S, T and U together can complete a certain piece of work in 30 days. If the ratio of their efficiencies (work rates) S : T : U is 20 : 15 : 12 respectively, then in how many days would U alone be able to complete the entire work individually?

Introduction / Context:
This question tests the core concept of work and time based on relative efficiency. When several people work together with a given ratio of efficiencies and we know the combined time for completion, we can find the individual time taken by any one of them working alone. Such problems are very common in competitive exams under the Time and Work topic.

Given Data / Assumptions:

• S, T and U together finish the work in 30 days.
• The ratio of their efficiencies S : T : U is 20 : 15 : 12.
• Each person works at a constant rate throughout.

Concept / Approach:
The idea is to convert the ratio of efficiencies into actual daily work rates using a common multiplier k. Then, using the total time taken by all three working together, we determine k. Finally, we use U's individual rate to find the time U alone would take to complete the full work. Time is always equal to total work divided by rate of work.

Step-by-Step Solution:
Let the daily work rates of S, T and U be 20k, 15k and 12k units per day respectively. Combined rate of S, T and U together = 20k + 15k + 12k = 47k units per day. They complete the entire work in 30 days, so total work W = (combined rate) * time. W = 47k * 30. We assume W = 1 unit (one complete job) for convenience. Therefore, 47k * 30 = 1. So, k = 1 / (47 * 30). Now, U's daily rate = 12k = 12 * [1 / (47 * 30)] = 12 / (1410) = 2 / 235 units per day. Time taken by U alone = total work / U's rate = 1 / (2 / 235) = 235 / 2 days. Numerically, 235 / 2 = 117.5 days.

Verification / Alternative check:
If U alone takes 235/2 days, then in 30 days U completes (30) * (2 / 235) = 60 / 235 of the work. Similarly, S and T together must complete the remaining fraction. Using their rates 20k and 15k with k = 1 / (47 * 30), you can verify that the sum of all three contributions over 30 days equals 1 complete job, which confirms the correctness of the calculation.

Why Other Options Are Wrong:
Option A (195/2 days) assumes a higher efficiency for U than actually implied by the ratio and combined time, leading to too small a time value.
Option C (225/2 days) is close but still smaller than the correct value; it does not satisfy the equation 47k * 30 = 1 with U's rate as 12k.
Option D (215/2 days) again does not give the correct relationship between total work and the combined work done in 30 days. Only 235/2 days maintains consistent ratios and time relationships.

Common Pitfalls:
A frequent mistake is to interpret the ratio 20 : 15 : 12 as time taken instead of efficiency. Remember, efficiency is inversely proportional to time. Another error is to forget that S, T and U working together for 30 days perform the whole work, which fixes the value of k. Some students also make arithmetic mistakes while simplifying fractions like 12 / 1410. Careful simplification and keeping track of reciprocal values is crucial.

Final Answer:
Thus, U alone will complete the entire work in 235/2 days, which is 117.5 days.