fraction. While the process itself isn't overly complex, a common mistake can lead to an incorrect answer. This post will guide you through the correct method, highlighting the pitfall to avoid and providing a clear understanding of the underlying principles.
Understanding Repeating Decimals
Repeating decimals, also known as recurring decimals, are numbers with digits that repeat infinitely. 0.737373... is a repeating decimal where the digits "73" repeat endlessly. This is often represented as 0.73̅ or 0.73. The bar or dot notation indicates the repeating block of digits.
The Common Mistake: Treating it Like a Terminating Decimal
A common error is treating the repeating decimal as if it were a terminating decimal (a decimal that ends). For instance, someone might incorrectly attempt to convert 0.73 to a fraction as 73/100. This is wrong because it ignores the repeating nature of the decimal. 73/100 represents 0.73, not 0.737373...
The Correct Method: Algebraic Manipulation
The correct way to convert a repeating decimal to a fraction involves algebraic manipulation. Here's a step-by-step guide using the example of 0.73:
-
Let x = the repeating decimal: Let x = 0.737373...
-
Multiply to shift the repeating block: Multiply both sides of the equation by 100 (because there are two digits in the repeating block): 100x = 73.737373...
-
Subtract the original equation: Subtract the original equation (x = 0.737373...) from the equation obtained in step 2:
100x - x = 73.737373... - 0.737373...
This simplifies to: 99x = 73
-
Solve for x: Divide both sides by 99:
x = 73/99
Therefore, the fraction representation of the repeating decimal 0.737373... is 73/99.
Why This Method Works
This method works because subtracting the original equation from the multiplied equation effectively cancels out the repeating decimal portion, leaving a simple algebraic equation to solve. The multiplier (100 in this case) is always a power of 10, where the power corresponds to the number of digits in the repeating block.
What if the Repeating Block Starts After a Non-Repeating Part?
If you have a repeating decimal with a non-repeating part, you will follow a slightly adjusted method. For instance, to convert 0.25737373... to a fraction, you first express it as the sum of the non-repeating part and the repeating part. Then, separately convert the repeating part to a fraction as described above. Finally, add the two fractions.
Frequently Asked Questions (FAQ)
How do I convert other repeating decimals to fractions?
The method described above applies to all repeating decimals. The only change is the multiplier you use in step 2. If the repeating block has n digits, multiply by 10n.
Can all repeating decimals be converted to fractions?
Yes, every repeating decimal can be expressed as a fraction. This is a fundamental property of rational numbers.
Are there any online tools to help with this conversion?
Yes, several online calculators can help convert repeating decimals to fractions. A simple search for "repeating decimal to fraction calculator" will yield many results. However, understanding the underlying process is crucial for solving these problems and avoiding potential mistakes.
By understanding this process and avoiding the common mistake of treating repeating decimals as terminating ones, you can accurately convert any repeating decimal into its equivalent fraction. Remember to always use algebraic manipulation to handle the infinitely repeating digits effectively.
