Questions: Express the repeating decimal as a fraction in lowest terms. 0 . 68=68/100+68/10,000+68/1,000,000+⋯ 0 . 68= (Type an integer or a simplified fraction.)

$Express the repeating decimal as a fraction in lowest terms. 0 . 68=68/100+68/10,000+68/1,000,000+⋯ 0 . 68= (Type an integer or a simplified fraction.)$

Transcript text: Express the repeating decimal as a fraction in lowest terms. \[ 0 . \overline{68}=\frac{68}{100}+\frac{68}{10,000}+\frac{68}{1,000,000}+\cdots \] $0 . \overline{68}=$ $\square$ (Type an integer or a simplified fraction.)

Solution

Solution Steps

Solution Approach

To express the repeating decimal $0.\overline{68}$ as a fraction, we can use the formula for converting repeating decimals to fractions. Let $x = 0.\overline{68}$. Then, multiply $x$ by 100 to shift the decimal point two places to the right, giving $100x = 68.\overline{68}$. Subtract the original $x = 0.\overline{68}$ from this equation to eliminate the repeating part, resulting in $99x = 68$. Solving for $x$ gives the fraction representation. Finally, simplify the fraction to its lowest terms.

Step 1: Define the Repeating Decimal

Let $ x = 0.\overline{68} $. This represents the repeating decimal we want to convert into a fraction.

Step 2: Multiply to Eliminate the Repeating Part

To eliminate the repeating part, multiply both sides of the equation by 100: \[ 100x = 68.\overline{68} \]

Step 3: Set Up the Equation

Now, we can set up the equation by subtracting the original $ x $ from this new equation: \[ 100x - x = 68.\overline{68} - 0.\overline{68} \] This simplifies to: \[ 99x = 68 \]

Step 4: Solve for $ x $

Now, solve for $ x $ by dividing both sides by 99: \[ x = \frac{68}{99} \]

Step 5: Simplify the Fraction

The fraction $ \frac{68}{99} $ is already in its simplest form, as 68 and 99 have no common factors other than 1.