The number of teenagers who attended the concert are:
150
Tickets to a concert cost $2 for children, $3 for teenagers, and $5 for adults.
Let number of children who attended the concert= x
Number of teenager who attended the concert= y
Number of adults who attended the concert= z
When 570 people attended the concert, the total ticket receipts were $1950.
i.e. the equations that are made with the help of above information is:
x+y+z= 570----------(1)
2x+3y+5z= 1950--------(2)
Also,Compared to the number of children in attendance, three-fourths as many teenagers attended.
i.e.
[tex]y=\dfrac{3}{4}x[/tex]
i.e.
[tex]x=\dfrac{4}{3}y--------(3)[/tex]
Now, from equation (1) we have:
[tex]z=570-(x+y)[/tex]
On putting the value of x from equation (3) we have:
[tex]z=570-(\dfrac{4}{3}y+y)\\\\i.e.\\\\z=570-\dfrac{7}{3}y[/tex]
on putting the value of z and x in terms of y in equation (2) we have:
[tex]2\times \dfrac{4}{3}y+3y+5\times (570-\dfrac{7}{3}y)+2850=1950\\\\i.e.\\\\\dfrac{8}{3}y+3y-\dfrac{35}{3}y+=1950\\\\i.e.\\\\\dfrac{8y+3y\times 3-35y}{3}+2850=1950\\\\i.e.\\\\\dfrac{-18y}{3}+2850=1950\\\\i.e.\\\\-6y+2850=1950\\\\i.e.\\\\6y=2850-1950\\\\i.e.\\\\6y=900\\\\i.e.\\\\y=150[/tex]
Hence, number of children who attended concert= 150
