.73 Repeating: Fraction Formula Unlocked

2 min read 10-03-2025

.73 Repeating: Fraction Formula Unlocked

The seemingly simple decimal .737373... (or 0.73 repeating) hides a fascinating mathematical concept: converting repeating decimals into fractions. Understanding this process not only helps in solving specific math problems but also deepens your understanding of number systems. This guide will unravel the mystery of converting .73 repeating into its fractional equivalent, providing a step-by-step approach and addressing common questions.

Understanding Repeating Decimals

Before diving into the conversion, let's clarify what a repeating decimal is. A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or a group of digits that repeat infinitely. In our case, the digits "73" repeat endlessly. We denote this with a bar over the repeating part: 0. $\overline{73}$ .

Converting .73 Repeating to a Fraction: The Step-by-Step Guide

Here's how to convert 0. $\overline{73}$ into a fraction:

Step 1: Assign a Variable

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

x = 0. $\overline{73}$

Step 2: Multiply to Shift the Decimal

We need to manipulate the equation to isolate the repeating part. Multiply both sides of the equation by 100 (because there are two repeating digits):

100x = 73. $\overline{73}$

Step 3: Subtract the Original Equation

Now, subtract the original equation (x = 0. $\overline{73}$ ) from the equation in Step 2:

100x - x = 73. $\overline{73}$ - 0. $\overline{73}$

This simplifies to:

99x = 73

Step 4: Solve for x

Divide both sides by 99 to solve for x:

x = 73/99

Therefore, the fraction equivalent of 0. $\overline{73}$ 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)

This section addresses common questions surrounding repeating decimals and their fractional equivalents.

How do you convert other repeating decimals to fractions?

The method above works for any repeating decimal. The key is to multiply by a power of 10 that shifts the repeating part to align with itself. If you have a single repeating digit, multiply by 10. If you have three repeating digits, multiply by 1000, and so on. Then subtract the original equation to eliminate the repeating part.

What if the repeating decimal starts after a non-repeating part?

For example, if you have a number like 1.2 $\overline{3}$ , you first handle the non-repeating part separately. You can rewrite this as 1.2 + 0.0 $\overline{3}$ . Convert 0.0 $\overline{3}$ to a fraction using the method above, and then add the fraction to 1.2 (or its fractional equivalent, 12/10 or 6/5).

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. A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. Repeating decimals are, therefore, rational numbers.

Are there any shortcuts for converting repeating decimals to fractions?

While the step-by-step method is reliable, there are no significant shortcuts that significantly reduce the number of steps. The core concept of multiplying and subtracting remains the foundation of the conversion.

This comprehensive guide explains how to convert 0.73 repeating to a fraction and provides answers to common questions, ensuring a solid understanding of this important mathematical concept. Remember, practice makes perfect; try converting other repeating decimals to fractions to solidify your understanding.