Problem 96 - Intermediate Algebra

Question

Find a fraction whose decimal equivalent is \(0 . \overline{61}\)

Step-by-Step Solution

Answer

The fraction is \(\frac{61}{99}\).

Step 1: Understand the Problem

We need to convert the repeating decimal \(0.\overline{61}\) into a fraction. This requires recognizing the repeating part, \(61\), and using algebra to express it as a fraction.

Step 2: Set Up the Equation

Let \(x = 0.\overline{61}\). This means \(x = 0.616161...\). We aim to isolate \(x\) in relation to the repeating segment.

Step 3: Eliminate the Repeating Decimal

Multiply \(x\) by 100 (since \(61\) has 2 digits) to shift the decimal: \[ 100x = 61.616161... \]

Step 4: Subtract to Remove the Repetition

Subtract the original \(x = 0.616161...\) from \(100x = 61.616161...\) to eliminate the repeating decimal: \[ 100x - x = 61.616161... - 0.616161... \] Which simplifies to: \[ 99x = 61 \]

Step 5: Solve for x

Solve for \(x\) by dividing both sides by 99: \[ x = \frac{61}{99} \] This fraction is already in its simplest form.

Key Concepts

Fraction Conversion

Converting a repeating decimal to a fraction can seem tricky, but with a structured approach, it's quite manageable. When we come across a recurring decimal like \(0.\overline{61}\), we aim to express it as a fraction in its simplest form. The repeating section, \(61\), plays a critical role here. Such conversions start by setting the repeating decimal equal to a variable, typically denoted as \(x\). This acts as our stepping stone. For \(0.\overline{61}\), we write: \(x = 0.616161...\). This setup allows us to handle the repetitive nature through multiplication and subtraction. Specifically, for decimals like \(0.\overline{61}\), since there are two digits in the repeating part, we multiply the entire equation by 100, transforming the equation into \(100x = 61.616161...\). By setting the groundwork with this equation, you create a path to eliminate the repeating decimal parts, plumbing them into a fractional form.

Algebraic Manipulation

Algebraic manipulation is key when dealing with repeating decimals and eventually converting them into fractions. Manipulating EquationsSubtracting equations is a fundamental move here. With equations like \(x = 0.616161...\) and \(100x = 61.616161...\), subtraction becomes a pivotal strategy:

• \(100x - x = 61.616161... - 0.616161...\)
• This simplifies to: \(99x = 61\)

Subtracting effectively eliminates the repetition from the equation, allowing you to handle only whole numbers and clean terms. This level of simplification is necessary for safely solving for \(x\), which represents the original repeating decimal in its new fraction format.

Simplification of Fractions

Once you've converted a repeating decimal into a fraction, you want the simplest form of that fraction. Simplifying fractions makes them easier to read and understand.In our example, after algebraic manipulation, we end up with the equation \(99x = 61\), leading us to find \(x = \frac{61}{99}\). At first glance, this fraction may seem as if it might not be simplified, but in truth, \(\frac{61}{99}\) is already in its simplest form.Understanding SimplificationTo ensure a fraction is as simple as possible, we check if the numerator and denominator share any common factors other than 1. Here, 61 and 99 do not share such factors:

• Verify by checking divisibility: 61 is a prime number, which narrows the possibilities.
• 99 is divisible by 3 and 9, neither of which divides 61.

Thus, \(\frac{61}{99}\) is the most straightforward representation of \(0.\overline{61}\) in fraction form, showing the satisfying conclusion of this mathematical process.