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CAT 2025 Slot 2 DILR · Set 1 of 5Logical ReasoningHard DifficultyQ1–Q5 · 1 MCQ + 4 TITA

CAT 2025 Slot 2 DILR Set 1: Math Puzzles (5 Questions) with Solutions

The Math Puzzles set from the official CAT 2025 Slot 2 DILR paper: the caselet, all 5 questions, the official answer key and a step-by-step solution for each. 100% free, no sign-up.

The Set: Read the Caselet First

CAT 2025 Slot 2 DILR · Questions 1–5
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.

In the exam this set competed with 4 others for 40 minutes. Budget about 8 minutes if you attempt it.

Q1. Math Puzzles

TITA · +3 / 0Solve on its own page
What is the PI of Whimshire?
Correct Answer
45
Step-by-step solution
📌 Core Concept
The single NUR-over-city exception fixes most PMs. The integer-PI condition then fixes the state groupings.
🔢 Step-by-Step Solution
1
Only one NUR beats any city, and it is Humbleset's NUR beating one of its own cities. So the other two NURs are below all six cities. They take the two lowest PMs, 10 and 20.
2
Humbleset's NUR beats exactly one city. That city must be the lowest city, Blusterburg, or the NUR would also beat a second city. So the order is Blusterburg 30, Humbleset NUR 40, then Noodleton 50, Splutterville 60, Quackford 70, Mumpypore 80, Zingaloo 90.
3
Humbleset's PI is 0.5(40)+0.25(30)+0.25x=27.5+0.25x0.5(40)+0.25(30)+0.25x = 27.5+0.25x, where xx is its other city. For an integer PI, xx is 50, 70 or 90, giving PI 40, 45 or 50.
4
For Whimshire and Fogglia, PI=2n+a+b4\text{PI}=\frac{2n+a+b}{4} with nn equal to 10 or 20. This is an integer only if a+ba+b is a multiple of 20.
5
If x=70x=70, the other cities pair as {50,90} and {60,80}, giving PIs 40 and 45. The 45 ties with Humbleset, so this fails.
6
If x=50x=50, the pairs are {60,80} and {70,90}. The PIs are 45 and 45 (tie), or 40 and 50. The first fails on the tie, and the second makes another state's PI equal to or above Humbleset's 40. So this fails.
7
If x=90x=90, Humbleset's PI is 50. The pairs are {50,70} and {60,80}. With NURs 20 and 10 respectively, the PIs are 40 and 40 (tie). With NURs 10 and 20, the PIs are 35 and 45, which are distinct and below 50.
8
Whimshire has NUR 20 and cities 60 and 80, so its PI is 0.5(20)+0.25(60+80)=10+35=450.5(20)+0.25(60+80)=10+35=45.
Quick Check
PIW=45\text{PI}_W=45 lies between Fogglia's 35 and Humbleset's 50, as required.
Correct Answer: 45

Q2. Math Puzzles

TITA · +3 / 0
What is the PI of Fogglia?
Correct Answer
35
Step-by-step solution
📌 Core Concept
The single NUR-over-city exception fixes most PMs. The integer-PI condition then fixes the state groupings.
🔢 Step-by-Step Solution
1
Only Humbleset's NUR beats any city, so the other two NURs are below all six cities. They take 10 and 20.
2
Humbleset's NUR must beat only its own lowest city, Blusterburg. So Blusterburg is 30 and Humbleset's NUR is 40. Then Noodleton is 50, Splutterville 60, Quackford 70, Mumpypore 80 and Zingaloo 90.
3
Humbleset's PI is 27.5+0.25x27.5+0.25x, where xx is its other city. For an integer PI, xx is 50, 70 or 90, giving PI 40, 45 or 50.
4
For the other two states, PI=2n+a+b4\text{PI}=\frac{2n+a+b}{4} with nn equal to 10 or 20. This needs a+ba+b to be a multiple of 20.
5
If x=70x=70, the pairs {50,90} and {60,80} give PIs 40 and 45. The 45 ties with Humbleset, so this fails.
6
If x=50x=50, the pairs {60,80} and {70,90} give PIs 45 and 45 (tie), or 40 and 50. The first fails on the tie, and the second makes another state's PI equal to or above Humbleset's 40. So this fails.
7
If x=90x=90, Humbleset's PI is 50. The pairs are {50,70} and {60,80}. Pairing {50,70} with NUR 20 and {60,80} with NUR 10 gives 40 and 40, a tie. So {50,70} must go with NUR 10, and {60,80} with NUR 20.
8
Fogglia has the lowest PI. It has NUR 10 and cities 50 and 70, so its PI is 0.5(10)+0.25(50+70)=5+30=350.5(10)+0.25(50+70)=5+30=35.
Quick Check
The PIs are 35, 45 and 50, which are distinct. Fogglia's is lowest and Humbleset's is highest.
Correct Answer: 35

