Convert .73 Repeating to a Fraction: Easy Guide

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12-03-2025

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Converting repeating decimals to fractions might seem daunting, but it's a straightforward process once you understand the steps. This guide will walk you through converting the repeating decimal 0.737373... (represented as 0.$\overline{73}$) into a fraction. We'll break down the method and explore similar problems to solidify your understanding.

Understanding Repeating Decimals

A repeating decimal, also known as a recurring decimal, is a decimal that has a digit or a group of digits that repeat infinitely. In our case, the digits "73" repeat endlessly. To represent this mathematically, we use a bar over the repeating part: 0.$\overline{73}$. This notation clearly indicates which digits repeat.

Converting 0.$\overline{73}$ to a Fraction: Step-by-Step

Here's the method to convert 0.$\overline{73}$ into its fractional equivalent:

Step 1: Set up an equation.

Let x = 0.737373...

Step 2: Multiply by a power of 10.

Since two digits repeat ("73"), we multiply both sides of the equation by 100 (10²):

100x = 73.737373...

Step 3: Subtract the original equation.

Subtract the equation from Step 1 (x = 0.737373...) from the equation in Step 2:

100x - x = 73.737373... - 0.737373...

This simplifies to:

99x = 73

Step 4: Solve for x.

Divide both sides by 99:

x = 73/99

Therefore, 0.$\overline{73}$ as a fraction 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 process remains the same for other repeating decimals, but the power of 10 you multiply by will depend on the number of repeating digits.

• One repeating digit: Multiply by 10.
• Two repeating digits: Multiply by 100.
• Three repeating digits: Multiply by 1000, and so on.

For example, to convert 0.$\overline{4}$ to a fraction:

1. Let x = 0.444...
2. 10x = 4.444...
3. 10x - x = 4.444... - 0.444...
4. 9x = 4
5. x = 4/9

And to convert 0.1$\overline{23}$ (note only the 23 repeats):

1. Let x = 0.1232323...
2. 10x = 1.232323...
3. 1000x = 123.232323...
4. 1000x - 10x = 123.232323... - 1.232323...
5. 990x = 122
6. x = 122/990 = 61/495

Frequently Asked Questions

What if the repeating decimal starts after some non-repeating digits?

If you have a decimal with a non-repeating part followed by a repeating part (e.g., 0.1$\overline{2}$), you still follow a similar process. You'll need to adjust the multiplication and subtraction steps to account for the non-repeating portion. It's usually easier to deal with the integer portion separately and focus on converting the repeating decimal portion to a fraction.

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).

Why does this method work?

This method works because it cleverly uses the properties of infinite geometric series. By subtracting the original equation from the multiplied equation, we eliminate the infinitely repeating part, leaving a solvable algebraic equation.

This guide provides a clear and comprehensive method for converting repeating decimals to fractions. By following these steps and practicing with different examples, you’ll master this essential math skill.