Let's call a child's ticket [tex]c[/tex] and an adult's ticket [tex]a[/tex]. From this, we can say:
[tex]c + a = 116[/tex],
since 116 tickets are sold in total.
Now, we are going to need to find another equation (the problem asks us to solve a systems of equations). This time, we are not going to base the equation on ticket quantity, but rather ticket price. We know that an adult's ticket is $17,000, and a child's ticket is thus
[tex]\$17000 \cdot (1 - \dfrac{1}{4}) = \$12750[/tex].
Given these values, we can say:
[tex]17000a + 12750c = 1653250[/tex],
since each adult ticket [tex]a[/tex] costs 17,000 and each child's ticket [tex]c[/tex] costs 12,750, and these costs sum to 1,653,250.
Now, we have two equations:
[tex]17000a + 12750c = 1653250[/tex]
[tex]c + a = 116[/tex]
Let's solve:
[tex]17000a + 12750c = 1653250[/tex]
[tex]a = 116 - c[/tex]
- Find [tex]a[/tex] on its own, which will allow us to substitute it into the first equation
[tex]17000(116 - c) + 12750a = 1653250[/tex]
- Substitute in [tex]116 - c[/tex] for [tex]a[/tex]
[tex]1972000 - 17000c + 12750c = 1653250[/tex]
- Apply the Distributive Property
[tex]1972000 - 4250c = 1653250[/tex]
[tex]4250c = 318750[/tex]
- Subtract 1972000 from both sides of the equation and multiply both sides by -1
[tex]c = 75[/tex]
We have now found that 75 child's tickets were sold. Thus,
[tex]75 + a = 116 \Rightarrow a = 116 - 75 = 41[/tex],
41 adult tickets were sold as well.
In sum, 41 adult tickets were sold along with 75 child tickets.
