7.33333 as a fraction

Daniel Parker

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28-02-2025

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Meta Description: Learn how to convert the repeating decimal 7.33333... into a fraction. This comprehensive guide provides a clear, step-by-step process, explaining the method and showing you how to solve similar problems. Master fraction conversion with simple techniques and practice examples.

The number 7.33333... is a decimal number with a repeating decimal part. Converting it into a fraction might seem tricky, but it's a straightforward process once you understand the method. This article will guide you step-by-step through the conversion.

Understanding Repeating Decimals

Before we begin, it's crucial to understand what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digit '3' repeats endlessly. We often represent this with a bar over the repeating digit(s): 7.3̅

Converting 7.33333... to a Fraction

Here's how to convert 7.33333... (or 7.3̅) into a fraction:

Step 1: Set up an Equation

Let's represent the repeating decimal as 'x':

x = 7.33333...

Step 2: Multiply to Shift the Decimal

Multiply both sides of the equation by 10. We choose 10 because only one digit is repeating. If more digits repeated, we would use a higher power of 10 (e.g., 100 for two repeating digits, 1000 for three, etc.).

10x = 73.33333...

Step 3: Subtract the Original Equation

Subtract the original equation (x = 7.33333...) from the equation we obtained in Step 2 (10x = 73.33333...):

10x - x = 73.33333... - 7.33333...

This simplifies to:

9x = 66

Step 4: Solve for x

Divide both sides of the equation by 9 to isolate 'x':

x = 66/9

Step 5: Simplify the Fraction

Finally, simplify the fraction by finding the greatest common divisor (GCD) of 66 and 9. The GCD of 66 and 9 is 3. Divide both the numerator and the denominator by 3:

x = 22/3

Therefore, 7.33333... as a fraction is 22/3.

How to Convert Other Repeating Decimals

The method described above works for other repeating decimals. The key is to multiply by the appropriate power of 10 to shift the repeating part of the decimal. Then, subtract the original equation to eliminate the repeating digits. Finally, simplify the resulting fraction.

Here's an example:

Convert 0.454545... to a fraction:

1. x = 0.454545...
2. 100x = 45.454545... (multiply by 100 since two digits repeat)
3. 100x - x = 45.454545... - 0.454545... => 99x = 45
4. x = 45/99
5. x = 5/11 (simplified)

Practice Problems

Try converting these repeating decimals into fractions using the method outlined above:

• 0.66666...
• 1.272727...
• 0.123123123...

Remember to always simplify your fractions to their lowest terms!

Conclusion

Converting repeating decimals to fractions is a valuable skill in mathematics. By following the step-by-step process described above, you can confidently handle any repeating decimal conversion. The key is understanding the concept of multiplying by the appropriate power of 10 and then subtracting the original equation to isolate the repeating part and simplifying the fraction at the end. Now you can confidently tackle similar problems and further your understanding of fractions and decimals.