Converting 0.8 Repeating to a Fraction: A Step-by-Step Guide
Have you ever encountered a decimal number like 0.8888... and wondered how to express it as a fraction? This is where the concept of repeating decimals comes into play. Let's break down how to convert 0.8 repeating (denoted as 0.8̅) into a fraction.
Understanding Repeating Decimals
A repeating decimal is a decimal number where one or more digits repeat infinitely after the decimal point. In our case, the digit "8" repeats indefinitely.
Setting Up the Equation
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Assign a Variable: Let's represent the repeating decimal 0.8̅ with the variable 'x'. So, x = 0.8̅.
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Multiply to Shift the Decimal: Multiply both sides of the equation by 10. This shifts the decimal one place to the right: 10x = 8.8̅
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Subtract the Original Equation: Now, subtract the original equation (x = 0.8̅) from the multiplied equation (10x = 8.8̅):
10x = 8.8̅ - x = 0.8̅ ---------------- 9x = 8 -
Solve for x: Isolate 'x' by dividing both sides by 9:
x = 8/9
The Result
Therefore, 0.8̅ (0.8 repeating) is equivalent to the fraction 8/9.
Verifying the Conversion
You can always verify your answer by dividing the numerator (8) by the denominator (9) using a calculator. The result will be 0.8888..., confirming that our conversion is correct.
Key Points to Remember
- Identifying the Repeating Block: When dealing with repeating decimals, clearly identify the block of digits that repeats.
- Multiplying by the Appropriate Power of 10: Choose the power of 10 that shifts the decimal to align the repeating blocks in the two equations.
- Subtracting to Eliminate the Repeating Part: The subtraction step is crucial to remove the repeating portion, leaving a simple equation to solve.
Conclusion
Converting a repeating decimal to a fraction involves a systematic approach of setting up an equation, shifting the decimal, and eliminating the repeating portion. This process allows you to express any repeating decimal as a simplified fraction, offering a more precise representation of the number.