Question

write the following decimal as a fraction of two integers in simplest form: $0.2overline{66}$
133/500
the two equations that need to be subtracted are $x = 0.2overline{66}$ and $100x = 26.overline{66}$. when you subtract $26.overline{66} - 0.2overline{66}$, the result is a terminating decimal.
hmm, that wasnt the right answer. lets give it one more try.
try again check my work beta

Explanation:

Step1: Define the repeating decimal

Let \( x = 0.2\overline{66} \), which means \( x = 0.26666\ldots \)

Step2: Multiply to align decimals

Multiply \( x \) by 10 to move the non - repeating part: \( 10x = 2.\overline{66} = 2.6666\ldots \)

Multiply \( x \) by 1000 to move past the repeating part: \( 1000x = 266.6666\ldots \)

Step3: Subtract to eliminate repeating decimals

Subtract \( 10x \) from \( 1000x \):
\[

$$\begin{align*} 1000x - 10x&=266.6666\ldots - 2.6666\ldots\\ 990x&=264 \end{align*}$$

\]

Step4: Solve for \( x \)

Divide both sides by 990: \( x=\frac{264}{990} \)

Step5: Simplify the fraction

Find the greatest common divisor (GCD) of 264 and 990. The GCD of 264 and 990 is 66.

Divide numerator and denominator by 66: \( \frac{264\div66}{990\div66}=\frac{4}{15} \)

Answer:

\(\frac{4}{15}\)