The rates of working of A and B are in the ratio 2 : 3. What is the ratio of the number of days taken by A and B, respectively, to finish the same work when each works alone?

Introduction / Context:
This is a conceptual time and work question focusing on the relationship between rate and time. We are given the ratio of working rates of A and B and asked for the ratio of the times they take to complete the same job individually. Such problems test understanding of the inverse proportionality between work rate and time taken.

Given Data / Assumptions:
- The ratio of work rates of A and B is 2 : 3 (rate of A : rate of B).
- Both A and B are assumed to be working on the same total amount of work individually.
- Total work is considered as a constant quantity for both workers.

Concept / Approach:
Time taken to complete a given piece of work is inversely proportional to the rate of working. If rate of A : rate of B is 2 : 3, then the time taken by A : time taken by B must be in the ratio 1 / 2 : 1 / 3. By simplifying this inverse ratio, we obtain the required ratio of days taken by A and B.

Step-by-Step Solution:
Step 1: Let the rate of A = 2 units per day and the rate of B = 3 units per day. Step 2: Time taken is inversely proportional to rate. So time taken by A ∝ 1 / 2, time taken by B ∝ 1 / 3. Step 3: Ratio of times = (1 / 2) : (1 / 3). Step 4: Simplify the ratio (1 / 2) : (1 / 3) by multiplying both terms by 6 (the LCM of 2 and 3). Step 5: We get (1 / 2) * 6 : (1 / 3) * 6 = 3 : 2. Step 6: Therefore, the ratio of days taken by A and B is 3 : 2 respectively.

Verification / Alternative check:
Assume a fixed work of 6 units. At a rate of 2 units per day, A will need 6 / 2 = 3 days. At a rate of 3 units per day, B will need 6 / 3 = 2 days. So the time ratio is clearly 3 : 2, which confirms our reasoning based on inverse proportionality.

Why Other Options Are Wrong:
- 2:3: This simply repeats the rate ratio, not the time ratio. Time is inversely, not directly, proportional to rate.
- 4:9 and 9:4: These are incorrect transformations and do not follow from basic inverse proportionality.
- 1:2: This would mean B takes twice the time of A, which contradicts the fact that B is faster due to a higher rate (3 versus 2).

Common Pitfalls:
The most common error is to mistakenly assume that the ratio of times is the same as the ratio of rates. In time and work problems, when the amount of work is fixed, time and rate always move in opposite directions: higher rate means less time and vice versa. Always remember that time ratio is the inverse of the rate ratio.

Final Answer:
The ratio of days taken by A and B to finish the work is 3:2.