Problem 10 - Algebra 2

Question

Write each repeating decimal as a fraction. \(0 . \overline{5}\)

Step-by-Step Solution

Answer

0.\overline{5} = \frac{5}{9} as a fraction.

Step 1: Let the Repeating Decimal be a Variable

Assign the repeating decimal \( x = 0.\overline{5} \). This means \( x = 0.5555... \) where the digit 5 repeats indefinitely.

Step 2: Multiply by a Power of 10

Since the repeating part is a single digit, multiply both sides of the equation by 10 to shift the decimal point: \( 10x = 5.5555... \). This equation represents the same repeating decimal.

Step 3: Subtract to Eliminate the Repeating Part

Subtract the original equation (\( x = 0.5555... \)) from this new equation \( 10x = 5.5555... \):\[10x - x = 5.5555... - 0.5555...\]This simplifies to \( 9x = 5 \).

Step 4: Solve for x

To solve for \( x \), divide both sides by 9:\[x = \frac{5}{9}\]So the repeating decimal \( 0.\overline{5} \) as a fraction is \( \frac{5}{9} \).

Key Concepts

Fraction Conversion

Converting a repeating decimal into a fraction can initially seem challenging, but it's actually a very systematic process. The fundamental aspect here is to express an infinite repeating decimal in a more familiar and manageable fractional form.

The method primarily revolves around assigning the decimal to a variable, say \( x \). This variable acts as a placeholding number that will be manipulated algebraically. For instance, if we start with \( x = 0.\overline{5} \), we're setting \( x \) to equal an endlessly repeating 0.5555... The goal is to "capture" this endlessly repeating sequence within a finite expression.

Once \( x \) is assigned, you essentially use algebraic manipulation (more on that soon) to simplify the expression until you neatly encapsulate the repeating decimal as a simple fraction, allowing for easier understanding and interchange between forms.

Algebraic Manipulation

Algebraic manipulation is key when converting repeating decimals to fractions. The basic idea is to creatively manipulate the expression to isolate the repeating sequence. This allows us to encapsulate it in finite terms.

Let's consider the sequence: \( x = 0.\overline{5} \). The first step here involves multiplying both sides by just enough powers of ten to move the decimal point right beyond the repeating part. That's why you multiply by 10, giving you: \( 10x = 5.5555... \).

Now, subtract the original equation, \( x = 0.5555... \), from the newly formed equation. This subtraction step is crucial because it eliminates the repeating sequence:

• After subtraction: \( 10x - x = 5.5555... - 0.5555... \)
• This results in \( 9x = 5 \)

What's left is the simplest form of the fraction: \( x = \frac{5}{9} \). This step, although simple, is powerful in smoothing out the repeating part into a concise fraction.

Decimals to Fractions

Transforming a decimal, especially a repeating decimal, into a fraction involves piecing together the principles of assigning, multiplying, and subtracting to "trap" the endless repetition.

For usual decimals, like 0.25, conversion into a fraction is straightforward as it directly equates to \( \frac{25}{100} \) or simplified, \( \frac{1}{4} \). However, repeating decimals require additional steps.

Assigning a variable, like setting \( x = 0.\overline{5} \), is the starting point. Subsequent multiplying by 10, a necessity to align the decimals, brings you into position to eventually subtract and address the repeating sequence.

The beauty lies in how sequence alignment and subtraction simplify into straightforward fractions. Such as in our example: by subtracting \( 0.5555\ldots \) from \( 5.5555\ldots \) results in the tidy \( \frac{5}{9} \). Through this method, we've shifted the cumbersome recurring decimal into an elegant, easy-to-read fraction.