Many find converting repeating decimals to fractions tricky. Let's break down how to transform the repeating decimal 0.08333... (where the 3 repeats infinitely) into its fractional equivalent. This process involves algebra and understanding the nature of repeating decimals.
Understanding Repeating Decimals
The key to solving this lies in recognizing the repeating part of the decimal. In 0.08333..., the digit '3' repeats infinitely. This is crucial for setting up the algebraic equation.
The Conversion Process
Here's a step-by-step guide to convert 0.08333... into a fraction:
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Let x equal the repeating decimal: We start by assigning a variable, typically 'x', to the repeating decimal:
x = 0.08333...
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Multiply to shift the repeating part: We need to manipulate the equation so the repeating part aligns. Multiply both sides of the equation by a power of 10 that shifts the repeating section to the left of the decimal point. Since the repeating part starts after two digits, we'll multiply by 100:
100x = 8.333...
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Multiply again to isolate the repeating part: Now we multiply by another power of 10 to create a second equation where the repeating part is again shifted. We only need to move the decimal point once to have the same repeating part as in step 2, so we multiply by 10:
1000x = 83.333...
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Subtract the equations: Subtracting the equation from step 2 from the equation in step 3 will eliminate the repeating decimal portion. This is the clever part that allows us to solve for x:
1000x - 100x = 83.333... - 8.333...
This simplifies to:
900x = 75
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Solve for x: Now we solve for 'x' by dividing both sides of the equation by 900:
x = 75/900
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Simplify the fraction: Finally, we simplify the fraction by finding the greatest common divisor (GCD) of 75 and 900, which is 75. Dividing both the numerator and the denominator by 75 gives us the simplified fraction:
x = 1/12
Therefore, the fraction equivalent of the repeating decimal 0.08333... is 1/12.
Verification: Decimal to Fraction Conversion Check
To double-check our work, divide 1 by 12. You'll find the result is indeed 0.08333..., confirming our conversion.
Key Takeaways and Further Practice
Converting repeating decimals to fractions requires careful attention to aligning the repeating part through multiplication and subtraction. Practice is key to mastering this technique. Try converting other repeating decimals to fractions to solidify your understanding. Remember to always simplify your final fraction to its lowest terms. Understanding this process will greatly improve your mathematical skills.
