CAT 2023 Slot 1 · DILR · Question 15

CAT 2023 Slot 1 previous year question on Caselet DI, with the answer and a step-by-step solution.

Caselet DIData InterpretationMultiple choiceMedium

Set 3: Restaurant Ratings of Gig Workers — this question belongs to a set. Read the full set with all 5 questions →

Five restaurants, coded R1, R2, R3, R4 and R5 gave integer ratings to five gig workers - Ullas, Vasu, Waman, Xavier and Yusuf, on a scale of 1 to 5.

The means of the ratings given by R1, R2, R3, R4 and R5 were 3.4, 2.2, 3.8, 2.8 and 3.4 respectively.

The summary statistics of these ratings for the five workers is given below.

	Ullas	Vasu	Waman	Xavier	Yusuf
Mean rating	2.2	3.8	3.4	3.6	2.6
Median rating	2	4	4	4	3
Modal rating	2	4	5	5	1 and 4
Range of rating*	3	3	4	4	3

* Range of ratings is defined as the difference between the maximum and minimum ratings awarded to a worker.

The following is partial information about ratings of 1 and 5 awarded by the restaurants to the workers.

(a) R1 awarded a rating of 5 to Waman, as did R2 to Xavier, R3 to Waman and Xavier, and R5 to Vasu.

(b) R1 awarded a rating of 1 to Ullas, as did R2 to Waman and Yusuf, and R3 to Yusuf.

Which among the following restaurants gave its median rating to exactly one of the workers?

A
R2
B
R5
C
R4
D
R3

Pick an option, then check.

Answer & solution

Answer: Option C

Solving the set

Step 1 – Each worker's five ratings. Sum = 5 × mean. Use the median, mode, range and the known 1s and 5s.

• Ullas: sum 11, has a 1 (R1), range 3 ⇒ maximum 4. Median 2 and mode 2 ⇒ 1, 2, 2, x, 4 with 1 + 2 + 2 + x + 4 = 11 ⇒ x = 2.

• Vasu: sum 19, has a 5 (R5), range 3 ⇒ minimum 2. Median 4 ⇒ 2, a, 4, b, 5 with a + b = 8 ⇒ a = b = 4.

• Waman: sum 17, two 5s (R1, R3) and a 1 (R2), median 4 ⇒ 1, x, 4, 5, 5 with x = 2.

• Xavier: sum 18, two 5s (R2, R3), range 4 ⇒ minimum 1. Median 4 ⇒ 1, x, 4, 5, 5 with x = 3.

• Yusuf: sum 13, two 1s (R2, R3), modes 1 and 4, median 3 ⇒ 1, 1, 3, 4, 4.

Worker	Sum (5 × mean)	Known 1s/5s	Ratings (from median, mode, range)
Ullas	11	1 (R1)	1, 2, 2, 2, 4
Vasu	19	5 (R5)	2, 4, 4, 4, 5
Waman	17	5 (R1, R3), 1 (R2)	1, 2, 4, 5, 5
Xavier	18	5 (R2, R3)	1, 3, 4, 5, 5
Yusuf	13	1 (R2, R3)	1, 1, 3, 4, 4

Step 2 – Restaurant totals (5 × mean): R1 = 17, R2 = 11, R3 = 19, R4 = 14, R5 = 17.

• R2: Waman 1 + Xavier 5 + Yusuf 1 = 7, so Ullas + Vasu = 4. Both are at least 2, so both are 2. Vasu's single 2 is at R2, so Vasu gets 4 from R1, R3 and R4.

• R3: Vasu 4 + Waman 5 + Xavier 5 + Yusuf 1 = 15, so Ullas = 4. Ullas then gets 2 from R4 and R5.

• R1: Ullas 1 + Vasu 4 + Waman 5 = 10, so Xavier + Yusuf = 7: (3, 4) or (4, 3).

• If Xavier = 4 and Yusuf = 3 at R1, Yusuf has 4 at both R4 and R5. R4 needs Waman + Xavier = 14 − 2 − 4 − 4 = 4, but Waman's options (2 or 4) and Xavier's (1 or 3) never add to 4. Not possible.

• So Xavier = 3 and Yusuf = 4 at R1. R4 needs Waman + Xavier + Yusuf = 8: only 4 + 1 + 3 works. R5 then gets Waman 2, Xavier 4, Yusuf 4 (total 17 ✓).

Restaurant	Ullas	Vasu	Waman	Xavier	Yusuf	Sum	Mean
R1	1	4	5	3	4	17	3.4
R2	2	2	1	5	1	11	2.2
R3	4	4	5	5	1	19	3.8
R4	2	4	4	1	3	14	2.8
R5	2	5	2	4	4	17	3.4

This question

Restaurant	Ratings in order	Median	Workers given the median
R2	1, 1, 2, 2, 5	2	2 (Ullas, Vasu)
R3	1, 4, 4, 5, 5	4	2 (Ullas, Vasu)
R4	1, 2, 3, 4, 4	3	1 (Yusuf)
R5	2, 2, 4, 4, 5	4	2 (Xavier, Yusuf)

R4's median is 3 and only Yusuf got a 3 from R4.

Hence, the answer is option 3.

CAT 2023 Slot 1 · DILR · Question 15

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