A theater has a seating capacity of 750 and charges $5 for children, $7 for students, and $9 for adults. at a certain screening with full attendance, there were half as many adults as children and students combined. the receipts totaled $5350. how many children attended the show?

To do this, we have to setup equations and then use substitution to get like terms. c=children, s=students, a=adults For total cost: 5350 = 5c + 7s + 9a Total capacity: 750 = c + s + a Data given: (.5) (c + s) = a Now, rearranging the data given equation for (c+s) gives us (c+s) = 2a. Using this and substituting into the total capacity equation for (c+s) will allow us to find a: 750 = 2a + a = 3a which gives us a = 750/3 = 250. Next, we set a = 250 in total cost and data given equation to get the following: Total Cost: 5350 = 5c + 7s + 2250 which gives 3100 = 5c + 7s Total Capacity: 750 = c + s + 250 which gives 500 = c + s Rearranging total capacity to solve for s: s = 500 - c Plugging that expression for s into total cost gives us: 3100 = 5c + 7 (500-c) = 5c - 7c + 3500. Solving for c gives us 400 = 2c. Total number of children is c = 200. Total number of students is s = 750 - 200 - 250 = 300.