A can complete a piece of work in 20 days, and B can complete the same work in 10 days. If they work together on the job for 5 days, what fraction of the total work is still left unfinished?

Introduction / Context:
This is a standard time and work problem where two workers, A and B, have different individual times to finish a job. They work together for some days, and we are asked to find what fraction of the work remains. Such questions test understanding of work rate addition and the relationship between time, work, and rate.

Given Data / Assumptions:
- A alone finishes the work in 20 days.
- B alone finishes the work in 10 days.
- They work together for 5 days.
- Total work is taken as 1 unit for ease of calculation.

Concept / Approach:
The basic idea is that work rate = 1 / time when total work is 1 unit. We find the individual daily rates of A and B, add them to get the combined rate, compute how much of the work is done in 5 days, and then subtract from 1 to find the remaining fraction of work.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Rate of A = 1 / 20 work per day. Step 3: Rate of B = 1 / 10 work per day. Step 4: Combined rate of A and B = 1 / 20 + 1 / 10. Step 5: Simplify combined rate: 1 / 20 + 1 / 10 = 1 / 20 + 2 / 20 = 3 / 20 work per day. Step 6: Work done in 5 days = 5 * (3 / 20) = 15 / 20 = 3 / 4 of the total work. Step 7: Remaining work = 1 - 3 / 4 = 1 / 4 of the total work.

Verification / Alternative check:
We can think of the job in terms of 20 equal parts. A does 1 part per day, B does 2 parts per day, so together they complete 3 parts per day. In 5 days, they finish 15 parts out of 20, leaving 5 parts. The remaining fraction is 5 / 20, which simplifies again to 1 / 4, confirming the previous calculation.

Why Other Options Are Wrong:
- 4/9 and 2/9: These do not correspond to the correct combined rate and time product for 5 days of work.
- 1/5: This would mean 80 percent of the work is done, which contradicts the computed 75 percent.
- 1/3: This implies 2/3 of the work is done, which does not match the combined rate of 3 / 20 per day over 5 days.

Common Pitfalls:
Common errors include taking the average of 20 and 10 instead of working with rates, or forgetting to multiply the combined rate by the number of days. Some students also incorrectly compute the remaining work by dividing instead of subtracting from 1. Always compute work done first, then subtract from total work to get the remaining fraction.

Final Answer:
The fraction of the work that is still left after 5 days is 1/4 of the total work.