cats wrote:
Tickets for next month’s production of
The Sea Gull at the local community theater went on sale yesterday. The ticket price for a person older than 12 is $
A, and the ticket price for a person 12 or under is $
C, where
A >
C. Yesterday, the sale of 33 tickets generated a total revenue of $323, and today the sale of 44 tickets generated a total revenue of $424.
In the table, select a value for
A and a value for
C that are jointly consistent with the given information. Make only two selections, one in each column.
Although official solution too talks of 11 tickets costing 101, I would take it with a pinch of salt.
All tickets are not of same value, so 33 tickets and 44 tickets would surely have different combination. Say 33 tickets consisted of 22 A and 11 C while 44 tickets consisted of 20A and 24 C. In that case the difference would be 13 of C and -2 of A.
But if 44 tickets had more of each of A and C, say 28A and 16C, then you will have 6 of A and 5 of C adding to 11.
Anyways, I would make equations and then work on number properties:
Yesterday: Let there be x of A and 33-x of C => x*A + (33-x) * C = 323.
(1) Surely exactly one of xA or (33-x)C would be odd to get odd+even = odd.
If x is odd, then 33-x is even and vice versa. Surely one of A and C will be odd.
Noe, there is only one odd option, so one of A or C would be 11.(2) If we take all to be of C value, we will get the least sum, of tickets => 33*10 = 330, which is greater than given sum 323.
Hence, C is less than 10. Only possibility 6 or 8. Also, A will be 11 for sure Just from yesterday sale, we know A=11 and C is either 8 or 6.
Although, you can substitute (11,8) and (11,6) here itself to check which is correct, let use today's sale to ease our calculations. After all, efforts have been made to add the details to the question.
Today: Let there be y of A and 44-y of C => y*A + (44-y) * C = 424.
Subtract the two equations to get (y-x)*A + 11*C -(y-x)*C = 101
(A-C) (y-x) = 101-11C
(
11,8): (11-8)(y-x) = 101-11*8 ........ 3(y-x) = 13 ...... y-x = \(\frac{13}{3}\), Not an integer.
(
11,6): (11-6)(y-x) = 101-11*6 ........ 5(y-x) = 35 ...... y-x = 7, Possible. Correct
Solution:
A=11 and C=6