In the Tigers Football Club, there are 9 players to each coach, while in the Eagles Football Club, there are 17 players to every 2 coaches. If the total number of players and coaches in the two clubs is 232, how many players and coaches are in each club?

9514 1404 393

Step-by-step explanation:

We can define a "basic unit" of each team to be the minimum number of players and coaches that can have the given ratio. For the Tigers, that is 1 coach and 9 players, for a total of 10 team members. For the Eagles, that is 2 coaches and 17 players, for a total of 19 team members. If we have t basic units of Tigers and e basic units of Eagles, then we want ...

  10t +19e = 232 . . . . . the total of players and coaches in the two clubs

We want t and e to be integers, so this is a Diophantine equation. It can be solved any of several ways, including use of the Extended Euclidean Algorithm. Here, we'll make some observations based on "number sense."

The value of e must be an even number between 0 and 232/19 = 12. The product of e and 19 must be a number that ends in 2, because 10t will end in 0, and the sum with 19e must end in 2.

The only multiple of 9 that ends in 2 is 9×8 = 72, so the value of e must be a positive number less than 12 that ends in 8. We must have e=8.

Then t=(232 -8×19)/10 = 8. This tells us there are 8 "basic units" of each team.

The Tigers have 8 coaches and 72 players.

The Eagles have 16 coaches and 136 players.