Ambivalent Fans

Two soccer teams are playing inside an arena which can hold up to 3000 fans. As they were arriving, the fans were asked which team they wanted to win. 5 14 \frac{5}{14} of the fans wanted team A to win the game, and 9 15 \frac{9}{15} of the fans wanted team B to win. If nobody wanted both teams to win, what is the most number of people that were at the game that did not care which team won?


The answer is 126.

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1 solution

Arron Kau Staff
May 13, 2014

We can write our two fractions as 5 14 = 25 70 \frac{5}{14} = \frac{25}{70} and 9 15 = 42 70 \frac{9}{15} = \frac{42}{70} . So in total 67 70 \frac{67}{70} of the people were cheering for one team or the other. That means that 3 70 \frac{3}{70} of the people were not cheering for any team. Since 3000 70 = 42 \left \lfloor \frac{3000}{70} \right \rfloor = 42 , so at most there were 42 × 70 = 2940 42 \times 70 = 2940 people in the stands, with 3 70 × 2940 = 126 \frac{3}{70} \times 2940 = 126 of them not cheering for either team.

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