Q3. Math Puzzles

TITA · +3 / 0
What is the PI of Humbleset?
Correct Answer
50
Step-by-step solution
📌 Core Concept
The single NUR-over-city exception fixes most PMs. The integer-PI and distinctness conditions then fix Humbleset's second city.
🔢 Step-by-Step Solution
1
Only Humbleset's NUR beats any city, so the other two NURs take 10 and 20, below all six cities.
2
Humbleset's NUR must beat only its own lowest city, Blusterburg. So Blusterburg is 30 and Humbleset's NUR is 40. Then Noodleton is 50, Splutterville 60, Quackford 70, Mumpypore 80 and Zingaloo 90.
3
Humbleset's PI is 0.5(40)+0.25(30)+0.25x=27.5+0.25x0.5(40)+0.25(30)+0.25x=27.5+0.25x, where xx is its other city. An integer PI needs xx to be 50, 70 or 90, giving PI 40, 45 or 50.
4
For the other two states, PI=2n+a+b4\text{PI}=\frac{2n+a+b}{4} with nn equal to 10 or 20. This needs a+ba+b to be a multiple of 20.
5
x=70x=70 leaves the pairs {50,90} and {60,80}, with PIs 40 and 45. The 45 ties with Humbleset, so this is out.
6
x=50x=50 leaves the pairs {60,80} and {70,90}. The PIs are 45 and 45 (tie), or 40 and 50. The first fails on the tie, and the second makes another state's PI equal to or above Humbleset's 40. So this is out.
7
So x=90x=90, which is Zingaloo. The pairs {50,70} and {60,80} then give PIs 35 and 45, both below Humbleset.
8
Humbleset's PI is 27.5+0.25(90)=27.5+22.5=5027.5+0.25(90)=27.5+22.5=50.
Quick Check
0.5(40)+0.25(30)+0.25(90)=20+7.5+22.5=500.5(40)+0.25(30)+0.25(90)=20+7.5+22.5=50.
Correct Answer: 50

Q4. Math Puzzles

MCQ · +3 / −1
Which pair of cities definitely belong to the same state?
  1. A) Blusterburg, Mumpypore
  2. B) Splutterville, Quackford
  3. C) Noodleton, Quackford
  4. D) Mumpypore, Zingaloo
Correct Answer
C) Noodleton, Quackford
Step-by-step solution
📌 Core Concept
The exception clue fixes every PM. The integer-PI and distinctness conditions then fix which cities form each state.
🔢 Step-by-Step Solution
1
Only Humbleset's NUR beats any city, so Whimshire's and Fogglia's NURs are 10 and 20. Humbleset's NUR must beat only its own lowest city. So Blusterburg is 30 and Humbleset's NUR is 40. Then Noodleton is 50, Splutterville 60, Quackford 70, Mumpypore 80 and Zingaloo 90.
2
Humbleset's PI is 27.5+0.25x27.5+0.25x, where xx is its other city. An integer PI needs xx to be 50, 70 or 90.
3
For the other two states, PI=2n+a+b4\text{PI}=\frac{2n+a+b}{4} needs a+ba+b to be a multiple of 20.
4
x=70x=70 gives Humbleset PI 45, and the other states then have PIs 40 and 45. This ties, so it fails.
5
x=50x=50 gives Humbleset PI 40. The other states then have PIs 45 and 45 (tie), or 40 and 50. The first fails on the tie, and the second makes another state's PI equal to or above Humbleset's 40. So it fails.
6
So x=90x=90, and Humbleset has Blusterburg and Zingaloo. The remaining cities split as {Noodleton, Quackford} and {Splutterville, Mumpypore}, as only these sums are multiples of 20.
7
Noodleton and Quackford are therefore always together.
Why Not the Others
A: Blusterburg is in Humbleset, while Mumpypore is not.
B: Splutterville and Quackford fall in different states.
D: Mumpypore and Zingaloo fall in different states.
Correct Answer: C

Q5. Math Puzzles

TITA · +3 / 0
For how many of the cities and NURs is it possible to identify their PM and the state they belong to?
Correct Answer
9
Step-by-step solution
📌 Core Concept
The exception clue fixes every PM. The integer-PI and distinctness conditions then fix every state assignment.
🔢 Step-by-Step Solution
1
Only Humbleset's NUR beats any city, so Whimshire's and Fogglia's NURs are 10 and 20. Humbleset's NUR must beat only its own lowest city. So Blusterburg is 30 and Humbleset's NUR is 40. Then Noodleton is 50, Splutterville 60, Quackford 70, Mumpypore 80 and Zingaloo 90. All nine PMs are fixed.
2
Humbleset's PI is 27.5+0.25x27.5+0.25x, where xx is its other city. An integer PI needs xx to be 50, 70 or 90.
3
The other two states need a+ba+b to be a multiple of 20 for an integer PI.
4
x=70x=70 leaves the pairs {50,90} and {60,80}, with PIs 40 and 45. The 45 ties with Humbleset's 45, so this fails.
5
x=50x=50 leaves the pairs {60,80} and {70,90}. The PIs are 45 and 45 (tie), or 40 and 50. The first fails on the tie, and the second makes another state's PI equal to or above Humbleset's 40. So this fails.
6
x=90x=90 works. Humbleset has NUR 40 with Blusterburg (30) and Zingaloo (90).
7
The pairs {50,70} and {60,80} must take NURs 10 and 20 respectively, to avoid a tie at 40. So Fogglia has NUR 10 with Noodleton and Quackford, and Whimshire has NUR 20 with Splutterville and Mumpypore.
8
Every city and NUR now has a known PM and state, so all 9 are identifiable.
Correct Answer: 9
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CAT 2025 Slot 2 DILR at a Glance

Sets in this section5
Questions in this section22
Section time40 minutes
Convenor IIMIIM Kozhikode

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