The decimal 0.737373... (where the "73" repeats infinitely) might seem tricky to convert into a fraction, but it's a straightforward process using a bit of algebra. This guide will walk you through the steps, explaining the method and providing a clear understanding of how repeating decimals are converted to fractions.
Understanding Repeating Decimals
Repeating decimals, also known as recurring decimals, are decimals where one or more digits repeat infinitely. They are rational numbers, meaning they can be expressed as a fraction of two integers. The key to converting them lies in manipulating the decimal to eliminate the repeating part.
How to Convert 0.737373... to a Fraction
Here's a step-by-step process:
-
Let x equal the repeating decimal: We'll start by assigning a variable to the repeating decimal:
x = 0.737373... -
Multiply to shift the repeating part: Multiply both sides of the equation by 100 (because there are two repeating digits):
100x = 73.737373... -
Subtract the original equation: Now, subtract the original equation (
x = 0.737373...) from the equation we just created:100x - x = 73.737373... - 0.737373...This simplifies to:
99x = 73 -
Solve for x: Divide both sides by 99 to isolate x:
x = 73/99
Therefore, the fraction equivalent of the repeating decimal 0.737373... is 73/99.
Can this fraction be simplified?
No, the fraction 73/99 cannot be simplified further. 73 is a prime number, and it does not share any common factors with 99 other than 1.
Frequently Asked Questions (FAQ)
This section answers some common questions related to converting repeating decimals to fractions.
How do you convert other repeating decimals to fractions?
The method described above works for any repeating decimal. The key is to multiply by a power of 10 that shifts the repeating part to the left of the decimal point, allowing you to subtract the original equation and eliminate the repeating digits. For example, if you had a repeating decimal with three repeating digits, you would multiply by 1000.
What if the repeating decimal doesn't start immediately after the decimal point?
If the repeating portion doesn't start immediately after the decimal, you can first handle the non-repeating portion as a separate fraction, and then apply the above method to the repeating part. Add the two resulting fractions to get the final answer.
Why does this method work?
This method works because it leverages the properties of infinite geometric series. The repeating decimal is essentially an infinite sum of terms, and the algebraic manipulation allows us to find a closed-form expression for that sum in the form of a fraction.
Are all repeating decimals rational numbers?
Yes, all repeating decimals are rational numbers. By definition, a rational number can be expressed as a fraction of two integers, and the method outlined above demonstrates how to convert any repeating decimal into such a fraction.
This comprehensive guide should give you a firm grasp on converting the repeating decimal 0.737373... to its equivalent fraction, 73/99, and also provide the knowledge to tackle other repeating decimal conversions. Remember, the key lies in carefully manipulating the decimal to remove the repeating sequence.
