Question

Tickets for a certain show cost ​$17​, ​$21​, ​or, for VIP​ seats, ​$40. If ten times as many ​$17 tickets were sold as VIP​ tickets, and the number of ​$17 tickets sold was 57 more than the sum of the number of ​$21 tickets and VIP​ tickets, sales of all three kinds of tickets would total ​$51 comma 471. How many of each kind of ticket would have been​ sold?

Answer 1

Tickets sold:

VIP
`=126`

$17 tickets
`=1,260`

$21 tickets
`=9\cdot 126+57=1,191`

Explanation:

Let x be the number of VIP tickets.

If ten times as many ​$17 tickets were sold as VIP​ tickets, then the number of $17 tickets is
`10x.`

If the number of ​$17 tickets sold was 57 more than the sum of the number of ​$21 tickets and VIP​ tickets, then
`10x+57=x+y` and the number y of $21 tickets is
`9x+57.`

Amounts earned:

VIP tickets
`=\$40x`

$17 tickets
`=\$17\cdot 10x=\$170x`

$21 tickets
`=\$21\cdot (9x+57)=\$(189x+1,197)`

Total
`=\$(40x+170x+189x+1,197)=\$(399x+1,197)`

The sales of all three kinds of tickets would total ​$51,471, so

`399x+1,197=51,471\\ \\399x=51,471-1,197\\ \\399x=50,274\\ \\x=126`

Tickets sold:

VIP
`=126`

$17 tickets
`=1,260`

$21 tickets
`=9\cdot 126+57=1,191`