CAT 2025 Slot 2 DILR Set 1 Q1–Q5

A DILR caselet from the Data Interpretation & Logical Reasoning (DILR) section of CAT 2025 Slot 2, with 5 questions (Quant-based Puzzles). Answers are CAT's official key.

The set

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.

Question 1

What is the PI of Whimshire?

TITA: type in the answer. There are no options and no negative marking.

Answer: 45 Quant-based Puzzles TITA

Solution

  1. Only one NUR-city pair has NUR > city, and it is within Humbleset. Any NUR above some city is above Blusterburg (the lowest city), so that city is Blusterburg, which belongs to Humbleset, and Humbleset's NUR is below Noodleton.
  2. The other two NURs are below every city, so they are 10 and 20; then Blusterburg \(= 30\), Humbleset NUR \(= 40\), and Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo \(= 50, 60, 70, 80, 90\).
  3. PI \(= \frac{2\,\text{NUR} + c_1 + c_2}{4}\) must be an integer. For NUR 10 or 20, \(2\,\text{NUR}\) is 20 or 40, so we need \(c_1 + c_2\) a multiple of 20; for Humbleset \(80 + 30 + x\) divisible by 4 means \(x \in \{50, 70, 90\}\).
  4. Humbleset with 50 gives PI 40, which cannot be the highest; with 70 gives 45, which ties another state. With 90: Humbleset PI \(= \frac{80+30+90}{4} = 50\).
  5. Remaining cities pair as \(\{50,70\}\) and \(\{60,80\}\). NUR 10 with \(\{50,70\}\) gives 35 and NUR 20 with \(\{60,80\}\) gives 45 (the other pairing ties at 40). So Fogglia \(= 35\) (NUR 10, Noodleton, Quackford), Whimshire \(= 45\) (NUR 20, Splutterville, Mumpypore), Humbleset \(= 50\) (NUR 40, Blusterburg, Zingaloo).
  6. PI of Whimshire \(= \frac{2(20) + 60 + 80}{4} = 45\).

Question 2

What is the PI of Fogglia?

TITA: type in the answer. There are no options and no negative marking.

Answer: 35 Quant-based Puzzles TITA

Solution

  1. Only one NUR-city pair has NUR > city, and it is within Humbleset. Any NUR above some city is above Blusterburg (the lowest city), so that city is Blusterburg, which belongs to Humbleset, and Humbleset's NUR is below Noodleton.
  2. The other two NURs are below every city, so they are 10 and 20; then Blusterburg \(= 30\), Humbleset NUR \(= 40\), and Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo \(= 50, 60, 70, 80, 90\).
  3. PI \(= \frac{2\,\text{NUR} + c_1 + c_2}{4}\) must be an integer. For NUR 10 or 20, \(2\,\text{NUR}\) is 20 or 40, so we need \(c_1 + c_2\) a multiple of 20; for Humbleset \(80 + 30 + x\) divisible by 4 means \(x \in \{50, 70, 90\}\).
  4. Humbleset with 50 gives PI 40, which cannot be the highest; with 70 gives 45, which ties another state. With 90: Humbleset PI \(= \frac{80+30+90}{4} = 50\).
  5. Remaining cities pair as \(\{50,70\}\) and \(\{60,80\}\). NUR 10 with \(\{50,70\}\) gives 35 and NUR 20 with \(\{60,80\}\) gives 45 (the other pairing ties at 40). So Fogglia \(= 35\) (NUR 10, Noodleton, Quackford), Whimshire \(= 45\) (NUR 20, Splutterville, Mumpypore), Humbleset \(= 50\) (NUR 40, Blusterburg, Zingaloo).
  6. PI of Fogglia \(= \frac{2(10) + 50 + 70}{4} = 35\).

Question 3

What is the PI of Humbleset?

TITA: type in the answer. There are no options and no negative marking.

