A and B together can complete a piece of work in 12 days. They work together for 5 days and then A alone finishes the remaining work in 14 more days. In how many days would A alone be able to complete the entire work from start to finish?

Introduction / Context:

This is a standard time and work problem where two people start working together and later only one of them continues. It tests the ability to use combined work rates, subtract completed work, and then infer the individual rate of one worker. Such problems are frequently seen in aptitude tests and help build intuition about how joint and individual efficiencies interact over time.

Given Data / Assumptions:

• A and B together can finish the entire work in 12 days.
• They work together for the first 5 days.
• After that, A works alone and finishes the remaining work in 14 more days.
• We assume the total work is of one unit and all rates remain constant over time.

Concept / Approach:

The strategy is to convert the combined completion time of 12 days into a joint daily work rate for A and B. Then we compute how much work A and B complete in the first 5 days. The rest of the work is completed by A alone in 14 days, which directly gives the daily work rate of A. The time A alone would take for the entire job is simply the reciprocal of this daily rate.

Step-by-Step Solution:

Verification / Alternative Check:

We can verify by reconstructing the time line: In 24 days A alone would do 1/24 per day times 24 days which equals 1 unit, so the rate is consistent. Also, if we use 1/24 as A's rate, the remaining rate for B when combined with A is 1/12 - 1/24 = 1/24. This means B is equally efficient as A, and if both have the same rate, together they should indeed complete the work in half of 24 days, which is 12 days, matching the original information.

Why Other Options Are Wrong:

Times like 20 or 18 days imply a higher daily rate for A which would cause the remaining 7/12 of the work to be finished in fewer than 14 days. Similarly, 22 days gives a rate that does not match the requirement of completing 7/12 of the work in exactly 14 days. Only 24 days aligns with both the joint work of 12 days and the solo completion of the remainder in 14 days.

Common Pitfalls:

An easy mistake is to subtract days instead of fractions of work, or to treat the 5 and 14 days together as a direct fraction of 12 days. It is essential to consistently work with rates and fractions of work completed, not just with raw day counts. Another error is to wrongly assume that A and B have equal efficiency without verifying from the rates.

Final Answer:

A alone can complete the entire work in 24 days when working from start to finish.