Manish alone can complete a piece of work in 20 days. After Manish has worked alone for 8 days, Sriram joins him to help finish the remaining work. The question asks: in how many days will the entire work be completed? You are given two data statements and must decide which statement or statements are sufficient to answer the question. Statement I: Sriram takes twice the time that Manish takes to complete the work alone. Statement II: Sriram can complete half the work in 15 days.

Introduction / Context:
This is a data sufficiency question based on work and time. Manish starts a job alone and Sriram joins later. Instead of asking you to compute the exact answer, the question asks which of the two given statements provide enough information to find the total time. For such questions you must not combine extra assumptions. You only check whether the details in each statement, taken separately, are adequate to uniquely determine the total number of days required to complete the work.

Given Data / Assumptions:

• Manish alone can complete the entire work in 20 days.
• Manish works alone for the first 8 days.
• After day 8, Sriram joins Manish and they work together until completion.
• Statement I: Sriram takes twice as long as Manish to finish the job alone.
• Statement II: Sriram completes half the work in 15 days when working alone.

Concept / Approach:
In work and time problems, if you know the time an individual takes to finish a job alone, you can find that person s work rate as 1 divided by time. When two people work together, their combined rate is the sum of individual rates. To judge sufficiency, we check whether each statement gives us enough information to determine Sriram s rate and therefore the combined rate, after which the remaining time can be computed. If one statement does that without needing the other, that statement alone is sufficient. If either statement alone allows the calculation, then the correct option is that either I or II is sufficient.

Step-by-Step Solution:
Step 1: From the basic data, Manish s time is 20 days, so his rate is 1/20 of the work per day.Step 2: In the first 8 days, Manish completes 8 * (1/20) = 8/20 = 0.4 of the work, so 0.6 remains.Step 3: Using Statement I, Sriram takes twice Manish s time, so Sriram s time alone is 40 days and his rate is 1/40 per day.Step 4: With Statement I alone, the combined rate is 1/20 + 1/40 = 3/40 of the work per day, so the remaining 0.6 work can be finished in 0.6 / (3/40) days, which is a unique value. Thus Statement I alone is sufficient.Step 5: Using Statement II alone, Sriram completes half the work in 15 days, so his rate is (1/2)/15 = 1/30 of the work per day. The combined rate of Manish and Sriram is then 1/20 + 1/30 = 5/60 = 1/12 of the work per day, from which the remaining time can again be uniquely determined. So Statement II alone is also sufficient.

Verification / Alternative check:
With Statement I, the remaining time after day 8 comes out to a specific number of days, so there is no ambiguity.With Statement II, a different but still unique total time can be computed, proving that Statement II also gives enough information by itself.Therefore each statement independently is sufficient to answer the question.

Why Other Options Are Wrong:
Option A is wrong because it ignores the fact that Statement II by itself also allows a complete solution.Option B is wrong because it ignores the sufficiency of Statement I.Option D is wrong because it claims that neither statement is sufficient, even though both are sufficient individually.

Common Pitfalls:
A common mistake is to actually compute final answers and then think that different results mean insufficiency, but data sufficiency only asks whether a single answer is possible, not to match both.Another pitfall is to think both statements together are needed whenever the question mentions two people, but here each statement alone provides Sriram s rate.

Final Answer:
Since each statement on its own is enough to compute the total time, the correct option is Either the statements I and II is sufficient to answer the question.