The decimal 0.737373... (or 0.73 repeating, often written as 0.73) might seem simple, but converting it to a fraction requires a specific method. This guide will walk you through the process step-by-step, explaining the underlying math and providing you with a solid understanding of how to handle repeating decimals. We'll also tackle some common related questions.
Understanding Repeating Decimals
Before we dive into the conversion, let's clarify what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digits "73" repeat endlessly. We represent this using a bar over the repeating part: 0.. Understanding this notation is crucial for the conversion process.
Converting 0.73 Repeating to a Fraction: A Step-by-Step Guide
Here's how to convert the repeating decimal 0.73 into its fractional equivalent:
Step 1: Set up an equation.
Let x = 0.737373...
Step 2: Multiply by 100.
We multiply both sides of the equation by 100 because two digits repeat (7 and 3). If only one digit repeated, we'd multiply by 10; if three digits repeated, we'd multiply by 1000, and so on.
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 = 73.737373...). Notice how the repeating part cancels out:
100x - x = 73.737373... - 0.737373...
This simplifies to:
99x = 73
Step 4: Solve for x.
Divide both sides of the equation by 99 to isolate x:
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.
Can 0.73 Repeating be Simplified Further?
No, 73/99 is the simplest form of the fraction. 73 is a prime number, meaning it's only divisible by 1 and itself. Since 73 and 99 share no common factors other than 1, the fraction cannot be reduced further.
What if I have a different repeating decimal?
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, allowing for subtraction to eliminate the repeating section. For example, if you had 0.123123..., you'd multiply by 1000.
How to convert a mixed repeating decimal to a fraction?
For mixed repeating decimals (like 0.12333...), you need a slightly different approach. You'll need to consider the non-repeating and repeating parts separately, working with them in a similar way as the example above but adjusted for the non-repeating part. Multiple equations might be needed. This situation will require more advanced algebraic steps.
This comprehensive guide should provide you with the knowledge and tools to confidently convert repeating decimals like 0.73 into their fractional equivalents. Remember the key steps: set up an equation, multiply by the appropriate power of 10, subtract, and solve for x.
