673 children, 11 students, and 336 adults attended the movie.
How many children attended the movie?
How many students attended the movie?
How many adults attended the movie?
How to calculate the total ticket sales?
How to use equations to solve a word problem?
How to check if the obtained solution is valid?
Let's begin by defining some variables:
Let C be the number of children attending the movie.
Let S be the number of students attending the movie.
Let A be the number of adults attending the movie.
We know that the theater has a seating capacity of 323, so we can write an equation that relates the number of people attending the movie to the seating capacity:
C + S + A = 323
We also know that the theater charges $5.00 for children, $7.00 for students, and $12.00 for adults, and that there are half as many adults as there are children. Using this information, we can write another equation that relates the total ticket sales to the number of people in each category:
5C + 7S + 12A = 2348
We can use the fact that there are half as many adults as children to express A in terms of C:
A = 0.5C
Substituting this into the first equation, we get:
C + S + 0.5C = 323
Simplifying, we get:
1.5C + S = 323
Now we have two equations with two unknowns (C and S), which we can solve to find the values of these variables:
1.5C + S = 323 (equation 1)
5C + 7S = 2348 (equation 2)
Multiplying equation 1 by 5 and subtracting it from equation 2, we can eliminate S and solve for C:
5(1.5C + S) - 7S = 7.5C + 5S - 7S = 2348 - 5(323) = 1683
2.5C = 1683
C = 673.2
Since C must be a whole number, we can round down to the nearest integer:
C = 673
Now we can use this value of C to find S:
1.5C + S = 323
1.5(673) + S = 323
S = 323 - 1010.5
S = 10.5
Again, since S must be a whole number, we round up to the nearest integer:
S = 11
Finally, we can use the equation A = 0.5C to find A:
A = 0.5C = 0.5(673) = 336.5
Rounding down to the nearest integer, we get:
A = 336
Therefore, the number of children, students, and adults who attended the movie are:
673 children, 11 students, and 336 adults.