Six Kirloskar pumps can fill a tank in 7 days, while two similar USHA pumps can fill the same tank in 18 days. What is the ratio of the efficiency (work rate) of one Kirloskar pump to one USHA pump?

Introduction / Context:
This is a work and time problem involving two different brands of pumps, Kirloskar and USHA. Multiple pumps of each type can fill the same tank in different times. We are asked to compare the efficiency of one pump of each type. Efficiency here means the rate at which a single pump can fill the tank, measured as a fraction of the tank per day. This type of question is common in aptitude tests because it combines proportional reasoning and work rate calculations.

Given Data / Assumptions:
- Six Kirloskar pumps fill the tank in 7 days.
- Two USHA pumps fill the same tank in 18 days.
- All pumps of the same brand have equal efficiency.
- The tank size is the same in both cases.
- We need the ratio of efficiency of one Kirloskar pump to one USHA pump.

Concept / Approach:
Work rate is defined as work done per unit time. If six Kirloskar pumps together fill one full tank in 7 days, their combined rate is 1 divided by 7 tanks per day. Therefore, a single Kirloskar pump does one sixth of that. Similarly, if two USHA pumps fill the tank in 18 days, their combined rate is 1 divided by 18 tanks per day. So a single USHA pump does half of that. Once we have individual rates, we can form their ratio and simplify.

Step-by-Step Solution:
Step 1: Combined rate of six Kirloskar pumps = 1 tank / 7 days = 1/7 tank per day. Step 2: Rate of one Kirloskar pump = (1/7) / 6 = 1 / 42 tank per day. Step 3: Combined rate of two USHA pumps = 1 tank / 18 days = 1/18 tank per day. Step 4: Rate of one USHA pump = (1/18) / 2 = 1 / 36 tank per day. Step 5: Efficiency ratio of one Kirloskar pump to one USHA pump = (1/42) : (1/36). Step 6: Simplify the ratio by taking reciprocals: (1/42) : (1/36) = 36 : 42. Step 7: Divide both terms by 6 to get 6 : 7.

Verification / Alternative check:
We can check the reasonableness of the ratio. Since a USHA pump alone fills the tank in 36 days and a Kirloskar pump in 42 days, a USHA pump is slightly faster. That means its efficiency is slightly higher, so the ratio of Kirloskar to USHA should be somewhat less than 1. Indeed, 6 : 7 corresponds to 6/7 which is less than 1, confirming that the direction and magnitude of the ratio make sense.

Why Other Options Are Wrong:
7 : 6 would suggest that one Kirloskar pump is more efficient than one USHA pump, which contradicts the filling times derived from the data. 7 : 54 is not consistent with any simple fraction obtained from the work rates 1/42 and 1/36. The option “None of these” is incorrect because 6 : 7 matches the correct simplified ratio of the two efficiencies.

Common Pitfalls:
Students sometimes mistakenly compare the total days taken by multiple pumps directly without converting to single pump rates. Another frequent error is to set the ratio using 42 : 36 instead of 36 : 42 or to forget to simplify the final ratio. Care must be taken when dividing the combined rate to obtain single pump efficiency and when simplifying the fractional ratio.

Final Answer:
The ratio of the efficiency of a Kirloskar pump to a USHA pump is 6 : 7.