N2025 P2 Q1

In a game, players have to collect tokens in the form of cars, trains and boats. A car is worth \(c\) points, a train \(t\) points and a boat \(b\) points. At the end of a game the total score of each player is worked out and the player with the highest score wins.

Pritti, Quentin, Ria and Sam played a game. At the end of the game:

• Pritti had \(8\) cars, \(11\) trains and \(5\) boats and scored \(114\) points.

• Quentin had \(5\) cars, \(14\) trains and \(7\) boats and scored \(112\) points.

• Ria had \(9\) cars, \(9\) trains and \(4\) boats and scored \(110\) points.

• Each player scored a different number of points.
• Sam came second.

1. Find the values of \(c\), \(t\) and \(b\).[3]

At the end of the game, Sam had at least \(6\) of each type of token, different numbers of each type of token, and more boats than trains.

2. Determine how many of each type of token Sam had at the end of the game. [You are given that there is only one solution.]

   [3]