CAT 2023 Slot 3 · DILR · Set 1: AC Sales by Four Dealers

The full set and all 5 questions (Q1–Q5) from CAT 2023 Slot 3, with answers and solutions.

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 ACs - 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.

Question 1 · Caselet DI

How many Split Inverter ACs did D2 sell?

Answer & solution

Answer: 14

Solving the set

Step 1 – City totals. Split Inverter = 36 and this is 80% of all Inverter ACs (20% are Window), so Inverter = 45 and Window Inverter = 9. Window = 9 + 6 = 15, which is 25% of the total, so the total is 60 and Split = 45, Split Non-inverter = 45 − 36 = 9.

	Inverter	Non-inverter	Total
Window	9	6	15 (25%)
Split	36	9	45 (75%)
Total	45	15	60

Step 2 – Window ACs by dealer. D3 = D4 = w and D2 = 3w (fact 5). D1 has the rest: 15 − 5w ≥ 2 (fact 1) ⇒ w = 2 (w = 1 is ruled out by fact 1, w = 3 leaves D1 nothing). So D1 = 5, D2 = 6, D3 = 2, D4 = 2. D1's Split = 2 × 5 = 10 (fact 4).

Step 3 – Window Non-inverter (6) is sold only by D2 and D3 with D2 = 2 × D3 ⇒ D2 4, D3 2. So D3's window ACs are both non-inverter, D2 has 2 window inverter, and D1 (5) and D4 (2) have only window inverter.

Step 4 – Split Inverter. D1 sold 13 inverter = 5 window + 8 split. D3 = D4 = s and D2 = 2s: 8 + 2s + s + s = 36 ⇒ s = 7. D2 = 14.

Step 5 – Split Non-inverter. D1 = 10 − 8 = 2. D3's non-inverter total is 5 = 2 window + 3 split. Remaining 9 − 2 − 3 = 4 split non-inverter go to D2 (a) and D4 (b), a + b = 4.

Dealer	Window Inv	Window Non-inv	Split Inv	Split Non-inv	Total
D1	5	0	8	2	15
D2	2	4	14	a	20 + a
D3	0	2	7	3	12
D4	2	0	7	b (= 4 − a)	9 + b
Total	9	6	36	9	60

This question

From Step 4, D3 and D4 each sold 7 split inverter ACs and D2 sold twice that: 14.

Hence, the answer is 14.

Question 2 · Caselet DI

What percentage of ACs sold were of Non-inverter type?

A
33.33%
B
75.00%
C
25.00%
D
20.00%

Pick an option, then check.

Answer & solution

Answer: Option C

Solving the set

	Inverter	Non-inverter	Total
Window	9	6	15 (25%)
Split	36	9	45 (75%)
Total	45	15	60

Step 4 – Split Inverter. D1 sold 13 inverter = 5 window + 8 split. D3 = D4 = s and D2 = 2s: 8 + 2s + s + s = 36 ⇒ s = 7. D2 = 14.

Step 5 – Split Non-inverter. D1 = 10 − 8 = 2. D3's non-inverter total is 5 = 2 window + 3 split. Remaining 9 − 2 − 3 = 4 split non-inverter go to D2 (a) and D4 (b), a + b = 4.

Dealer	Window Inv	Window Non-inv	Split Inv	Split Non-inv	Total
D1	5	0	8	2	15
D2	2	4	14	a	20 + a
D3	0	2	7	3	12
D4	2	0	7	b (= 4 − a)	9 + b
Total	9	6	36	9	60

This question

From the city table, Non-inverter ACs = 15 out of 60, i.e., 25%.

Hence, the answer is option 3.

Question 3 · Caselet DI

What was the total number of ACs sold by D2 and D4?

Answer & solution

Answer: 33

Solving the set

	Inverter	Non-inverter	Total
Window	9	6	15 (25%)
Split	36	9	45 (75%)
Total	45	15	60

Step 4 – Split Inverter. D1 sold 13 inverter = 5 window + 8 split. D3 = D4 = s and D2 = 2s: 8 + 2s + s + s = 36 ⇒ s = 7. D2 = 14.

