Converting repeating decimals, like 0.737373..., into fractions can seem daunting, but it's a straightforward process once you understand the underlying method. This guide will walk you through the steps, explaining the logic behind the formula and providing examples to solidify your understanding. We'll also address some common questions people have about this type of conversion.
Understanding Repeating Decimals
A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, we're dealing with 0.73 repeating, which is written as 0.73̅ or 0.7̅3̅. The bar over the digits indicates the repeating part. The key to converting this to a fraction lies in manipulating algebraic equations.
The Formula: Converting Repeating Decimals to Fractions
The general formula for converting a repeating decimal to a fraction is:
x = 0.dddddd...
where 'x' represents the repeating decimal and 'd' represents the repeating digit(s). To solve:
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Set x equal to the repeating decimal: x = 0.737373...
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Multiply x by 10n: where 'n' is the number of digits that repeat. In our case, n = 2 (since '73' repeats), so we multiply by 100: 100x = 73.737373...
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Subtract the original equation (step 1) from the result of step 2:
100x - x = 73.737373... - 0.737373...
This simplifies to: 99x = 73
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Solve for x: x = 73/99
Therefore, the fraction equivalent of 0.73 repeating is 73/99.
How to Simplify Fractions
While 73/99 is the correct fraction, it might not always be in its simplest form. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. In this instance, the GCD of 73 and 99 is 1, meaning the fraction is already in its simplest form.
What if the Repeating Decimal Doesn't Start Immediately?
Let's say you had a number like 0.2737373... This involves a slightly different approach:
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Identify the non-repeating part: The non-repeating part is 0.2.
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Isolate the repeating part: The repeating part is 0.0737373...
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Apply the formula to the repeating part: Following the steps above, you would find that 0.0737373... converts to 73/990
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Add the non-repeating part: 0.2 converts to 2/10 or 1/5
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Add the fractions: (1/5) + (73/990) = (198 + 73)/990 = 271/990
Therefore, 0.2737373... converts to 271/990
Why Does This Method Work?
This method works because it leverages the properties of infinite series. By multiplying the decimal by 10n, you effectively shift the decimal point, allowing you to subtract the repeating part and isolate a whole number. This manipulation creates an equation that can be solved to find the fractional equivalent.
Can all repeating decimals be converted to fractions?
Yes! Every repeating decimal can be expressed as a fraction. This is a fundamental principle in mathematics.
Are there any online tools to help with this conversion?
Several online calculators can convert repeating decimals to fractions. Searching for "repeating decimal to fraction calculator" will provide various options.
This comprehensive guide provides you with the knowledge and tools to confidently convert any repeating decimal to its fractional equivalent. Remember the core formula and the steps involved, and you'll be able to tackle these conversions with ease.
