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CAT 2023 Slot 3 DILR · Set 1 of 4Logical ReasoningModerate DifficultyQ1–Q5 · 2 MCQ + 3 TITA

CAT 2023 Slot 3 DILR Set 1: Arrangements (5 Questions) with Solutions

The Arrangements set from the official CAT 2023 Slot 3 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 2023 Slot 3 DILR · Questions 1–5
An air conditioner (AC) company has four dealers, D1, D2, D3 and D4 in a city. It is evaluating sales performances of these dealers. The company sells two variants of AC's Window and Split. Both these variants can be either Inverter type or Non-inverter type. It is known that of the total number of ACs sold in the city, 25% were of Window variant, while the rest were of Split variant. Among the Inverter ACs sold, 20% were of Window variant.
The following information is also known:
1
Every dealer sold at least two window ACs.
2
D1 sold 13 inverter ACs, while D3 sold 5 Non-inverter ACs.
3
A total of six Window Non-inverter ACs and 36 Split Inverter ACs were sold in the city.
4
The number of Split ACs sold by D1 was twice the number of Window ACs sold by it.
5
D3 and D4 sold an equal number of Window ACs and this number was one-third of the number of similar ACs sold by D2.
6
D2 and D3 were the only ones who sold Window Non-inverter ACs. The number of these ACs sold by D2 was twice the number of these ACs sold by D3.
7
D3 and D4 sold an equal number of Split Inverter ACs. This number was half the number of similar ACs sold by D2.

In the exam this set competed with 3 others for 40 minutes. Budget about 10 minutes if you attempt it.

Q1. Arrangements

TITA · +3 / 0
How many Split Inverter ACs did D2 sell?
Correct Answer
14
Step-by-step solution
📌 Core Concept
Build the totals from the percentage conditions, then use the dealer clues to split Window and Split Inverter ACs among the dealers.
🔢 Step-by-Step Solution
1
Split Inverter ACs are 80% of all Inverter ACs, since 20% of Inverter ACs are Window. By clue 3 there are 36 Split Inverter ACs, so total Inverter ACs = 36÷0.8=4536 \div 0.8 = 45 and Window Inverter ACs = 9.
2
Window ACs = Window Inverter + Window Non-inverter = 9+6=159 + 6 = 15. This is 25% of all ACs, so the total is 60.
3
Let D3 and D4 each sell xx Window ACs. By clue 5, D2 sells 3x3x. Let D1 sell ww. Then w+5x=15w + 5x = 15.
4
Clue 1 gives x≥2x \ge 2. If x=3x = 3, then w=0w = 0, which breaks clue 1. So x=2x = 2, D2 sells 6 and D1 sells w=5w = 5 Window ACs.
5
Clue 4 gives D1 sold 2×5=102 \times 5 = 10 Split ACs.
6
Clue 6 says only D2 and D3 sold Window Non-inverter ACs, so D1 sold none. All 5 of D1's Window ACs are therefore Inverter.
7
D1 sold 13 Inverter ACs (clue 2), so its Split Inverter ACs = 13−5=813 - 5 = 8.
8
By clue 7, D3 and D4 each sold yy Split Inverter ACs and D2 sold 2y2y. The total gives 8+y+y+2y=368 + y + y + 2y = 36, so 4y=284y = 28 and y=7y = 7.
9
D2 sold 2y=142y = 14 Split Inverter ACs.
Quick Check
Split Inverter ACs by dealer are 8+14+7+7=368 + 14 + 7 + 7 = 36, which matches clue 3.
Correct Answer: 14

