P, Q and R can together complete a job in 12 days. Their efficiencies are in the ratio 3 : 8 : 5 respectively. In how many days can Q alone complete the same job?

Introduction / Context:
This question is a time and work problem where workers have efficiencies in a given ratio. Knowing the total time taken by all of them together, we can deduce the total combined rate and then extract the individual rate for Q using the ratio. Finally, we take the reciprocal of Q's rate to obtain the time taken by Q alone.

Given Data / Assumptions:

• P, Q and R together complete the job in 12 days.
• Their efficiencies are in the ratio 3 : 8 : 5.
• All three work at constant efficiencies.
• We need the time taken by Q alone.

Concept / Approach:
If efficiencies are in the ratio 3 : 8 : 5, we can assume that P, Q and R do 3k, 8k and 5k units of work per day respectively for some constant k. Together their daily work rate is (3k + 8k + 5k) = 16k. Since they finish the job in 12 days, the total work equals 12 * 16k units. The daily work rate of Q is 8k, so Q's time alone is total work divided by 8k. This gives a simple ratio based calculation.

Step-by-Step Solution:
Assume P's daily rate = 3k, Q's daily rate = 8k, R's daily rate = 5k. Together, their daily rate = 3k + 8k + 5k = 16k. They complete the work in 12 days ⇒ Total work W = 16k * 12 = 192k units. Q's daily rate = 8k. Time taken by Q alone, T = W / Q's rate = 192k / (8k). Simplify: 192k / 8k = 24 days.

Verification / Alternative check:
We can check that P and R alone would take W / (3k) = 192k / 3k = 64 days and W / (5k) = 192k / 5k = 38.4 days respectively, which are longer than Q's 24 days, as expected because Q has the highest efficiency (8k) among the three. The numbers are consistent and reflect the efficiency ratio correctly.

Why Other Options Are Wrong:
36 days and 30 days would correspond to smaller daily rates for Q than 8k and thus would not maintain the 3 : 8 : 5 ratio while still allowing all three together to finish in 12 days. 22 days is not a proper multiple stemming from the ratio and the total work calculation. Only 24 days fits the calculations based strictly on the given ratio and combined time.

Common Pitfalls:
Students sometimes mistakenly divide 12 by 8 or attempt to assign direct times from the ratio instead of working with rates. Always remember that efficiency ratios refer to rates (work per day), and total time must be derived by dividing total work by an individual rate.

Final Answer:
Q alone can complete the job in 24 days.