Question:medium

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.
 UllasVasuWamanXavierYusuf
Mean rating2.23.83.43.62.6
Median rating24443
Model rating24551 and 4
Range of rating33443
* 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.
How many individual ratings cannot be determined from the above information? [This question was asked as TITA]

Updated On: Jan 13, 2026
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The Correct Option is A

Solution and Explanation

The objective is to ascertain if all individual ratings can be definitively established given the mean ratings of restaurants and workers, alongside a few specified ratings.

Step 1: Total ratings per restaurant
Each restaurant provided ratings for 5 workers, resulting in 5 ratings per restaurant.  

Step 2: Total sum of ratings for each restaurant
 

  • R1: Mean = 3.4 → Total Sum = \( 3.4 \times 5 = 17 \)
  • R2: Mean = 2.2 → Total Sum = \( 2.2 \times 5 = 11 \)
  • R3: Mean = 3.8 → Total Sum = \( 3.8 \times 5 = 19 \)
  • R4: Mean = 2.8 → Total Sum = \( 2.8 \times 5 = 14 \)
  • R5: Mean = 3.4 → Total Sum = \( 3.4 \times 5 = 17 \)

Step 3: Incorporating known ratings
 

  • R1 rated Waman 5 → Remaining sum for R1 = \( 17 - 5 = 12 \)
  • R2 rated Xavier 5, Waman 1, Yusuf 1 → Remaining sum for R2 = \( 11 - 7 = 4 \)
  • R3 rated Waman 5, Xavier 5, Yusuf 1 → Remaining sum for R3 = \( 19 - 11 = 8 \)
  • R5 rated Vasu 5 → Remaining sum for R5 = \( 17 - 5 = 12 \)


Step 4: Worker ratings analysis

WorkerKnown RatingsMean
Ullas1 (from R1), ?, ?, ?, ?2.2
Vasu?, ?, ?, ?, 5 (from R5)3.8
Waman5 (from R1), 1 (from R2), 5 (from R3), ?, ?3.4
Xavier?, 5 (from R2), 5 (from R3), ?, ?3.6
Yusuf?, 1 (from R2), 1 (from R3), ?, ?2.6


Step 5: Uniqueness of determined ratings
Yes, all ratings can be uniquely determined because:

  • The total ratings for each restaurant are fixed.
  • The mean ratings for each worker establish equations for the sum of their ratings.
  • Sufficient known ratings exist to resolve the system of equations definitively.

Final Answer: \( \boxed{0} \) ratings remain undetermined.

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