TheCATExam
Sign in

XAT 2011 — QA & DI Question 21

Simple EquationsEasy
Passage / Data

Answer the following question based on the information given below.

A man standing on a boat south of a light house observes his shadow to be 24 meters long, as measured at the sea level. On sailing 300 meters eastwards, he finds his shadow as 30 meters long, measured in a similar manner. The height of the man is 6 meters above sea level.

The football league of a certain country is played according to the following rules: 

  • Each team plays exactly one game against each of the other teams.
  • The winning team of each game is awarded l point and the losing team gets 0 point.
  • If a-match ends in a draw, both the teams get 1/2 point.

After the league was over, the teams were ranked according to the points that they earned at the end of the tournament. Analysis of the points table revealed the following:

  • Exactly half of the points earned by each team were earned in games against the ten teams which finished at the bottom of the table.
  • Each of the bottom ten teams earned half of their total points against the other nine teams in the bottom ten.
  • How many teams participated in the league?

Answer & solution

  • A

    16

  • B

    18

  • C

    19

  • 25

  • E

    30

Solution

Let there be 10 + n teams.

One point is awarded for each match.

The bottom 10 teams will play 10C2 = 45 matches against each other and 10 × n matches against the top n teams.

Similarly top n teams will play nC2 matches against each other & 10n matches against the bottom 10 teams.

Now, each of the bottom ten teams earned half of their total points against each other.

Hence they earned 45 points against the top n teams.

Hence, top n teams earned 10n – 45 points against the bottom 10 teams.

Now, half of the points earned by each team were earned in games against the ten teams which finished at the bottom 10.

Hence, top n team earned half of their points against the bottom 10 teams.

Hence, nC2 = 10n – 45

∴n(n+1)2 = 10n - 45

∴ n2 - 21n + 90 = 0

∴ (n - 6) (n - 15) = 0

∴ n = 6 or n = 15

Now, If n = 6 then top 6 teams will earn 6C2 + 10n – 45 = 30 points, with an average of 5 points.

The bottom 10 will earn an average of 9 points.

But this is not possible, as the average score of the top n should be greater than the bottom 10.

Hence, n = 15

∴ Total number of teams = 10 + n = 10 + 15 = 25

Hence, option (d).

XAT 2011 QA & DI Q21: The football league of a certain country is played according to the following rules: Each team plays exactly o — Solution | TheCATExam