Lasya alone can complete a certain work in 16 days. Srimukhi works at a speed that is 20 percent less than Lasya, and Rashmi working together with Srimukhi can complete the same work in 12 days. What is the ratio of Rashmi's efficiency to Lasya's efficiency?

Introduction / Context:
This problem involves comparing efficiencies of three workers when one worker's speed is a fixed percentage less than another, and the combined time of two workers is known. We must determine the ratio of Rashmi's efficiency to Lasya's efficiency, which is a common type of time and work ratio question.

Given Data / Assumptions:

• Lasya alone can complete the work in 16 days.
• Srimukhi works at a speed that is 20 percent less than Lasya's speed.
• Rashmi working together with Srimukhi completes the work in 12 days.
• All workers maintain constant work rates and the total work is taken as one unit.

Concept / Approach:
We first convert Lasya's completion time into a daily work rate. Then we reduce that rate by 20 percent to obtain Srimukhi's work rate. Knowing the combined completion time for Srimukhi and Rashmi allows us to compute their combined rate. Subtracting Srimukhi's rate from this combined rate yields Rashmi's individual rate. Finally, we compare Rashmi's rate to Lasya's rate to obtain the required efficiency ratio.

Step-by-Step Solution:
Let the total work be 1 job. Lasya's time alone = 16 days, so Lasya's rate = 1 / 16 job per day. Srimukhi works at 20 percent less speed than Lasya. So Srimukhi's rate = 80 percent of Lasya's rate = 0.8 * (1 / 16) = 1 / 20 job per day. Srimukhi and Rashmi together complete the job in 12 days. So combined rate of Srimukhi and Rashmi = 1 / 12 job per day. Let Rashmi's rate be R job per day. Then R + (1 / 20) = 1 / 12. Therefore, R = 1 / 12 - 1 / 20. Compute R: with common denominator 60, 1 / 12 = 5 / 60 and 1 / 20 = 3 / 60. So R = (5 / 60) - (3 / 60) = 2 / 60 = 1 / 30 job per day. Lasya's rate was 1 / 16, Rashmi's rate is 1 / 30. Efficiency ratio Rashmi : Lasya = (1 / 30) : (1 / 16) = 16 : 30. Simplify 16 : 30 by dividing both sides by 2 to get 8 : 15.

Verification / Alternative check:
We can verify by directly checking the combined rate of Srimukhi and Rashmi: 1 / 20 + 1 / 30 = (3 / 60 + 2 / 60) = 5 / 60 = 1 / 12 job per day, which matches the given time of 12 days for them to complete the work together. This confirms that our rates and thus the efficiency ratio are correct.

Why Other Options Are Wrong:
19 : 7 and 31 : 17 are arbitrary looking ratios that do not relate to the underlying rates of 1 / 16 and 1 / 30. 30 : 19 does not simplify to the correct ratio even if inverted. 15 : 8 is the inverse of the correct ratio and would imply Rashmi is more efficient than Lasya, which conflicts with their times.

Common Pitfalls:
The main trap is mishandling the phrase "20 percent less," which some learners interpret as dividing by 1.2 or performing other incorrect operations. The correct approach is to multiply the original rate by 0.8. Another common mistake is to compare completion times directly instead of converting them to rates before forming ratios. Always use work rates when dealing with efficiency questions.

Final Answer:
The ratio of Rashmi's efficiency to Lasya's efficiency is 8 : 15.