The seemingly simple decimal 0.737373... (or 0.73 repeating) hides a fascinating mathematical relationship. Understanding how to convert repeating decimals to fractions is a key skill in arithmetic, and this article will guide you through the process, unveiling the fraction equivalent of 0.73 repeating. We'll also explore related concepts and answer frequently asked questions.
Understanding Repeating Decimals
Repeating decimals, also known as recurring decimals, are decimals where one or more digits repeat infinitely. They're represented by a bar over the repeating digits. In our case, 0.73 repeating is written as 0.7̅3̅. This notation clearly indicates that "73" repeats endlessly. It's crucial to distinguish repeating decimals from terminating decimals (like 0.75), which have a finite number of digits.
Converting Repeating Decimals to Fractions: A Step-by-Step Guide
Here's the method to convert 0.7̅3̅ into a fraction:
Step 1: Set up an equation.
Let x = 0.737373...
Step 2: Multiply by a power of 10.
We need to multiply x by a power of 10 that shifts the repeating part to the left of the decimal point. Since the repeating block is two digits long ("73"), we'll multiply by 100:
100x = 73.737373...
Step 3: Subtract the original equation.
Subtracting the original equation (x = 0.737373...) from the equation in Step 2 eliminates the repeating part:
100x - x = 73.737373... - 0.737373...
This simplifies to:
99x = 73
Step 4: Solve for x.
Divide both sides of the equation by 99:
x = 73/99
Therefore, the fraction equivalent of 0.73 repeating is 73/99. This fraction is in its simplest form because 73 is a prime number and doesn't share any common factors with 99.
How to Convert Other Repeating Decimals to Fractions
The method above can be applied to any repeating decimal. The key is to multiply by the appropriate power of 10 to align the repeating part before subtraction. For instance, if you had a decimal with three repeating digits, you would multiply by 1000.
Frequently Asked Questions (FAQ)
What is the simplest form of the fraction for 0.73 repeating?
As shown above, the simplest form of the fraction for 0.73 repeating is 73/99.
Can all repeating decimals be expressed as fractions?
Yes, all repeating decimals can be expressed as fractions using the method described above. This is a fundamental property of rational numbers (numbers that can be expressed as a fraction of two integers).
What if the repeating decimal starts after some non-repeating digits?
If there are non-repeating digits before the repeating block, you'll need to adjust the process slightly. You still multiply by the power of 10 to align the repeating block, but the subtraction will be more complex, involving potentially removing the non-repeating digits as well.
Are there any shortcuts for converting repeating decimals to fractions?
While the method outlined above is generally the most straightforward, familiarity with common repeating decimal-fraction pairs can speed up the process. For instance, knowing that 1/9 = 0.1̅ and 1/11 = 0.0̅9̅ can often help you determine the fraction equivalent quickly for related repeating decimals.
This comprehensive guide provides a clear understanding of how to convert repeating decimals to fractions, equipping you with the skills to tackle similar problems effectively. Remember to practice to solidify your understanding!
