A is three times as fast as B at doing a piece of work. A alone finishes the entire work in 32 days less than B alone. In how many days will A and B, working together at their respective constant rates, complete the whole work?

Introduction / Context:
This is a classic time and work aptitude question where two workers, A and B, have different efficiencies. We are told that A is three times as fast as B and that A takes 32 days less than B to finish the same job alone. The goal is to determine how long they will take to complete the work when they both work together at their constant individual rates.

Given Data / Assumptions:

• A is three times as fast as B.
• A alone finishes the work 32 days faster than B.
• Both A and B work at constant speeds with no breaks.
• The total work is treated as one complete job.

Concept / Approach:
In time and work problems, efficiency is measured as work done per day. If a person takes T days to do a job alone, the daily work rate is 1 / T. The phrase three times as fast means the rate of A is three times the rate of B. Using the relationship between their times and the 32 day difference, we can find their individual times, then compute the combined rate and finally the total time when they work together.

Step-by-Step Solution:
Let the time taken by B alone be TB days, so B's rate is 1 / TB.A is three times as fast as B, so A's rate is 3 * (1 / TB) = 3 / TB.Thus A's time TA = 1 / (3 / TB) = TB / 3.Given that A finishes 32 days sooner: TA = TB - 32, so TB / 3 = TB - 32.Multiply both sides by 3: TB = 3TB - 96, so 2TB = 96 and TB = 48 days.Then TA = TB / 3 = 48 / 3 = 16 days.Combined rate of A and B together = 1 / 16 + 1 / 48 = (3 + 1) / 48 = 4 / 48 = 1 / 12.Time taken together = 1 / (1 / 12) = 12 days.

Verification / Alternative check:
If B takes 48 days, B's rate is 1 / 48. A is three times as fast, so A's rate is 3 / 48 = 1 / 16, matching the earlier time of 16 days. The difference in times is 48 - 16 = 32 days, which agrees with the condition. Their combined rate 1 / 16 + 1 / 48 equals 1 / 12, giving 12 days for the joint work. All conditions fit consistently, so the solution is verified.

Why Other Options Are Wrong:
16 days would be the time taken by A alone, not the time taken together. Twenty four days and 32 days are both larger than the time of the faster worker A, so they cannot be the result of combining two positive work rates. Any combined effort must finish sooner than the fastest worker alone, which immediately rules out those larger values.

Common Pitfalls:
A common mistake is to confuse three times as fast with taking three times as many days, which is actually the opposite. Another typical error is to subtract rates instead of using the 32 day difference in time. Some learners also forget that when two people work together, their rates add, which must always give a completion time less than that of the faster individual worker.

Final Answer:
The required time for A and B working together to finish the entire work is 12 days.