Problem 28 - Algebra 2

Question

Write each repeating decimal as a fraction. \(0 . \overline{7}\)

Step-by-Step Solution

Answer

The repeating decimal \( 0.\overline{7} \) as a fraction is \( \frac{7}{9} \).

Step 1: Set Up an Equation

Let \( x \) equal the repeating decimal. So, set \( x = 0.\overline{7} \). This means \( x = 0.7777\ldots \)

Step 2: Eliminate the Repeating Part

Multiply both sides of the equation by 10 to shift the decimal point: \[ 10x = 7.7777\ldots \]

Step 3: Subtraction to Solve for x

Subtract the original equation (Step 1) from this new equation (Step 2): \[ 10x - x = 7.7777\ldots - 0.7777\ldots \] which simplifies to \[ 9x = 7 \]

Step 4: Solve for x

Divide both sides by 9 to isolate \( x \): \[ x = \frac{7}{9} \]

Step 5: Verify the Fraction

Convert \( \frac{7}{9} \) back to a decimal to check: \( \frac{7}{9} = 0.\overline{7} \) confirming the repeating decimal is equivalent to the fraction.

Key Concepts

Repeating Decimals

Repeating decimals are a fascinating concept in mathematics. They are decimals that have one or more repeating digits following the decimal point. For instance, in the repeating decimal \(0.\overline{7}\), the digit '7' repeats indefinitely. Understanding repeating decimals is crucial because they are part of the world of real numbers and often appear in everyday calculations and applications.

• Repeating decimals always have a predictable pattern of one or more digits repeating forever.
• They are not finite, meaning they never end or terminate like decimals such as 0.5 or 0.25.
• Learning to identify and work with repeating decimals is essential for converting them efficiently.

By recognizing these endless sequences, you develop a deeper comprehension of their behavior and how they relate to fractions.

Fractions

Fractions represent parts of whole numbers and express ratios between two integers. In the case of repeating decimals, fractions can be used to represent these infinite decimals succinctly and exactly. As seen in the example, the repeating decimal \(0.\overline{7}\) is equivalent to the fraction \(\frac{7}{9}\).

• Fractions consist of a numerator (top number) and a denominator (bottom number).
• They can represent repeating decimals, providing an exact value rather than an approximation.
• By practicing the conversion of repeating decimals to fractions, you can verify your understanding with precision.

The process involves setting up an algebraic equation to solve for the fraction, introducing a world where infinite decimals are tamed into finite ratios.

Decimal Representation

Decimal representation refers to expressing numbers in a base-10 system, where each position has a power of 10. When it comes to repeating decimals, they have a special representation seen through their repetitive sequence. For example, \(0.\overline{7}\) indicates the digit '7' repeating after the decimal point continuously.

• Each number after the decimal point in a decimal representation signifies a fraction of a whole, decreasing by a factor of 10 progressively.
• Repeating decimals showcase how limitless a decimal representation can be, repeating into infinity.
• Different repeating cycles tell us where exactly our cycle starts and ends in such decimals.

Understanding decimal representation equips you with the ability to visualize how repeating decimals behave on a number line and relate to simple fractions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying algebraic expressions to solve equations. It's a powerful tool, particularly useful in the conversion of repeating decimals to fractions.

• Start by representing the repeating decimal with a variable, say \( x = 0.\overline{7}\).
• Multiply by 10, or a power of 10, to shift the decimal point and create an equivalent but shifted equation.
• Subtract the original equation from this new equation to eliminate the repeating sequence.
• Finally, solve for \( x \) by dividing through any coefficients left, determining the fractional equivalent.

This algebraic technique systematically transforms the infinite decimal into a concrete fraction, allowing clear understanding and verification of the original sequence with its exact fractional form.