Question

write 0.818181... as a fraction in simplest form. select one: a. \\(\frac{81}{100}\\) b. \\(\frac{4}{5}\\) c. \\(\frac{9}{11}\\) d. \\(\frac{81}{1000}\\)

Explanation:

Step1: Identify the repeating decimal

The decimal is \(0.818181\ldots\), where "81" repeats. Let \(x = 0.818181\ldots\).

Step2: Multiply by a power of 10 to shift the decimal

Since the repeating cycle is 2 digits, multiply \(x\) by \(100\): \(100x = 81.818181\ldots\).

Step3: Subtract the original equation

Subtract \(x\) from \(100x\): \(100x - x = 81.8181\ldots - 0.8181\ldots\), which simplifies to \(99x = 81\).

Step4: Solve for \(x\)

Divide both sides by 99: \(x = \frac{81}{99}\). Simplify the fraction by dividing numerator and denominator by 9: \(x = \frac{9}{11}\).

Step5: Match with the options

Option c is \(\frac{9}{11}\), so this is the correct fraction.

Answer:

c. \(\frac{9}{11}\)