CAT 2024 Slot 2 — DILR Question 20

DistributionEasy

Passage / Data

Answer the following questions based on the information given below.

Eight gymnastics players numbered 1 through 8 underwent a training camp where they were coached by three coaches - Xena, Yuki, and Zara. Each coach trained at least two players. Yuki trained only even numbered players, while Zara trained only odd numbered players. After the camp, the coaches evaluated the players and gave integer ratings to the respective players trained by them on a scale of 1 to 7, with 1 being the lowest rating and 7 the highest.

The following additional information is known.
1. Xena trained more players than Yuki.
2. Player-1 and Player-4 were trained by the same coach, while the coaches who trained Player-2, Player-3 and Player-5 were all different.
3. Player-5 and Player-7 were trained by the same coach and got the same rating. All other players got a unique rating.
4. The average of the ratings of all the players was 4.
5. Player-2 got the highest rating.
6. The average of the ratings of the players trained by Yuki was twice that of the players trained by Xena and two more than that of the players trained by Zara.
7. Player-4's rating was double of Player-8's and less than Player-5's.

What was the rating of Player-6?

Answer & solution

Answer: 5

Solution

Hard

With $r=4$ , Yuki's two ratings sum to 12. Clue 7 fixes P8 and P4; Clue 5 makes P2 the unique highest. That pins Yuki = {P2, P6} and hence P6.

sum(Yuki) $= 2r+4 = 12$ . Clue 7: $P4 = 2\,P8$ and $P4 < P5 = 4$ . Clue 5: P2 is the highest rating. Ratings are integers 1–7; P5 = P7 = 4, all other players' ratings are distinct.

Find P8 and P4. $P4=2P8$ with $P4<4$ forces $P4=2,\ P8=1$ . Then the even player joining Yuki cannot be P8 (P8→Xena), so Yuki = {P2, P6}.

\begin{aligned} &P4<4,\ P4=2P8\ \Rightarrow\ P8=1,\ P4=2\\ &\Rightarrow\ \text{Yuki}=\{P2,P6\} \end{aligned}

Split Yuki's sum of 12. P2 is the unique highest ( $\le 7$ ) and distinct from P6.

\begin{aligned} &P2+P6=12,\ P2>P6,\ P2\le7\\ &\Rightarrow\ P2=7,\ P6=5 \end{aligned}

5