CAT 2025 Slot 2 · DILR · Question 22
CAT 2025 Slot 2 previous year question on Caselet Based DI, with the answer and a step-by-step solution.
Set 5: Pollution Index of Three States — this question belongs to a set. Read the full set with all 5 questions →
The two most populous cities and the non-urban region (NUR) of each of three states, Whimshire, Fogglia, and Humbleset, are assigned Pollution Measures (PMs). These nine PMs are all distinct multiples of 10, ranging from 10 to 90. The six cities in increasing order of their PMs are: Blusterburg, Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo.
The Pollution Index (PI) of a state is a weighted average of the PMs of its NUR and cities, with a weight of 50% for the NUR, and 25% each for its two cities.
There is only one pair of an NUR and a city (considering all cities and all NURs) where the PM of the NUR is greater than that of the city. That NUR and the city both belong to Humbleset.
The PIs of all three states are distinct integers, with Humbleset and Fogglia having the highest and the lowest PI respectively.
For how many of the cities and NURs is it possible to identify their PM and the state they belong to?
Answer & solution
Answer: 9
The PM ladder is forced: the two lowest NURs are 10 and 20, Blusterburg is 30, Humbleset's NUR is 40, and the cities Noodleton to Zingaloo are 50 to 90. Every PM is known.
The integer and ordering conditions force Humbleset = NUR 40 + Blusterburg + Zingaloo. The other cities must pair as Noodleton–Quackford and Splutterville–Mumpypore, and the only distinct assignment is Fogglia = NUR 10 + Noodleton + Quackford (35) and Whimshire = NUR 20 + Splutterville + Mumpypore (45).
All 3 NURs and all 6 cities have a known PM and a known state: 9 in total.
Shortcut: once the arrangement turns out to be unique, all nine places are fully identified.
Hence, the answer is 9.
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