Question

6. In a four team softball league, each team has played every other team 4 times. Each team earned 3
points for a win, 1 point for a tie and no points for a loss. The total accumulated points were:
Lions: 22 Tigers 19 Mounties 14 Royals 12
How many games ended in a win and how many games ended in a tie?

Answer 1

$$\begin{gathered} \mathrm{total}\mathrm{points}=22+19+14+12 \\ =67 \\ \mathrm{here}\mathrm{are}4\mathrm{teams},\mathrm{so}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{games}\mathrm{is}4\mathrm{x}3\mathrm{x}4/2=24\mathrm{games} \\ (\mathrm{There}\mathrm{are}3\mathrm{games}\mathrm{by}\mathrm{each}\mathrm{team}\mathrm{in}\mathrm{each}\mathrm{round}\mathrm{and}\mathrm{there}\mathrm{are}4\mathrm{rounds}.\mathrm{But}\mathrm{each}\mathrm{game}\mathrm{is}\mathrm{played}\mathrm{by}2\mathrm{teams}\mathrm{so}\mathrm{we}\mathrm{divide}\mathrm{by}2.) \\ \mathrm{So}\mathrm{if}\mathrm{a}=\mathrm{number}\mathrm{of}\mathrm{games}\mathrm{which}\mathrm{end}\mathrm{in}\mathrm{a}\mathrm{win}\mathrm{or}\mathrm{a}\mathrm{loss},\mathrm{b}=\mathrm{number}\mathrm{of}\mathrm{games}\mathrm{which}\mathrm{are}\mathrm{drawn},\mathrm{then} \\ \mathrm{a}+\mathrm{b}=24\mathrm{and} \\ \mathrm{b}=24-\mathrm{a} \\ \mathrm{Also}3\mathrm{a}+2\mathrm{b}=67(\mathrm{remember}\mathrm{that}\mathrm{both}\mathrm{teams}\mathrm{earn}\mathrm{one}\mathrm{point}\mathrm{in}\mathrm{a}\mathrm{draw}) \\ 3\mathrm{a}+48-2\mathrm{a}=67 \\ \mathrm{a}=19 \\ \mathrm{So}\mathrm{there}\mathrm{were}19\mathrm{games}\mathrm{in}\mathrm{which}\mathrm{one}\mathrm{other}\mathrm{team}\mathrm{won} \\ \mathrm{and}\mathrm{hence}5\mathrm{games}\mathrm{were}\mathrm{tied} \end{gathered}$$