Two-worker collaboration: X alone completes a job in 12 days. Together, X and Y complete the job in 6 2/3 days. In how many days can Y alone finish the job?

Introduction / Context:
Here we know one person's solo time and the pair's time. We first find the joint rate, subtract X’s rate to get Y’s rate, and finally invert to determine Y’s solo completion time.

Given Data / Assumptions:

• X alone = 12 days ⇒ rate_X = 1/12.
• X + Y together = 6 2/3 days = 20/3 days ⇒ joint rate = 3/20.
• Total work = 1 job; constant rates.

Concept / Approach:
rate_Y = joint rate − rate_X. Then Y’s time = 1 / rate_Y. Keep mixed numbers consistent by converting to improper fractions where needed.

Step-by-Step Solution:
Joint rate = 3/20; X’s rate = 1/12. rate_Y = 3/20 − 1/12 = (9 − 5)/60 = 4/60 = 1/15. Hence, Y alone takes 15 days.

Verification / Alternative check:
Check: 1/12 + 1/15 = (5 + 4)/60 = 9/60 = 3/20 ⇒ 20/3 days together, as given.

Why Other Options Are Wrong:
10 or 12 days are too fast for Y given the combined time; 18 days would make the pair slower than stated.

Common Pitfalls:
Mishandling the mixed fraction 6 2/3; always convert to an improper fraction to avoid arithmetic slips.

Final Answer:
15 days