Fundraiser Ticket Revenue Reconciliation
A theater group sold 340 tickets to a fundraiser: adult tickets at $12 each and child tickets at $7 each. The treasurer reports total ticket revenue of $3,285. Before the group finalizes its budget, a volunteer notices that the $3,285 deposit included a $45 check from the concession stand, which is not ticket revenue and must be recorded separately. Nobody has questioned the count of 340 tickets or the two ticket prices. The group's grant application requires the actual number of adult and child tickets sold, stated as whole numbers consistent with the corrected revenue.
1. Set up and solve a system of equations for the corrected counts of adult and child tickets.
Read the worked answer
Corrected ticket revenue: $3,285 - 45 = 3,240$.
Let a = adult tickets, c = child tickets.
$$a + c = 340$$ $$12a + 7c = 3240$$
Multiply the first equation by 7:
$$7a + 7c = 2380$$
Subtract from the revenue equation:
$$(12a + 7c) - (7a + 7c) = 3240 - 2380 \Rightarrow 5a = 860 \Rightarrow a = 172$$
Then $c = 340 - 172 = 168$.
Check: $172(12) = 2064$ and $168(7) = 1176$; $2064 + 1176 = 3240$. Correct - 172 adult and 168 child tickets.
2. Solve the same system using the uncorrected $3,285 and describe exactly how the error distorts each count.
Read the worked answer
With $3,285$: $5a = 3285 - 2380 = 905 \Rightarrow a = 181$, and $c = 159$.
Compare with the corrected answer: 181 vs 172 adults, 159 vs 168 children. The error moves 9 tickets from child to adult.
Why 9: each dollar is fine, but swapping one child ticket for one adult ticket raises revenue by $12 - 7 = 5$. The stray $45 is $45 \div 5 = 9$ such swaps. So any unaccounted revenue that is a multiple of 5 will silently convert into a whole number of "adult" tickets and the wrong solution will still check out - which is why the arithmetic alone cannot catch this; only reconciling the revenue total to its source can.
3. The grant requires adults to be at least 55% of attendees. Does the group qualify, and what is the correct percentage?
Read the worked answer
Adult share: $172/340 = 0.50588...$ which is about 50.6%.
The group does not qualify. $0.55 \times 340 = 187$ adults would be needed; the group has 172, which is 15 short.
Had the uncorrected revenue been used, the share would have been $181/340 \approx 53.2%$ - still below the threshold, but close enough to invite a mistaken appeal. The correction matters: it changes the grant answer from a near miss to a clear miss, and it also lowers reported ticket revenue by $45, which affects the budget total the grant is meant to support.