Two football teams were playing in a league last season. Team A won (1)/(10) of their games drew (3)/(5) of their games and lost the rest of their games. The ratio of wins to draws to losses for Team B was 1:4:1 Both teams played the same number of games in the season. A win was worth 3 points a draw was worth 1 point and a loss was worth 0 points. a) What is the smallest possible number of games that each team could have played? b) What is the smallest possible number of points that Team A could have had at the end of the season?

a) To calculate the least possible number of matches played, we look for the least common multiple (LCM) of the denominators of fractions for Team A (which are 10 and 5 from and ), and from Team B's ratio (which is 1 : 4 : 1 equating to 6 parts total). By finding the least common multiple, we can find the minimum number of games played that still ensures each team's win, lose, and draw fractions or ratio can all be represented with a whole number. b) The number of wins, draws, and losses directly translates to the total points earned by the team. Each point derives from the number of matches multiplied by the points each match represents: Win: \( \frac{1}{10} * \text{(3 points)} * \text{(No. of games)} \) Draw: \( \frac{3}{5} * \text{(1 point)} * \text{(No. of games)} \) Loss: (remaining games after win and draw, i.e., ) multiplied by 0 (points) obviously renders 0 points