A theater has a seating capacity of 450 and charges $2 for children, $3 for students, and $5 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $1,550. How many children attended the show?
A) 300 children
B) 190 children
C) 10 children
D) 390 children
E) 100 children

Question

A theater has a seating capacity of 450 and charges $2 for children, $3 for students, and $5 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totaled $1,550. How many children attended the show? ​
A) 300 children
B) 190 children
C) 10 children
D) 390 children
E) 100 children

Quizplus · Accepted Answer

Let the number of children be C, students be S, and adults be A. We have three equations based on the given information:1. $C + S + A = 450$ (total seats)2. $A = \frac{1}{2}(C + S)$ (half as many adults as children and students combined)3. $2C + 3S + 5A = 1550$ (total receipts)From equation 2, we can express A as $A = \frac{1}{2}C + \frac{1}{2}S$. Substituting this into equation 1 gives us $C + S + \frac{1}{2}C + \frac{1}{2}S = 450$, which simplifies to $1.5C + 1.5S = 450$. This simplifies further to $C + S = 300$.Substituting $A = \frac{1}{2}C + \frac{1}{2}S$ into equation 3 gives us $2C + 3S + 5(\frac{1}{2}C + \frac{1}{2}S) = 1550$, which simplifies to $2C + 3S + 2.5C + 2.5S = 1550$, and further to $4.5C + 5.5S = 1550$.Using $C + S = 300$, we can find the number of children by solving these equations together, which ultimately reveals that there were 100 children attending the show.

A Theater Has a Seating Capacity of 450 and Charges