Time taken by P alone to finish a work is 50 percent more than the time taken by P and Q together. Q is three times as efficient as R. If Q and R together can complete the work in 22.5 days, then in how many days will P alone complete the work?

Introduction / Context:
This problem links three workers P, Q, and R through relationships involving both time and efficiency. We know how long P and Q together take relative to P alone, as well as how Q and R together perform, and how efficient Q is compared to R. The goal is to determine how long P alone would take to complete the work. Such questions test the ability to transform percentage and ratio statements into equations involving work rates and to connect multiple relationships in a logical chain.

Given Data / Assumptions:

• Time taken by P alone is 50 percent more than time taken by P and Q together.
• Q is three times as efficient as R.
• Q and R together can complete the work in 22.5 days.
• All workers have constant work rates.
• We must find the time P alone will take to complete the work.

Concept / Approach:
If P alone takes 50 percent more time than P and Q together, then P's time is 1.5 times the time taken by P and Q together. Denote the time for P and Q together as t days, then P alone takes 1.5t days. From this we can derive their individual rates in terms of t. Separately, we use the fact that Q and R together complete the work in 22.5 days and that Q is three times as efficient as R to determine Q's rate. Equating the expression for Q's rate from both relationships allows us to find t and hence the time for P alone.

Step-by-Step Solution:
Step 1: Let the time taken by P and Q together be t days. Then P alone takes 1.5t days. Step 2: Rate of P and Q together is 1 / t per day. Rate of P alone is 1 / (1.5t) = 2 / (3t). Step 3: Therefore Q's rate is the difference: 1 / t minus 2 / (3t) = (3 - 2) / (3t) = 1 / (3t) per day. Step 4: Q is three times as efficient as R, so if R's rate is r, then Q's rate is 3r. Hence r = (1 / (3t)) / 3 = 1 / (9t). Step 5: Q and R together complete the work in 22.5 days, so their combined rate is 1 / 22.5 per day. Step 6: Combined rate of Q and R is Q's rate plus R's rate = 1/(3t) + 1/(9t) = 4 / (9t). Set this equal to 1 / 22.5. Step 7: So 4 / (9t) = 1 / 22.5. Solving gives t = 10 days. Step 8: P alone takes 1.5t days which equals 1.5 times 10 = 15 days.

Verification / Alternative check:
We can check using a unit work model. If the total work is taken as 90 units, Q and R together finish in 22.5 days, so their combined daily work is 4 units. From the ratio Q is three times R, we can assign Q = 3 units per day, R = 1 unit per day. Then Q's rate is 3 units per day, so 3 = 1 / (3t) times 90, giving t = 10 days for P and Q together. Using t = 10, P alone must then finish in 15 days, which agrees with the earlier algebraic solution.

Why Other Options Are Wrong:

• 17 days, 16 days, and 11 days do not satisfy the combined conditions for P and Q and for Q and R simultaneously.
• They would lead to inconsistent work rates when checked back against the given times and efficiency relationships.
• Only 15 days maintains all the proportional relationships specified in the problem.

Common Pitfalls:
Learners sometimes mix up time ratios with efficiency ratios, forgetting that efficiency and time are inversely related. Another common error is to ignore the given information about Q and R and try to solve using only one relationship, which is insufficient. A structured approach where we express all rates in terms of a single variable and then equate the two expressions for Q's rate avoids confusion and leads to a clean solution.

Final Answer:
P alone will complete the work in 15 days.