Two persons K and L can complete a piece of work in 30 days and 45 days respectively when working alone. If they work together, what fraction of the work will be completed in 3 days?

Introduction / Context:
This is a direct Time and Work question about two people working together for a given number of days. We are given the individual times taken by K and L to complete the work alone, and asked to find the fraction of the work done when they work together for 3 days. The key is to convert the times into daily work rates and then scale by the number of days worked together.

Given Data / Assumptions:

• K alone can finish the work in 30 days.
• L alone can finish the work in 45 days.
• They work together for 3 days.
• Total work is considered as 1 unit.
• Both K and L have constant daily work rates.

Concept / Approach:
If a person takes T days to complete a job alone, their daily work rate is 1 / T of the job per day. For K and L, we find 1 / 30 and 1 / 45 as their rates. When they work together, their combined rate is the sum of these individual rates. Multiplying this combined rate by 3 (the number of days worked) gives the fraction of the job completed in those 3 days.

Step-by-Step Solution:
Let total work W = 1 unit. K's daily rate = 1 / 30 units per day. L's daily rate = 1 / 45 units per day. Combined rate of K and L = 1 / 30 + 1 / 45. LCM of 30 and 45 is 90. 1 / 30 = 3 / 90 and 1 / 45 = 2 / 90. Combined rate = (3 + 2) / 90 = 5 / 90 = 1 / 18 units per day. In 3 days, work done = 3 * (1 / 18) = 3 / 18 = 1 / 6. Therefore, they complete 1 / 6 of the work in 3 days.

Verification / Alternative check:
You can also think in terms of how long it would take them together to complete the entire job: time together T satisfies (1 / 30 + 1 / 45) * T = 1. Since the combined rate is 1 / 18, T = 18 days. In 3 days, the fraction completed is 3 / 18 = 1 / 6, exactly matching our earlier calculation.

Why Other Options Are Wrong:
Option B (1/3) would imply they complete 1/3 of the work in 3 days, meaning their daily rate would be 1/9, which contradicts the actual combined rate of 1/18. Option C (2/3) and Option D (1/18) are also inconsistent with the rates derived from 30 and 45 days. Only 1/6 is supported by the rate and time calculations.

Common Pitfalls:
Students sometimes wrongly average the times 30 and 45, or they forget to compute the LCM correctly when adding fractions. Another error is to multiply individual times instead of converting times to rates. Always remember: work rate is 1 / time, and combined work rate is the sum of individual rates, not the average of times.

Final Answer:
Together, K and L will complete 1/6 of the work in 3 days.