In a group of 800 participants planning a community event, some opted for the role of organizers in a Science Fair, while others chose to organize a Talent Show, with an initial ratio of 6:11, respectively. Some participants were initially undecided. As the event planning progressed, all participants took on roles, and the distribution between the Science Fair and Talent Show became 13:19, respectively. The undecided participants then divided themselves between the Science Fair and Talent Show in a 7:8 ratio. How many participants were initially undecided about their roles in the event?

Let, the number of participants who choose to organize a Science Fair be '6x'. Therefore, the number of participants who choose to organize a Talent Show is  = (11/6) × 6x = '11x'. Let, the number of participants who were initially undecided is '15y'. ATQ, The number of undecided participants who decided to organize a Science Fair = (7/15) × 15y = '7y' And, number of undecided participants who decided to organize Talent Show = 15y - 7y = '8y' Total number of participants who choose to organize a Science Fair = (6x + 7y) Total number of participants who choose to organize a Talent Show = (11x + 8y) ATQ; (6x + 7y)/(11x + 8y) = 13/19 Or,(19 × (6x+7y)) = 13 × (11x + 8y)  Or,114x + 133y = 143x + 104y Or,143x-114x = 133y-104y Or,29x = 29y x = y So, the number of participants who were initially undecided= 15y = 15x So, total number of participants = 6x + 11x + 15x = 32x = 800 Or, x = 800/32 x = 25 Therefore, the number of participants who were initially undecided is = 15x = 15 × 25 = 375