Question

A and B work at digging a ditch alternately for a day each. If A can dig a ditch in $a$ days and B can dig that ditch in $b$ days, will work get done faster if A begins the work?
I. $n$ is a positive integer such that $n\left(\frac{1}{a} + \frac{1}{b}\right) = 1$.
II. $b>a$.

• A. If the question can be answered with the help of statement I alone.
• B. If the question can be answered with the help of statement II alone.
• C. If both statements I and II are needed to answer the question.
• D. If the question cannot be answered even with the help of both statements.

Hint:

In alternating work problems, always check if the completion day falls on the faster worker’s turn. Statement giving rates + ordering of speeds together can answer the question.

Answer 1

The Correct Option is C

We must decide whether starting with A will finish the ditch faster than starting with B. This depends on how much work remains after alternating days.

Step 1: Understanding Statement I
Statement I says: \[ n\left(\frac{1}{a} + \frac{1}{b}\right) = 1 \] This means that in $n$ days, A and B together working in alternation will complete the entire ditch exactly. However, $n$ here represents the number of \emph{pairs of days} (one A-day + one B-day). The equation tells us total work rate over $n$ such pairs.
But this alone does not tell us whether starting with A is faster than starting with B, because we don’t know the relative speeds (except through $a$ and $b$) nor whether a partial day’s work at the end changes the completion day count.
So Statement I alone is not sufficient.

Step 2: Understanding Statement II
Statement II says: \[ b>a \] This means A is faster than B (since fewer days means higher rate). This gives us the speed ordering but not the exact relationship of how the alternation affects total time, especially near completion. We still can’t determine if starting with A will complete earlier without knowing if the last day falls on A’s turn or B’s. So Statement II alone is insufficient.

Step 3: Combining Statements I and II
From Statement I, we know how much total work is done in each A+B pair of days and how many such pairs ($n$) are needed to complete the ditch. From Statement II, we know A is faster than B.
Combining them: If work is completed in an integer number of pairs, starting with A or B may take the same time. But if completion requires part of an extra day, then starting with A will result in earlier completion since A is faster. Knowing $n$ and that A is faster allows us to deduce the answer to “Will starting with A be faster?” conclusively.
Thus both statements together are sufficient.
Hence the correct answer is (c).