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Converting a repeating decimal to a fraction may seem tricky at first, but it's a systematic process. Start by setting the repeating decimal equal to a variable, typically denoted as \( x \). For example, if you have \( 0.\overline{38} \), you let \( x = 0.\overline{38} \). The goal is to transform this into an equation that lets you work with whole numbers.
Next, consider the length of the repeating sequence; in our case, it's 2 digits (38). Multiply both sides of your equation by a power of 10 that matches the length of the repeating sequence. So here, multiply by 100: \( 100x = 38.\overline{38} \).
By manipulating the equation this way, we've shifted the decimal point to the right, making it easier to subtract and eventually isolate \( x \).

Algebraic manipulation is key to solving equations involving repeating decimals. Once you've set up your variable as \( x = 0.\overline{38} \) and multiplied both sides by 100, you get two equations:

• \( x = 0.\overline{38} \)
• \( 100x = 38.\overline{38} \)

The next step involves subtraction to eliminate the repeating decimal. Subtract the first equation from the second: \( 100x - x = 38.\overline{38} - 0.\overline{38} \). This simplifies to \( 99x = 38 \). Subtraction helps in cancelling out the repeating part, leaving you with a simpler equation. Now, divide both sides by 99 to isolate \( x \), giving \( x = \frac{38}{99} \).

Once you have your fraction, it’s important to simplify it to its lowest terms. For \( \frac{38}{99} \), check if there are any common factors between the numerator (38) and the denominator (99). Use the greatest common divisor (GCD) to do this. Here, the GCD of 38 and 99 is 1, meaning \( \frac{38}{99} \) is already in its simplest form.
Always ensure to check if the fraction can be broken down further to make calculations easier and answers more elegant. Simplifying fractions is a crucial step because it helps you communicate your answer clearly and concisely.