Recurring s are an interesting but sometimes challenging aspects of mathematics. While some people may find it easy to convert a to a decimal, the opposite process might prove difficult for some. This article will outline a guide on how to convert a recurring decimal into a fraction.

Before we begin, it is important to note that recurring decimals are also known as repeating decimals. They are decimals that keep repeating a sequence of digits after the decimal point. For example, the decimal representation of one-third is 0.3333…. Here, the digit “3” repeats infinitely, making it a recurring decimal.

Step One: Identify the Whole Number and Repeating Part of the Decimal

The first step to a recurring decimal to a fraction is to identify the whole and the repeating part. The whole number is easy to spot, and it’s the part of the decimal that appears before the decimal point. For example, in the recurring decimal 3.5454…, the whole number is 3.

The repeating part is the tricky bit. It’s the part of the decimal that keeps repeating after the decimal point. In the example above, 0.54 is the repeating part. To make it easier to identify, some people enclose the repeating part in brackets. Thus, 3.5454… can be written as 3.(54).

Step Two: Multiply Both Sides by 10^n

Next, we need to remove the decimal point from the number we’re working with. To do this, multiply both sides of the equation by 10^n, where n is the number of digits in the repeating part of the decimal. In the example above, we have two repeating digits in our decimal, so n is equal to 2.

Multiplying both sides by 10^2 gives us the following equation:

354.54… = 3,545.454…

Step Three: Subtract Both Equations

Now that we’ve eliminated the decimals, we can subtract both equations. That is, we subtract the equation we got in step two from the original equation. This gives us:

3.(54) – 0.(54) = 3,545.454… – 354.54…
3.(54) – 0.(54) = 3,190.909…

Step Four: Simplify the Equation

Next, we need to simplify the equation we got in step three. To do this, we need to isolate the repeating part on one side of the equation. In this case, we want 0.(54) on one side of the equation.

3.(54) – 0.(54) = 3,190.909…
100 * 3.(54) – 1 * 0.(54) = 3,190.909… * 100
354 – 0.54 = 319.09
353.46 = 319.09

Step Five: Convert into a Fraction

Finally, we can convert the decimal result into a fraction by placing its value over 1. In the example above, 353.46 can be written as the fraction (353.46/1). To simplify the fraction, we need to find the greatest common factor (GCF) of both the numerator and denominator. In this case, the GCF is 1.

Thus, we can conclude that the recurring decimal 3.(54) is equal to the fraction 353.46/100. This process works for any recurring decimal, no matter how many repeating digits there are. By following these steps, you can easily convert a recurring decimal to a fraction and, in the process, deepen your understanding of mathematics.

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