Ramesh and Rahman can complete a piece of work in 20 and 25 days respectively. They work together for 10 days and then leave the work due to illness. Suresh completes the remaining work in 3 days. In how many days can Suresh alone complete the entire work?

Introduction / Context:
In this problem, two workers, Ramesh and Rahman, start a job together but stop after some time, and a third worker, Suresh, completes the remaining portion. We are asked to find Suresh’s full-job time using the information about how much work is left for him and how long he takes to finish that remainder.

Given Data / Assumptions:
- Ramesh alone can complete the work in 20 days.
- Rahman alone can complete the work in 25 days.
- They work together for 10 days.
- Suresh completes the remaining work in 3 days.
- Total work is considered as 1 unit.

Concept / Approach:
First, we compute the combined work rate of Ramesh and Rahman and use this to determine how much work they complete in 10 days. Subtracting this from 1 gives the remaining work. Since Suresh finishes that remaining work in 3 days, we can find his rate by dividing the remaining work by 3. Finally, we take the reciprocal of Suresh’s rate to obtain the number of days he would need to complete the full job alone.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Rate of Ramesh = 1 / 20 work per day. Step 3: Rate of Rahman = 1 / 25 work per day. Step 4: Combined rate of Ramesh and Rahman = 1 / 20 + 1 / 25. Step 5: LCM of 20 and 25 is 100, so 1 / 20 = 5 / 100 and 1 / 25 = 4 / 100. Step 6: Combined rate = 5 / 100 + 4 / 100 = 9 / 100 work per day. Step 7: Work done by them in 10 days = 10 * 9 / 100 = 90 / 100 = 0.9 of the work. Step 8: Remaining work = 1 - 0.9 = 0.1 = 1 / 10 of the total work. Step 9: Suresh completes this 1 / 10 in 3 days. Step 10: Rate of Suresh = (1 / 10) / 3 = 1 / 30 work per day. Step 11: Time taken by Suresh for the entire work = 1 / (1 / 30) = 30 days.

Verification / Alternative check:
Check the consistency: If Suresh works at 1 / 30 work per day, in 3 days he completes 3 * 1 / 30 = 1 / 10 of the work. Adding this to 0.9 completed by Ramesh and Rahman gives 0.9 + 0.1 = 1, which is the full job. This confirms that Suresh’s full-job time of 30 days is correct.

Why Other Options Are Wrong:
- 28, 29, 32 days: These values yield slightly different rates for Suresh and do not reproduce exactly 1 / 10 of the work being finished in 3 days.
- 25 days: This implies a larger rate for Suresh and would result in more than 1 / 10 being completed in 3 days, which contradicts the data.

Common Pitfalls:
Some learners miscalculate the work done by Ramesh and Rahman in 10 days, either by not using the correct combined rate or by making mistakes in fraction addition. Others forget to subtract from 1 to find the remaining work. A clear separation of steps — finding combined rate, computing work done, finding remainder, and then deriving Suresh’s rate — helps avoid these errors.

Final Answer:
Suresh alone can complete the entire work in 30 days.