Directions (61-63): The following questions are accompanied by two statements i.e. statement (I) and statement (II). You have to determine which statement (s) is/are sufficient/necessary to answer the questions.

The ratio of the efficiency of A to B to do a certain work is 5: 2 respectively . A takes X hour to complete the work and C destroys the work in X + 5 hour. If all them started doing the work at 7 a.m. and completed the work at 1 p.m. & C works only for A hours, then find the time taken by C alone to destroy the completed work completely.
Statement (I): A can do the $\frac{8}{9}$th of the work in 16 hrs.
Statement (II): Efficiency of B to C is 5:1.

The correct option is: Statement (I) alone is sufficient to answer the question but statement (II) alone is not sufficient to answer the questions

Step-by-step Explanation:

The problem asks us to find the time taken by C alone to destroy the completed work completely. From the problem description, we are given that:
- A takes X hours to complete the work.
- C destroys the work in X + 5 hours.

Therefore, the time taken by C alone to destroy the work completely is:
$\mathrm{Time}=X+5$ hours.

To find this time, we need to determine the value of X.

Analyzing Statement (I):
Statement (I) states: A can do $\frac{8}{9}$th of the work in 16 hours.
If A completes $\frac{8}{9}$ of the work in 16 hours, the time taken by A to complete the whole work (which is X hours) is:
$X=16\times \frac{9}{8}=18$ hours.

Since we have found X = 18 hours, the time taken by C alone to destroy the work is:
$X+5=18+5=23$ hours.

Thus, Statement (I) alone is sufficient to answer the question.

Analyzing Statement (II):
Statement (II) states: Efficiency of B to C is 5:1.
While this gives the ratio of efficiencies between B and C, we do not know the actual amount of work, the value of X, or the clear meaning of "C works only for A hours" (since 'A' represents a person and using it as a duration 'A hours' introduces ambiguity). Without knowing the absolute time or the exact relation to X, Statement (II) alone is not sufficient to find a unique value for X.

Therefore, Statement (I) alone is sufficient to answer the question, but Statement (II) alone is not sufficient.