Converting repeating decimals, like 0.737373..., into fractions might seem daunting, but it's a straightforward process with a simple formula. This tutorial provides a step-by-step guide, clarifying the method and addressing common questions. Understanding this process enhances your mathematical skills and provides a deeper understanding of the relationship between decimals and fractions.
Understanding Repeating Decimals
Before diving 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, we're dealing with 0.737373..., where "73" repeats endlessly. We often denote this with a bar over the repeating digits: 0..
How to Convert 0.73 Repeating to a Fraction
Here's a step-by-step method to convert 0. into a fraction:
Step 1: Assign a Variable
Let's represent the repeating decimal with a variable, 'x':
x = 0.737373...
Step 2: 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...
Step 3: Subtract the Original Equation
Subtract the original equation (x = 0.737373...) from the equation obtained in Step 2:
100x - x = 73.737373... - 0.737373...
This simplifies to:
99x = 73
Step 4: 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.
Can All Repeating Decimals Be Converted to Fractions?
Yes! Every repeating decimal can be converted to a fraction using a similar method. The key is identifying the repeating block of digits and multiplying by the appropriate power of 10. For example, if the repeating block has three digits, you'd multiply by 1000.
What if the Repeating Decimal Starts After a Non-Repeating Part?
For example, consider 0.2. The method is similar, but requires a slightly different approach:
- Let x = 0.2333...
- Multiply by 10: 10x = 2.333...
- Multiply by 100: 100x = 23.333...
- Subtract: 100x - 10x = 23.333... - 2.333... which simplifies to 90x = 21
- Solve for x: x = 21/90 = 7/30
How Do I Simplify a Fraction After Conversion?
Once you've converted the repeating decimal to a fraction, always simplify it to its lowest terms. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by the GCD. Many online calculators or tools can help with this process.
Why is this Conversion Important?
Understanding this conversion highlights the interconnectedness of different number systems. It demonstrates that seemingly infinite decimals can be expressed as precise fractions, which is fundamental in many areas of mathematics, science, and engineering.
By following these steps and understanding the underlying principles, you can confidently convert any repeating decimal into its equivalent fraction. Remember, practice makes perfect! Try converting other repeating decimals to solidify your understanding.
