3.2 repeating as a fraction

Art Scpae

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10-10-2024

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Unmasking the Mystery: 3.2 Repeating as a Fraction

Have you ever encountered a decimal like 3.2222... and wondered how to express it as a fraction? It might seem like a tricky task, but there's a straightforward method to convert repeating decimals into their fractional counterparts. Let's dive into the process and demystify the conversion of 3.2 repeating.

Understanding the Problem

The decimal 3.2 repeating, denoted as 3.2̅, signifies that the digit '2' repeats infinitely after the decimal point. Our aim is to find a fraction that represents this value precisely.

The Conversion Method

The core idea involves setting up an equation and manipulating it to isolate the desired fractional value. Here's how we do it:

1. Assign a Variable: Let 'x' be equal to the decimal we're working with, i.e., x = 3.2̅.

2. Multiply to Shift the Decimal: Multiply both sides of the equation by 10. This shifts the decimal one place to the right: 10x = 32.2̅.

3. Subtract the Original Equation: Now, subtract the original equation (x = 3.2̅) from the modified equation (10x = 32.2̅):

   10x = 32.2̅
   - x = 3.2̅
   -----------------
   9x = 29 
   

4. Solve for 'x': Divide both sides by 9 to isolate 'x':

   x = 29/9 
   

Therefore, 3.2 repeating is equivalent to the fraction 29/9.

Additional Insights

• Simplifying Fractions: The fraction 29/9 is already in its simplest form, as 29 and 9 share no common factors other than 1.
• Generalization: This method can be applied to any repeating decimal. The key is to multiply the equation by a power of 10 that shifts the decimal to align the repeating block.

Practical Example

Let's say we have a repeating decimal like 0.565656...

1. Let x = 0.56̅.
2. Multiply by 100: 100x = 56.56̅.
3. Subtract the original equation: 99x = 56.
4. Solve for x: x = 56/99.

Conclusion

Converting a repeating decimal to a fraction may seem daunting initially, but with the right approach, it becomes a simple algebraic manipulation. By understanding the steps involved, you can easily express any repeating decimal as a fraction, unlocking its hidden mathematical representation.

Attribution:

The core method described in this article is a standard technique widely used in mathematics. The provided examples and explanations are original content created for this article, building upon the foundational method.