Finding a partner’s rate from partial collaboration: Ramesh can finish a job in 20 days. He works alone for 10 days, then completes the remaining work in 2 additional days working together with Dinesh. In how many days would Ramesh and Dinesh together complete the whole job from the start?

Introduction / Context:
We infer Dinesh’s rate using the segment where he works with Ramesh, then sum both rates to get the combined completion time.

Given Data / Assumptions:

• Ramesh alone: 20 days ⇒ rate = 1/20 per day.
• First 10 days: Ramesh alone.
• Next 2 days: Ramesh + Dinesh finish the rest.

Concept / Approach:
Compute remaining work after 10 days, divide by the 2-day joint period to get the pair’s rate, then subtract Ramesh’s rate to get Dinesh’s.

Step-by-Step Solution:
Work done by Ramesh in 10 days = 10 * (1/20) = 1/2.Remaining = 1/2, finished in 2 days ⇒ pair rate = (1/2)/2 = 1/4.Dinesh’s rate = 1/4 − 1/20 = 1/5.Together from start: 1/20 + 1/5 = 1/4 ⇒ time = 4 days.

Verification / Alternative check:
Check consistency: in the last 2 days, pair does 2 * 1/4 = 1/2 of the job, matching the remainder.

Why Other Options Are Wrong:
5, 6, 8, 10 days contradict the deduced combined rate 1/4.

Common Pitfalls:
Using 10 days of both workers by mistake; forgetting that the pair works only for the last 2 days.

Final Answer:
4 days