Three-person collaboration: P can do a work in 12 days, Q alone in 8 days, and with the help of R they finish in 4 days. How many days would R alone take to complete the work?

Introduction / Context:
We are given two solo rates and the combined rate with a third person. Subtract the known solo rates from the team rate to isolate R’s rate, then invert to get R’s solo time for the complete job.

Given Data / Assumptions:

• P’s time = 12 days ⇒ rate_P = 1/12.
• Q’s time = 8 days ⇒ rate_Q = 1/8.
• (P + Q + R) time = 4 days ⇒ joint rate = 1/4.

Concept / Approach:
rate_R = (1/4) − (1/12) − (1/8). Compute numerically and invert for R’s time alone.

Step-by-Step Solution:
1/4 − 1/12 − 1/8 = (6 − 2 − 3)/24 = 1/24. Therefore, R alone takes 24 days.

Verification / Alternative check:
Check by addition: 1/12 + 1/8 + 1/24 = 2/24 + 3/24 + 1/24 = 6/24 = 1/4, consistent with 4 days together.

Why Other Options Are Wrong:
25 or 34 are arbitrary; 14 is too fast and fails the rate sum check.

Common Pitfalls:
Arithmetic with fractions; always use a common denominator to avoid mistakes.

Final Answer:
24 days