Two Workers — Partial Contribution Then Solo Finish A can do the work in 80 days. He works alone for 10 days, and then B alone finishes the remaining work in 42 days. If A and B worked together from the start, in how many days would they finish?

Introduction / Context:
This question blends partial work and rate-finding. By using the given phases, we determine B’s solo rate and then add A’s rate to find a combined completion time.

Given Data / Assumptions:

• A alone: 80 days ⇒ rate = 1/80.
• A works 10 days initially.
• B alone finishes in 42 days.

Concept / Approach:
Compute how much work A completes in 10 days. The remainder is done by B; from this we get B’s rate. Then add rates to get the together time.

Step-by-Step Solution:
Work done by A in 10 days = 10*(1/80) = 1/8.Remaining = 1 − 1/8 = 7/8.B completes 7/8 in 42 days ⇒ B's rate = (7/8)/42 = 1/48.Combined rate = 1/80 + 1/48 = (3 + 5)/240 = 8/240 = 1/30.Together time = 1 / (1/30) = 30 days.

Verification / Alternative check:
Check B's time alone: 1/48 implies 48 days per job; 7/8 of a job in 42 days fits.

Why Other Options Are Wrong:
25, 24, 35, and 40 do not match the computed combined rate of 1/30.

Common Pitfalls:
Using 42 as B’s full-time duration for a whole job without adjusting for the fractional workload.

Final Answer:
30 days