Which of the following decimals has the longest repeating cycle?

Correct option is B

Given
Four fractions: $$\frac{1}{6}, \frac{1}{7}, \frac{1}{9}, \frac{1}{11}$$​​
Formula Used
Convert fractions into decimals using division to observe the repeating block.

Solution
Let us expand each fraction into its decimal form:
Option A: $$\frac{1}{6}$$​ = 0.1666... (The repeating cycle is "6", length = 1)

Option B:  $$\frac{1}{7}$$​= 0.142857142857... (The repeating cycle is "142857", length = 6)

Option C: $$\frac{1}{9}$$​ = 0.111... (The repeating cycle is "1", length = 1)

Option D: $$\frac{1}{11}$$​= 0.0909... (The repeating cycle is "09", length = 2)
Comparing the lengths, $$\frac{1}{7}$$​ has the longest repeating cycle.

Final Answer
So the correct answer is (b)