A can complete a work alone in 20 days and B can complete the same work alone in 50 days. If they work together on this job for 5 days, what fraction of the total work will still remain incomplete at the end of the fifth day?

Introduction / Context:
This problem is another application of the time and work concept, where two workers A and B have known individual completion times. You are asked to calculate how much of the job is still incomplete after they work together for a certain number of days. This type of question is common in quantitative aptitude exams.

Given Data / Assumptions:

A alone can finish the work in 20 days.
B alone can finish the work in 50 days.
A and B work together for exactly 5 days at constant speed.
The entire job is treated as one unit of work.

Concept / Approach:
We use the fact that a person who finishes a work in T days does 1/T of the work per day. The combined daily rate of two workers is the sum of their individual rates. After computing how much fraction of the work they complete in 5 days, we subtract that from 1 to obtain the fraction of work that remains unfinished.

Step-by-Step Solution:
Let total work = 1 unit. Daily rate of A = 1/20 of the work per day. Daily rate of B = 1/50 of the work per day. Combined daily rate of A and B = 1/20 + 1/50. Find the LCM of 20 and 50, which is 100. Convert fractions: 1/20 = 5/100 and 1/50 = 2/100. Combined rate = 5/100 + 2/100 = 7/100 of the work per day. In 5 days, work completed = 5 * (7/100) = 35/100. Simplify 35/100 to 7/20 of the work completed. Remaining work = 1 - 7/20 = 13/20 of the work.

Verification / Alternative check:
To verify, imagine the job consists of 100 equal units. A does 5 units per day (since 20 days for 100 units) and B does 2 units per day (since 50 days for 100 units). Together they do 7 units per day. In 5 days they complete 7 * 5 = 35 units. That leaves 100 - 35 = 65 units, which is 65/100 = 13/20 of the work, matching our earlier calculation. This confirms that 13/20 of the job is still incomplete.

Why Other Options Are Wrong:
Option 1/3 is too small; it would mean that two-thirds of the work is already finished, which contradicts the calculated 7/20 completed portion.
Option 1/6 is even smaller and suggests that most of the job has been done, which is not supported by the rates.
Option 2/9 does not align with any natural fraction from these numbers and does not result from correct calculations.
Option 7/10 is closer but still incorrect; this would require that only 3/10 of the work had been done, not 7/20.

Common Pitfalls:
Learners often confuse the fraction of work completed with the fraction remaining. Some students also incorrectly average 20 and 50 instead of adding the rates 1/20 and 1/50. Always remember to work with rates (reciprocals of time), carefully find a common denominator and then sum the fractions. Subtract the completed fraction from 1 to get what is left.

Final Answer:
Thus, after working together for 5 days, the fraction of the work still left incomplete is 13/20, so the correct option is 13/20.