Q2. Arrangements

TITA · +3 / 0
What percentage of ACs sold were of Non-Inverter type?
Correct Answer
25
Step-by-step solution
📌 Core Concept
The percentage split of Inverter and Non-inverter ACs follows from the Split Inverter count and the 20% Window share among Inverter ACs.
🔢 Step-by-Step Solution
1
Among Inverter ACs, 20%20\% are Window, so 80%80\% are Split. Clue 3 gives 36 Split Inverter ACs, so the number of Inverter ACs is 360.8=45\dfrac{36}{0.8} = 45.
2
Window Inverter ACs are 0.2×45=90.2 \times 45 = 9. Clue 3 gives 6 Window Non-inverter ACs, so Window ACs total 9+6=159 + 6 = 15.
3
Window ACs are 25%25\% of the total, so the total is 15÷0.25=6015 \div 0.25 = 60.
4
Non-inverter ACs are 60−45=1560 - 45 = 15.
5
The percentage is 1560×100=25%\dfrac{15}{60} \times 100 = 25\%.
Quick Check
Split ACs are 60−15=4560 - 15 = 45. Of these, 36 are Inverter and 9 are Non-inverter. Non-inverter ACs then total 6+9=156 + 9 = 15, which is consistent.
Correct Answer: 25

Q3. Arrangements

TITA · +3 / 0
What was the total number of ACs sold by D2 and D4?.
Correct Answer
33
Step-by-step solution
📌 Core Concept
Fix the Window AC count of each dealer, then use the Split totals to find D2's and D4's overall sales.
🔢 Step-by-Step Solution
1
Inverter ACs total 45, from 0.8I=360.8I = 36 (Split Inverter ACs are 80% of Inverter ACs). Window ACs total 9+6=159 + 6 = 15, and the grand total is 60. Split ACs total 45.
2
Let D3 and D4 each sell ww Window ACs (clue 5). D2 sells 3w3w and D1 sells 15−5w15 - 5w. Clue 1 requires at least 2 each, which forces w=2w = 2. So D1, D2, D3 and D4 sell 5, 6, 2 and 2 Window ACs.
3
By clue 4, D1 sold 2×5=102 \times 5 = 10 Split ACs.
4
D3 sold 5 Non-inverter ACs (clue 2). Two of these are Window (clue 6), so 3 are Split Non-inverter. D1 sold 13 Inverter ACs, of which 5 are Window, so 8 are Split Inverter.
5
Clue 7 gives Split Inverter ACs of 8+s+s+2s=368 + s + s + 2s = 36, so s=7s = 7. D3 and D4 each sold 7 and D2 sold 14.
6
D3 sold 7+3=107 + 3 = 10 Split ACs. D1 sold 10, so D2 and D4 together sold 45−10−10=2545 - 10 - 10 = 25 Split ACs.
7
D2 and D4 sold 6+2=86 + 2 = 8 Window ACs.
8
Total for D2 and D4 is 25+8=3325 + 8 = 33.
Quick Check
D1 sold 15 and D3 sold 12. These add to 27, and 60−27=3360 - 27 = 33.
Correct Answer: 33

