The decimal .73 repeating (written as 0.737373...) might seem tricky to convert into a fraction, but with a simple mathematical trick, it becomes surprisingly straightforward. This guide will show you how to convert repeating decimals like this into fractions quickly and easily, providing you with a clear understanding of the process.
Understanding Repeating Decimals
Repeating decimals, also known as recurring decimals, are numbers that have a digit or a group of digits that repeat infinitely after the decimal point. In the case of 0.737373..., the digits "73" repeat endlessly. Understanding this repeating pattern is key to converting it into a fraction.
The Conversion Method: A Step-by-Step Guide
Here's how to convert 0.737373... into a fraction:
-
Assign a Variable: Let's represent the repeating decimal with a variable, say 'x':
x = 0.737373... -
Multiply to Shift the Decimal: Multiply both sides of the equation by 100 (because two digits repeat). This shifts the repeating block to the left of the decimal point:
100x = 73.737373... -
Subtract the Original Equation: Now, subtract the original equation (x) from the new equation (100x):
100x - x = 73.737373... - 0.737373...This simplifies to:
99x = 73 -
Solve for x: Divide both sides by 99 to isolate 'x':
x = 73/99
Therefore, the fraction equivalent of the repeating decimal 0.737373... 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.
Frequently Asked Questions (FAQs)
Here are some common questions about converting repeating decimals to fractions:
How do you convert other repeating decimals to fractions?
The method described above works for any repeating decimal. The key is to multiply by a power of 10 that shifts the repeating block to the left of the decimal point. If only one digit repeats (e.g., 0.333...), multiply by 10. If three digits repeat (e.g., 0.123123...), multiply by 1000, and so on. Then, subtract the original equation and solve for the variable.
What if the repeating decimal has a non-repeating part?
If the decimal has a non-repeating part before the repeating part (e.g., 0.2737373...), you'll need a slightly modified approach. You'll still use the same principle of multiplying and subtracting, but you'll need to account for the non-repeating part in your calculations.
Can all repeating decimals be expressed as fractions?
Yes, all repeating decimals can be expressed as fractions. This is a fundamental property of rational numbers (numbers that can be expressed as a fraction of two integers).
Are there any online tools to help with this conversion?
Yes, several online calculators can convert repeating decimals to fractions. Simply search for "repeating decimal to fraction calculator" on your preferred search engine.
This method provides a clear and efficient way to convert repeating decimals into fractions. Remember the steps, and you'll be able to tackle any repeating decimal with confidence.
