X can complete a work in 10 days and Y can complete the same work in 15 days. If both of them are hired together for 5 days, what fraction of the work will remain unfinished after those 5 days?

Introduction / Context:
This is a basic work and time question involving two workers, X and Y, who have different completion times for the same job. We are asked to find the fraction of work unfinished after they work together for a given number of days. Such questions are often used early in aptitude tests to check whether candidates can handle simple rate addition and fraction calculations correctly.

Given Data / Assumptions:

• X can complete the work alone in 10 days.
• Y can complete the same work alone in 15 days.
• X and Y work together for 5 days.
• Work rates are constant, and there is no change in efficiency over time.
• We need the fraction of the work that remains unfinished after 5 days.

Concept / Approach:
The core idea is to convert individual completion times into daily work rates and then add these rates to obtain the combined rate. Once we know how much of the work is completed per day by both workers together, multiplying that rate by the number of days worked gives the total fraction of the work completed. The unfinished fraction is then one minus the completed fraction. This straightforward method avoids any unnecessary complexity.

Step-by-Step Solution:
Step 1: Rate of X is 1 / 10 of the work per day. Step 2: Rate of Y is 1 / 15 of the work per day. Step 3: Combined daily rate of X and Y is 1/10 + 1/15. Step 4: Compute 1/10 + 1/15 = (3 + 2) / 30 = 5 / 30 = 1 / 6 of the work per day. Step 5: In 5 days, X and Y together complete 5 times 1/6 = 5/6 of the work. Step 6: Fraction of work remaining unfinished is 1 - 5/6 = 1/6 of the work.

Verification / Alternative check:
We can assign a convenient total work amount, such as 30 units. X then completes 3 units per day and Y completes 2 units per day, together completing 5 units per day. In 5 days they complete 25 units, leaving 5 units out of 30. The remaining fraction is 5/30 which simplifies to 1/6. This unit method exactly matches the fraction based solution and confirms the answer.

Why Other Options Are Wrong:

• 1/3 of the work remaining would mean they completed only 2/3, which does not agree with the calculated 5/6 completion.
• 2/3 of the work remaining would mean very little progress, which is unrealistic given their combined rate.
• 5/6 of the work remaining would mean almost no work done, which is clearly inconsistent with working together for 5 days.

Common Pitfalls:
Some learners mistakenly add the times instead of the rates or forget to multiply the combined rate by the number of days worked. Others confuse the completed fraction with the remaining fraction and answer 5/6 instead of 1/6. Always compute what is completed first, then subtract from 1 to obtain what remains.

Final Answer:
After working together for 5 days, the fraction of work that remains unfinished is 1/6 of the work.