Answer: 50 Quant-based Puzzles TITA

Solution

  1. Only one NUR-city pair has NUR > city, and it is within Humbleset. Any NUR above some city is above Blusterburg (the lowest city), so that city is Blusterburg, which belongs to Humbleset, and Humbleset's NUR is below Noodleton.
  2. The other two NURs are below every city, so they are 10 and 20; then Blusterburg \(= 30\), Humbleset NUR \(= 40\), and Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo \(= 50, 60, 70, 80, 90\).
  3. PI \(= \frac{2\,\text{NUR} + c_1 + c_2}{4}\) must be an integer. For NUR 10 or 20, \(2\,\text{NUR}\) is 20 or 40, so we need \(c_1 + c_2\) a multiple of 20; for Humbleset \(80 + 30 + x\) divisible by 4 means \(x \in \{50, 70, 90\}\).
  4. Humbleset with 50 gives PI 40, which cannot be the highest; with 70 gives 45, which ties another state. With 90: Humbleset PI \(= \frac{80+30+90}{4} = 50\).
  5. Remaining cities pair as \(\{50,70\}\) and \(\{60,80\}\). NUR 10 with \(\{50,70\}\) gives 35 and NUR 20 with \(\{60,80\}\) gives 45 (the other pairing ties at 40). So Fogglia \(= 35\) (NUR 10, Noodleton, Quackford), Whimshire \(= 45\) (NUR 20, Splutterville, Mumpypore), Humbleset \(= 50\) (NUR 40, Blusterburg, Zingaloo).
  6. PI of Humbleset \(= \frac{2(40) + 30 + 90}{4} = 50\).

Question 4

Which pair of cities definitely belong to the same state?
  1. A
    Blusterburg, Mumpypore
  2. B
    Splutterville, Quackford
  3. C
    Mumpypore, Zingaloo
  4. D
    Noodleton, Quackford

Answer: (D) Noodleton, Quackford Quant-based Puzzles MCQ

Solution

  1. Only one NUR-city pair has NUR > city, and it is within Humbleset. Any NUR above some city is above Blusterburg (the lowest city), so that city is Blusterburg, which belongs to Humbleset, and Humbleset's NUR is below Noodleton.
  2. The other two NURs are below every city, so they are 10 and 20; then Blusterburg \(= 30\), Humbleset NUR \(= 40\), and Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo \(= 50, 60, 70, 80, 90\).
  3. PI \(= \frac{2\,\text{NUR} + c_1 + c_2}{4}\) must be an integer. For NUR 10 or 20, \(2\,\text{NUR}\) is 20 or 40, so we need \(c_1 + c_2\) a multiple of 20; for Humbleset \(80 + 30 + x\) divisible by 4 means \(x \in \{50, 70, 90\}\).
  4. Humbleset with 50 gives PI 40, which cannot be the highest; with 70 gives 45, which ties another state. With 90: Humbleset PI \(= \frac{80+30+90}{4} = 50\).
  5. Remaining cities pair as \(\{50,70\}\) and \(\{60,80\}\). NUR 10 with \(\{50,70\}\) gives 35 and NUR 20 with \(\{60,80\}\) gives 45 (the other pairing ties at 40). So Fogglia \(= 35\) (NUR 10, Noodleton, Quackford), Whimshire \(= 45\) (NUR 20, Splutterville, Mumpypore), Humbleset \(= 50\) (NUR 40, Blusterburg, Zingaloo).
  6. Noodleton and Quackford are both in Fogglia; none of the other pairs share a state.

Question 5

For how many of the cities and NURs is it possible to identify their PM and the state they belong to?

TITA: type in the answer. There are no options and no negative marking.

Answer: 9 Quant-based Puzzles TITA

Solution

  1. Only one NUR-city pair has NUR > city, and it is within Humbleset. Any NUR above some city is above Blusterburg (the lowest city), so that city is Blusterburg, which belongs to Humbleset, and Humbleset's NUR is below Noodleton.
  2. The other two NURs are below every city, so they are 10 and 20; then Blusterburg \(= 30\), Humbleset NUR \(= 40\), and Noodleton, Splutterville, Quackford, Mumpypore, Zingaloo \(= 50, 60, 70, 80, 90\).
  3. PI \(= \frac{2\,\text{NUR} + c_1 + c_2}{4}\) must be an integer. For NUR 10 or 20, \(2\,\text{NUR}\) is 20 or 40, so we need \(c_1 + c_2\) a multiple of 20; for Humbleset \(80 + 30 + x\) divisible by 4 means \(x \in \{50, 70, 90\}\).
  4. Humbleset with 50 gives PI 40, which cannot be the highest; with 70 gives 45, which ties another state. With 90: Humbleset PI \(= \frac{80+30+90}{4} = 50\).
  5. Remaining cities pair as \(\{50,70\}\) and \(\{60,80\}\). NUR 10 with \(\{50,70\}\) gives 35 and NUR 20 with \(\{60,80\}\) gives 45 (the other pairing ties at 40). So Fogglia \(= 35\) (NUR 10, Noodleton, Quackford), Whimshire \(= 45\) (NUR 20, Splutterville, Mumpypore), Humbleset \(= 50\) (NUR 40, Blusterburg, Zingaloo).
  6. The arrangement is unique, so all 9 (six cities, three NURs) have known PM and state.

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