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Which of the following decimals has the longest repeating cycle?
Question

Which of the following decimals has the longest repeating cycle?

A.

16\frac{1}{6} ​​

B.

17\frac{1}{7} ​​

C.

19\frac{1}{9} ​​

D.

111\frac{1}{11} ​​

Correct option is B

Given
Four fractions: 16,17,19,111 \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: 16\frac{1}{6}​ = 0.1666... (The repeating cycle is "6", length = 1)

Option B:  17\frac{1}{7} ​= 0.142857142857... (The repeating cycle is "142857", length = 6)

Option C: 19\frac{1}{9}​ = 0.111... (The repeating cycle is "1", length = 1)

Option D: 111\frac{1}{11} ​= 0.0909... (The repeating cycle is "09", length = 2)
Comparing the lengths, 17\frac{1}{7}​ has the longest repeating cycle.

Final Answer
So the correct answer is (b)

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