Q4. Arrangements

MCQ · +3 / −1
Which of the following statements is necessarily false?
  1. A) D1 and D3 together sold more ACs as compared to D2 and D4 together.
  2. B) D4 sold more Split ACs as compared to D3.
  3. C) D2 sold the highest number of ACs.
  4. D) D1 and D3 sold an equal number of Split ACs.
Correct Answer
A) D1 and D3 together sold more ACs as compared to D2 and D4 together.
Step-by-step solution
🔢 Clue Breakdown
1
Window ACs are 25% of all ACs sold; Window ACs are 20% of all Inverter ACs sold.
2
6 Window Non-inverter ACs were sold, only by D2 and D3, in the ratio 2 : 1, so D2 sold 4 and D3 sold 2.
3
36 Split Inverter ACs were sold: D3 and D4 sold the same number, and D2 sold twice that.
4
D3 and D4 sold the same number of Window ACs, one-third of D2's; every dealer sold at least 2 Window ACs.
5
D1 sold 13 Inverter ACs and twice as many Split ACs as Window ACs; D3 sold 5 Non-inverter ACs.
🔢 Step-by-Step Deduction
Totals: Split Inverter ACs are 80% of Inverter ACs, so Inverter = 36 / 0.8 = 45 and Window Inverter = 9. Window = 9 + 6 = 15 = 25% of all, so 60 ACs were sold: 15 Window and 45 Split (36 Inverter, 9 Non-inverter).
Window ACs by dealer: with D3 = D4 = ww and D2 = 3w3w, D1 = 15−5w15 - 5w. Every dealer sold at least 2, so w=2w = 2: D1 5, D2 6, D3 2, D4 2. D1 and D4 sold no Window Non-inverter ACs, so their Window ACs are all Inverter; D2 sold 2 Window Inverter and D3 none (5 + 2 + 0 + 2 = 9).
Split Inverter: D1 sold 13 - 5 = 8. Then 8+4y=368 + 4y = 36 gives y=7y = 7: D2 14, D3 7, D4 7.
Split Non-inverter: D1 sold 2 × 5 = 10 Split ACs, so 2 Non-inverter. D3's 5 Non-inverter ACs are 2 Window + 3 Split. D2 and D4 share the remaining 9 - 2 - 3 = 4 (say aa and bb, a+b=4a + b = 4).
DealerWindowSplit InverterSplit Non-inverterTotal
D158215
D2614aa20+a20 + a
D327312
D427bb9+b9 + b
Checking the options
A) D1 and D3 together sold more ACs than D2 and D4 together: D1 + D3 = 27, while D2 + D4 = 29 + (a+ba + b) = 33. Always false.
B) D4 sold more Split ACs than D3: D4 sold 7+b7 + b, D3 sold 10. True when b=4b = 4, so not necessarily false.
C) D2 sold the highest number of ACs: D2 sold at least 20, while D1 sold 15, D3 12 and D4 at most 13. Always true.
D) D1 and D3 sold an equal number of Split ACs: both sold 10. Always true.
Correct Answer: A

Q5. Arrangements

MCQ · +3 / −1
If D3 and D4 sold an equal number of ACs, then what was the number of Non-Inverter ACs sold by D2?
  1. A) 5
  2. B) 7
  3. C) 6
  4. D) 4
Correct Answer
A) 5
Step-by-step solution
📌 Core Concept
Establish the base table from the clues. The extra condition that D3 and D4 sold equal totals then fixes D4's Split Non-inverter ACs, which in turn fixes D2's.
🔢 Step-by-Step Solution
1
Totals are 60 ACs: 15 Window (6 Non-inverter, 9 Inverter), 45 Split (36 Inverter, 9 Non-inverter).
2
Window ACs by dealer are D1 5, D2 6, D3 2 and D4 2. Clues 1 and 5 force these values.
3
Window Non-inverter ACs are D2 4 and D3 2 (clue 6). So D2 has 2 Window Inverter ACs, D4 has 2 Window Inverter ACs, and D1 has 5.
4
Split Inverter ACs are D1 8, D3 7, D4 7 and D2 14 (clues 2 and 7).
5
Split Non-inverter ACs total 9. D1 has 10−8=210 - 8 = 2 (its Split count is 10 by clue 4). D3 has 5−2=35 - 2 = 3 (clue 2). That leaves 4 for D2 and D4 combined.
6
D3's total is 2+7+3=122 + 7 + 3 = 12. For D4 to also sell 12, 2+7+y=122 + 7 + y = 12, so D4 has y=3y = 3 Split Non-inverter ACs.
7
D2 then has 4−3=14 - 3 = 1 Split Non-inverter AC.
8
D2's Non-inverter ACs are 4+1=54 + 1 = 5, counting Window and Split.
Why Not the Others
B (7), C (6) and D (4) cannot be reached, because D2's Window Non-inverter count is fixed at 4 and its Split Non-inverter count is 1.
Correct Answer: A
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CAT 2023 Slot 3 DILR at a Glance

Sets in this section4
Questions in this section20
Section time40 minutes
Convenor IIMIIM Lucknow

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