Understanding 1.33333 as a Fraction
The decimal 1.33333 represents a repeating decimal, which means the '3' after the decimal point goes on forever. It might seem tricky to convert this repeating decimal into a fraction, but it's actually a straightforward process. Here's how:
1. Set up an Equation:
- Let 'x' equal the decimal:
- x = 1.33333...
2. Multiply to Shift the Decimal:
- Multiply both sides of the equation by 10:
- 10x = 13.33333...
3. Subtract the Original Equation:
- Subtract the original equation (x = 1.33333...) from the equation in step 2:
- 10x - x = 13.33333... - 1.33333...
- This simplifies to 9x = 12
4. Solve for x:
- Divide both sides by 9:
- x = 12/9
5. Simplify:
- The fraction 12/9 can be simplified by dividing both numerator and denominator by their greatest common factor, which is 3.
- x = 4/3
Therefore, 1.33333 as a fraction is 4/3.
Understanding Repeating Decimals
The process we used works because repeating decimals represent fractions with denominators that are powers of 10 (like 10, 100, 1000, etc.) minus 1. This is because multiplying by 10 shifts the decimal place, and subtracting the original equation eliminates the repeating part.
Example:
- 0.33333... represents 3/9 (which simplifies to 1/3)
- 0.66666... represents 6/9 (which simplifies to 2/3)
- 0.142857142857... represents 142857/999999
Tips for Converting Repeating Decimals to Fractions
- Identify the repeating block: Determine the digits that repeat. In 1.33333..., the repeating block is '3'.
- Count the digits in the repeating block: The number of digits in the repeating block tells you what to multiply by (10 for one digit, 100 for two digits, etc.).
- Use the subtraction method: Subtract the original equation from the multiplied equation to eliminate the repeating part.
- Simplify the fraction: Always simplify the fraction to its lowest terms.
Conclusion
Converting a repeating decimal like 1.33333 to a fraction might seem complicated at first, but it follows a simple and consistent method. By understanding the process and practicing a few examples, you can confidently convert any repeating decimal into its equivalent fractional form.