Step 5 – Split Non-inverter. D1 = 10 − 8 = 2. D3's non-inverter total is 5 = 2 window + 3 split. Remaining 9 − 2 − 3 = 4 split non-inverter go to D2 (a) and D4 (b), a + b = 4.

Dealer	Window Inv	Window Non-inv	Split Inv	Split Non-inv	Total
D1	5	0	8	2	15
D2	2	4	14	a	20 + a
D3	0	2	7	3	12
D4	2	0	7	b (= 4 − a)	9 + b
Total	9	6	36	9	60

This question

From the dealer table, D2 = 20 + a and D4 = 9 + b with a + b = 4. D2 + D4 = 29 + 4 = 33 (a and b need not be known).

Hence, the answer is 33.

Question 4 · Caselet DI

Which of the following statements is necessarily false?

A
D2 sold the highest number of ACs.
B
D4 sold more Split ACs as compared to D3.
C
D1 and D3 sold an equal number of Split ACs.
D
D1 and D3 together sold more ACs as compared to D2 and D4 together.

Pick an option, then check.

Answer & solution

Answer: Option D

Solving the set

	Inverter	Non-inverter	Total
Window	9	6	15 (25%)
Split	36	9	45 (75%)
Total	45	15	60

Step 4 – Split Inverter. D1 sold 13 inverter = 5 window + 8 split. D3 = D4 = s and D2 = 2s: 8 + 2s + s + s = 36 ⇒ s = 7. D2 = 14.

Step 5 – Split Non-inverter. D1 = 10 − 8 = 2. D3's non-inverter total is 5 = 2 window + 3 split. Remaining 9 − 2 − 3 = 4 split non-inverter go to D2 (a) and D4 (b), a + b = 4.

Dealer	Window Inv	Window Non-inv	Split Inv	Split Non-inv	Total
D1	5	0	8	2	15
D2	2	4	14	a	20 + a
D3	0	2	7	3	12
D4	2	0	7	b (= 4 − a)	9 + b
Total	9	6	36	9	60

This question

Check each statement with the dealer table (a + b = 4):

• D2 (20 + a ≥ 20) sold more than D1 (15), D3 (12) and D4 (≤ 13) – always true.

• D4's split = 7 + b, D3's split = 10: D4 > D3 when b = 4 – possible.

• D1's split = 10 = D3's split (7 + 3) – always true.

• D1 + D3 = 27 < D2 + D4 = 33 – the statement 'D1 and D3 together sold more' is always false.

Hence, the answer is option 4.

Question 5 · Caselet DI

If D3 and D4 sold an equal number of ACs, then what was the number of Non-inverter ACs sold by D2?

Pick an option, then check.

Answer & solution

Answer: Option B

Solving the set

	Inverter	Non-inverter	Total
Window	9	6	15 (25%)
Split	36	9	45 (75%)
Total	45	15	60

Step 4 – Split Inverter. D1 sold 13 inverter = 5 window + 8 split. D3 = D4 = s and D2 = 2s: 8 + 2s + s + s = 36 ⇒ s = 7. D2 = 14.

Step 5 – Split Non-inverter. D1 = 10 − 8 = 2. D3's non-inverter total is 5 = 2 window + 3 split. Remaining 9 − 2 − 3 = 4 split non-inverter go to D2 (a) and D4 (b), a + b = 4.

Dealer	Window Inv	Window Non-inv	Split Inv	Split Non-inv	Total
D1	5	0	8	2	15
D2	2	4	14	a	20 + a
D3	0	2	7	3	12
D4	2	0	7	b (= 4 − a)	9 + b
Total	9	6	36	9	60

This question

D3 = 12 and D4 = 9 + b. Equal totals ⇒ b = 3 ⇒ a = 1. D2's Non-inverter = 4 (Window) + 1 (Split) = 5.

Hence, the answer is option 2.

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